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[B]: Books, [P]: Preprints, [J]: Journal articles, [BC]: Book Chapters,
[CP]: Conference Proceedings.
Books
[B1] 
S. Adly.
A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics,
SpringerBriefs in Mathematics (2019).
About this book.
Abstract
Abstract:
This brief examines mathematical models in nonsmooth mechanics and nonregular electrical circuits, including evolution variational
inequalities, complementarity systems, differential inclusions, secondorder dynamics, Lur'e systems and Moreau's sweeping
process.
The field of nonsmooth dynamics is of great interest to mathematicians, mechanicians, automatic controllers and engineers. The
present volume acknowledges this transversality and provides a multidisciplinary view as it outlines fundamental results in
nonsmooth dynamics and explains how to use them to study various problems in engineering. In particular, the author explores the
question of how to redefine the notion of dynamical systems in light of modern variational and nonsmooth analysis.
With the aim of bridging between the communities of applied mathematicians, engineers and researchers in control theory and
nonlinear systems, this brief outlines both relevant mathematical proofs and models in unilateral mechanics and electronics.

Submitted Papers
[P4] 
S. Adly.
Understanding qualitative robust stability of composite variational systems,
Submitted in September 2024 .
Abstract
Abstract:
The main objective of this paper is to study the Lipschitzlike stability property,
in particular the Aubin property, of the solution map for potentially nonmonotone variational
systems with composite structure. Using the Mordukhovich coderivative criterion and secondorder subdifferential analysis, we derive simple and geometrical characterizations of this property based on the data involved in the problem.
Applications of these theoretical results are given in the context of market pricing strategies
in economics. Furthermore, we explore Julia and Ipopt solver to compute the Lipschitz modulus of
the solution map around a reference point, showing the prcatical implementation of our results.

[P3] 
S. Adly, L. Bourdin.
The derivative of a positively homogeneous variational inequality is a complementarity problem,
Submitted in June 2024 .
Abstract
Abstract:
Using advanced techniques from convex and variational analysis, we establish, under a set of appropriate conditions
(including a polyhedricity assumption), that the solution to a parameterized variational inequality associated with a
positively homogeneous function is differentiable, and furthermore that its derivative is the solution to a corresponding
complementarity problem. We illustrate our main result through a series of examples and counterexamples.
In particular we introduce an example of projection operator onto a nonempty closed convex cone that is not directionally
differentiable, which is, to our best knowledge, new in the literature.

[P2] 
S. Adly, Jun Huang and Ba Khiet Le.
Sliding Mode Observers for Setvalued Lur'e Systems with Uncertainties Beyond Observational Range,
Submitted in February 2024 .
Abstract
Abstract:
In this paper, we introduce a new approximate sliding mode observer for Lur'e setvalued dynamical systems,
particularly addressing challenges posed by uncertainties not within the standard range of observation.
Traditionally, Luenbergerlike observers and sliding mode observer have been designed only for uncertainties in
the range of observation. Central to our approach is the treatment of the uncertainty term which we decompose into two
components: the first part in the observation subspace and the second part in its complemented subspace. We establish that when the second part converges to zero,
an exact sliding mode observer for the system can be obtained.} In scenarios where this convergence
does not occur, our methodology allows for the estimation of errors between the actual state and the observer
state. This leads to a practical interval estimation technique, valuable in situations where part of the
uncertainty lies outside the observable range. Finally, we show that our observer is also an $H^\infty$ observer

[P1] 
S. Adly, H. Attouch and J. M. Fadili.
Comparative Analysis of Accelerated Gradient Algorithms for Convex Optimization:
High and Super Resolution ODE Approach,
Submitted in May 2023 .
Abstract
Abstract:
We investigate convex differentiable optimization and explore the temporal discretization of
damped inertial dynamics driven by the gradient of the objective function. This leads to three
accelerated gradient algorithms: Nesterov Accelerated Gradient (NAG), Ravine Accelerated Gradient
(RAG), and (IGAHD). Attouch, Chbani, Fadili, and Riahi introduced (IGAHD) by discretizing inertial
dynamics with Hessiandriven damping to attenuate inherent oscillations in inertial methods.
By analyzing the highresolution ODEs of order $p=0,1,2$ for these algorithms, we gain insights
into their similarities and differences. All three algorithms share the same lowresolution
ODE of order $0$, which is the dynamic proposed by Su, Boyd, and Cand\`es as a continuous
surrogate for (NAG). To differentiate Nesterov from Ravine, we refine the comparison and
demonstrate distinct highresolution ODEs of order $2$ in $h$ (termed superresolution).
The corresponding Taylor expansions in $h$ reveal matching terms of order $1$ but differing
terms of order $2$. To the best of our knowledge, this result is completely new and emphasizes
the need to avoid confusion between the Ravine and Nesterov methods in the literature.
We present numerical experiments to illustrate our theoretical results. Performance profiles, measuring the
number of iterations, indicate that (IGAHD) outperforms both (NAG) and (RAG) methods. (RAG) exhibits a slight
advantage over (NAG) in terms of the average number of iterations. When considering CPUtime, both (RAG) and
(NAG) outperform (IGAHD). All three algorithms exhibit similar behavior when evaluating based on gradient norms.

Refereed Publications
2024
[J118] 
S. Adly and H. Attouch.
Accelerated Optimization through Timescale Analysis of Inertial Dynamics with Asymptotic Vanishing and Hessiandriven Dampings,
Accepted for publication in Optimization Journal (Taylor and Francis publisher) .
Abstract
Abstract:
Gradient optimization algorithms can be effectively studied from the perspective of ordinary differential equations (ODEs), where these algorithms are obtained through the temporal
discretization of the continuous dynamics. This approach provides a powerful tool to comprehend acceleration phenomena in optimization, thanks to timescale techniques and Lyapunov analysis.
In the context of a Hilbert setting, our study focuses on the rapid optimization properties of inertial dynamics, which combine asymptotically vanishing viscous damping with Hessiandriven damping.
These dynamics are derived as highresolution ODEs from both the Nesterov accelerated gradient method and the Ravine method. For a general differentiable convex function $f$, choosing the viscous
damping coefficient in the form of $\alpha/t$ with $\alpha > 3$ guarantees an inverse quadratic convergence rate of the values, i.e., $o(1/t^2)$, the weak convergence of trajectories towards optimal
solutions, and the fast convergence of gradients towards zero.
By carefully scaling the dynamics temporally and based on the tuning of the damping coefficient in front of the Hessian term, we identify the limit dynamic as $\alpha$ becomes large. In particular,
we examine the case where the limit dynamic corresponds to the LevenbergMarquardt regularization of Newton's continuous method. This explanation accounts for the aforementioned fast convergence
properties and provides fresh insights into the complexity of these methods, which play a central role in optimization for highdimensional problems. The interplay between numerical algorithms and continuoustime ODEs plays a fundamental role in our analysis for the design and understanding of accelerated optimization algorithms.
Numerical experiments are presented to illustrate and support the theoretical results.

[J117] 
S. Adly, M. G. Cojocaru and B. K. Le .
StateDependent Sweeping Processes: Asymptotic Behavior and Algorithmic Approaches.
Accepted for publication in JOTA, Journal of Optimization Theory and Applications .
Abstract
Abstract:
In this paper, we investigate the asymptotic properties of a particular class of statedependent sweeping processes. While extensive research has been conducted on the existence
and uniqueness of solutions for sweeping processes, there is a scarcity of studies addressing their behavior in the limit of large time. Additionally, we introduce novel
algorithms designed for the resolution of quasivariational inequalities.
As a result, we introduce a new derivativefree algorithm to find zeros of nonsmooth Lipschitz continuous mappings with a linear convergence rate. This algorithm can be
effectively used in nonsmooth and nonconvex optimization problems that do not possess necessarily secondorder differentiability conditions of the data.

[J116] 
S. Adly, Nguyen Nang Thieu and Nguyen Dong Yen.
Noncoercive Convex Sweeping Processes with Velocity Constraints,
Accepted for publication in Optimization Journal (Taylor and Francis publisher) .
Abstract
Abstract:
In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously
investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result
for coercive convex sweeping processes with velocity constraints and a useful fact on measurable singlevalued mappings, we prove the
solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable
conditions.
These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work
[S.~Adly, N.~N.~Thieu, N.~D.~Yen, Convex and nonconvex sweeping processes with velocity constraints: wellposedness and insights,
Appl. Math. Optim., 88 (2023), Paper No. 45]. Additionally, we resolve two other open questions from the same paper concerning the behavior of
the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.

[J115] 
S. Adly and H. Attouch.
Complexity analysis based on tuning the viscosity parameter of the SuBoydCandès inertial gradient dynamics,
Accepted for publication in Setvalued and Variational Analysis .
Abstract
Abstract:
In a Hilbert setting, our study focuses on the dynamical system introduced by SuBoydCand\`es as a low resolution ODE of Nesterov's
accelerated gradient method (NAG). This inertial system, denoted by ${\rm (AVD)}_{\alpha}$, is driven by the gradient of the function $f$
to be minimized, and is damped with an asymptotic vanishing coefficient of the form $\alpha/t$, with $\alpha\geq 3$. Taking $\alpha$ large
enough plays a crucial role in the asymptotic convergence properties of the trajectories. For a general convex function $f$, taking $\alpha >3$
guarantees the asymptotic convergence rate of the values $o \left( 1/t^2 \right)$, as well as the convergence of the trajectories towards optimal
solutions. For strongly convex $f$, the asymptotic rate of convergence is of order $ 1/t^{\frac{2\alpha}{3}} $, which increases with $\alpha$.
To analyze the effect of the parameter $\alpha$ in the convergence properties of ${\rm (AVD)}_{\alpha}$, we show that a judicious time scaling
of ${\rm (AVD)}_{\alpha}$ produces trajectories close to those of the continuous steepest descent method associated with $f$ {when $\alpha$ is
sufficiently large}. This limiting process involves a singular perturbation property, as we move from a secondorder evolution equation to a
firstorder one. This transition enables us to understand the change in the rate of convergence from $1/t$ to $1/t^2$ between the steepest descent
method and (NAG). Based on a complexity analysis over a finite time interval, new results are obtained regarding the optimal tuning of the
parameter $\alpha$ and the involved constants $C_\alpha$ in the estimations. Numerical experiments have been conducted to illustrate and confirm the theoretical results.

[J114] 
S. Adly, H. Attouch and Manh Hung Le.
A doubly nonlinear evolution system with threshold effects associated with dry friction,
Accepted for publication in Journal of Optimization Theory and Applications (JOTA) .
Abstract
Abstract:
In this paper, we investigate the asymptotic behavior of inertial dynamics with dry friction within the context of a Hilbert framework for convex differentiable optimization. Our study focuses on a doubly nonlinear firstorder evolution inclusion that encompasses two potentials. {In our analysis, we specifically focus on two main components:
the differentiable function $f$ that needs to be minimized,
which influences the system's state through its gradient,
and the nonsmooth dry friction potential denoted as$\varphi = r\\cdot\$.
It's important to note that the dry friction term acts on a linear combination of the velocity vector and the gradient of $f$.
Consequently, any stationary point in our system corresponds to a critical point of $f$,
unlike the case where only the velocity vector is involved in the dry friction term, resulting in an approximate critical point of $f$}.
To emphasize the crucial role of $\nabla f(x)$, we also explore the dual formulation of this dynamic, which possesses a Riemannian gradient structure.
To address these dynamics, we employ the recently developed generic acceleration approach by Attouch, Bot, and Nguyen. {This approach involves the time scaling of a continuous firstorder
differential equation, followed by the application of the method of averaging}. By applying this methodology, we derive fast convergence results for secondorder timeevolution systems with
dry friction, asymptotically vanishing viscous damping, and implicit Hessiandriven damping.

2023
[J113] 
S. Adly and Ba Khiet Le.
Sliding Mode Observer for Setvalued Lur'e Systems and Chattering Removing,
Nonlinear Analysis: Hybrid Systems 50 (2023), 101406.
Abstract
Abstract:
In this paper, we study} a sliding mode observer for a class of setvalued Lur'e systems
subject to uncertainties. We show that our approach has obvious advantages than the existing
Luenbergerlike observers. {Furthermore, we provide an effective continuous approximation to
eliminate the chattering effect in the sliding mode technique.

[J112] 
S. Adly, H. Attouch and Manh Hung Le.
Firstorder inertial optimization algorithms with threshold effects associated with dry friction,
Computational Optimization and Applications (COAP), Vol. 86, pp 801843 (2023).
Abstract
Abstract:
In a Hilbert space setting, we consider a new first order optimization algorithm which is obtained by temporal discretization of a damped inertial dynamic involving
dry friction. The function $f$ to be minimized is assumed to be differentiable (not necessarily convex).
The dry friction potential function $\phi$, which has a sharp minimum at the origin, enters the algorithm via its proximal mapping, which acts as a soft
thresholding operator on the sum of the velocity and the gradient terms. After a finite number of steps, the structure of the algorithm changes, losing its
inertial character to become the steepest descent method. The geometric damping driven by the Hessian of $f$ makes it possible to control and attenuate the
oscillations.
The algorithm generates convergent sequences when $f$ is convex, and in the nonconvex case when $f$ satisfies the KurdykaLojasiewicz property. As a remarkable
property, the convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence towards zero. The study is then
extended to the case of a nonsmooth convex function $f$, in which case the algorithm involves the proximal operators of $f$ and $\phi$ separately.
Then, applications are given to the Lasso problem and nonsmooth d.c. programming.

[J111] 
S. Adly, L. Bourdin, F. Caubet and A. Jacob de Cordemoy.
Shape Optimization for Variational Inequalities: The Scalar Tresca Friction Problem,
SIAM Journal on Optimization, Vol. 33, Iss. 4 (2023) .
Abstract
Abstract:
This paper investigates, without any regularization or penalization procedure, a shape optimization problem involving a simplified friction
phenomena modeled by a scalar Tresca friction law. Precisely, using tools from convex and variational analyses such as proximal operators and the
notion of twice epidifferentiability, we prove that the solution to a scalar Tresca friction problem admits a directional derivative with respect to the
shape which moreover coincides with the solution to a boundary value problem involving Signorinitype unilateral conditions. Then we explicitly characterize
the shape gradient of the corresponding energy functional and we exhibit a descent direction. Finally numerical simulations are performed to solve the corresponding
energy minimization problem under a volume constraint which shows the applicability of our method and our theoretical results.

[J110] 
S. Adly and H. Attouch.
Accelerated dynamics with dry friction via time scaling and averaging of doubly nonlinear evolution equations,
Nonlinear Analysis: Hybrid Systems 50 (2023), 101402.
Abstract
Abstract:
In a Hilbert framework, for convex differentiable optimization, we analyze the longtime behavior
of inertial dynamics with dry friction. In classical approaches based on asymptotically vanishing
viscous damping (in accordance with Nesterov's method), the results are expressed in terms of rapid
convergence of the values of the function
to its minimum value. On the other hand, dry friction induces convergence towards an approximate minimizer,
typically the system stops at $x$ when a given threshold $\ \nabla f (x)\ \leq r$ is satisfied.
We will obtain rapid convergence results in this direction.
In our approach, we start from a doubly nonlinear firstorder evolution equation involving two potentials:
one is the differentiable function $f$ to be minimized, which acts on the state of the system via its gradient,
and the other is the nonsmooth potential dry friction $\varphi (x) =r \x\$ which acts on the velocity vector
via its subdifferential.
To highlight the central role played by $ \nabla f (x)$, we will also argue with the dual formulation of this
dynamics, which has a Riemannian gradient structure. We then rely on the general acceleration method recently
developed by Attouch, Bot and Nguyen, which consists in applying the method of time scaling and then averaging
to a continuous differential equation of the first order in time.
We thus obtain fast convergence results for secondorder timeevolution systems involving dry friction,
asymptotically vanishing viscous damping, and Hessiandriven damping in the implicit form.

[J109] 
S. Adly, Nguyen Nang Thieu and Nguyen Dong Yen.
Convex and Nonconvex Sweeping Processes with Velocity Constraints: wellposedness and insights,
Applied Mathematics and Optimization, Vol. 66, Number 45, (2023).
Abstract
Abstract:
In this paper, we study some classes of sweeping processes with velocity constraints in the moving set. In addition to the solution existence and the
solution uniqueness for the case of a moving convex constraint set, some results on the solution existence and the solution multiplicity where the
constraint set is a finite union of disjoint convex sets are also obtained. Our main tool is a theorem on the solution sensitivity of parametric variational
inequalities. Beside the traditional requirement that the constraint set moves continuously in the Hausdorff distance sense, we intensively use a new assumption
on the local Lipschitzlikeness of the constraint setvalued mapping. The obtained results are compared with the existing ones and analyzed by several examples.

[J108] 
S. Adly, H. Attouch and Van Nam Vo.
Convergence of inertial dynamics driven by sums of potential and nonpotential operators and with implicit Newtonlike damping,
Journal of Optimization Theory and Applications, VoL. 198, pp 290331 (2023).
Abstract
Abstract:
We propose and study the convergence properties of the trajectories generated by a damped inertial dynamic which is driven by the sum of potential and nonpotential operators.
Precisely, we seek to reach asymptotically the zeros of sums of potential term (the gradient of a continuously differentiable convex function) and nonpotential monotone and cocoercive operator.
As an original feature, in addition to viscous friction, the dynamic involves implicit Newtontype damping. This contrasts with the authors' previous study where explicit Newtontype damping was
considered, which, for the potential term, corresponds to Hessiandriven damping. We show the weak convergence, as time goes to infinity, of the generated trajectories towards the zeros of the sum of
the potential and nonpotential operators. Our results are based on Lyapunov analysis and appropriate setting of damping parameters. The introduction of geometric dampings allows to control and
attenuate the oscillations known for the viscous damping of inertial methods. Rewriting the secondorder evolution equation as a system involving only first order derivative in time and space allows
us to extend the convergence analysis to nonsmooth convex potentials. Our study concerns the autonomous case with positive fixed parameters. These results open the door to their extension to the
nonautonomous case and to the design of new firstorder accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms.
The proofs and techniques are original due to the presence of the nonpotential term.

[J107] 
S. Adly and Manh Hung Le.
Solving Inverse Pareto Eigenvalue Problems,
Optimization Letters, Vol. 17, pp 829849 (2023).
Abstract
Abstract:
We compare in this note a variety of methods for solving inverse Pareto eigenvalue problems which are aimed at constructing matrices
whose Pareto spectrum contains a prescribed set of distinct reals. We choose to deal with such problems by first formulating them as nonlinear systems of
equations which can be smooth or nonsmooth, depending on the chosen approach, and subsequently adopt Newton type methods to solve the corresponding systems.
Our smooth approach includes the Squaring Trick (ST) and the socalled Mehrotra Predictor Corrector Method (MPCM), adapted in this context to inverse Pareto eigenvalue
complementarity problems. For the nonsmooth approach, we consider the Lattice Projection Method (LPM), and two other nonsmooth methods using complementarity function
techniques, namely $\text{SNM}_{\text{FB}}$ and $\text{SNM}_{\text{min}}$ (with FischerBurmeister and minimum complementarity functions respectively).
We compare the five methods using the performance profiles (Dolan, Mor\'e), where the average number of iterations and the percentage of failures are the performance
measures. Numerical tests show that among the methods considered, $\text{SNM}_{\text{FB}}$ performs the best in terms of the number of failures whereas LPM surpasses
all other methods with respect to the number of iterations. Finally, we point out possible extensions of the discussed methods to the inverse quadratic pencil eigenvalue
complementarity problem.

[J106] 
S. Adly, H. Attouch and R.T. Rockafellar.
Preservation or not of the maximally monotone property by graphconvergence,
Journal of Convex Analysis, Vol. 30, No. 2, pp 413440 (2023).
Abstract
Abstract:
In a general real Hilbert space $H$, given a sequence $(A_n)_{n\in\N}$ of maximally monotone operators $A_n: H \rightrightarrows H$,
which graphically converges to an operator $A$ whose domain is nonempty, we analyze
if the limit operator $A$ is still maximally monotone.
This question is justified by the fact that, as we show on an example in infinite dimension, the graph limit in the sense of Painlev\'eKuratowski of a sequence of maximally monotone operators may not be maximally monotone.
Indeed, the answer depends on the type of graph convergence which is considered. In the case of the Painlev\'eKuratowski convergence, we give a positive answer under a local compactness assumption on the graphs of the operators $A_n$. Under this assumption, the sequence $(A_n)_{n\in\N}$ turns out to be convergent for the bounded Hausdorff topology. Inspired by this result, we show that, more generally, when the sequence $(A_n)_{n\in\N}$ of maximally monotone operators converges for the bounded Hausdorff topology to an operator whose domain is nonempty, then the limit is still maximally monotone.
The answer to these questions plays a crucial role in the analysis of the sensitivity of monotone variational inclusions, and makes it possible to understand these questions in a unified way thanks to the concept of protodifferentiability.
It also leads to revisit several notions which are based on the convergence of sequences of maximally monotone operators, in particular the notion of variational sum of maximally monotone operators.

2022
[J105] 
S. Adly, M. Haddou and Manh Hung Le.
Interior point methods for solving coneconstrained eigenvalue problems,
Optimization Methods and Software, 38, no. 3, 543569, (2023).
Abstract
Abstract:
In this paper, we propose to solve coneconstrained eigenvalue problems by using interiorpoint methods. Precisely, we focus the study on an adaptation of the Mehrotra
Predictor Corrector Method (MPCM) and a NonParametric Interior Point Method (NPIPM).
We compare these two methods with the Lattice Projection Method (LPM) and the SoftMax Method (SM).
The performance profiles, on a set of data generated from the MatrixMarket, highlight the efficiency of MPCM and NPIPM for solving eigenvalue complementarity problems.
We also consider an application to a concrete and large size situation corresponding to a geomechanical fracture problem.
Finally, we discuss the extension of MPCM and NPIPM methods to solve quadratic pencil eigenvalue problems under conic constraints.

[J104] 
S. Adly, L. Bourdin and F. Caubet.
The derivative of a parameterized mechanical contact problem with a Tresca's friction law involves Signorini unilateral conditions,
ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV), 28, Paper No. 29, 29 pp (2022).
Abstract
Abstract:
The present paper investigates the sensitivity analysis, with respect to righthand source term perturbations, of a mechanical
contact problem involving a Tresca's friction law. The weak formulation of this problem leads to a variational inequality of
the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using
the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex
subdifferential of the friction functional on the Sobolev space~$\H^1(\Omega)$ and show that all its subgradients satisfy a
PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the
twice epidifferentiability, concept introduced and thoroughly studied by R.T. Rockafellar, we show the differentiability
of the parameterized Tresca's solution and that its derivative satisfies Signorini unilateral conditions. Some numerical
simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first
time that the concept of twice epidifferentiability is applied in the context of mechanical contact problems, which makes this
contribution new and original in the literature.

[J103] 
S. Adly, H. Attouch and Van Nam Vo.
Newtontype inertial algorithms for solving monotone equations governed by sums of potential and nonpotential operators,
Journal of Applied Mathematics and Optimization, 85 , no. 3, Paper No. 31, 33 pp (2022).
Abstract
Abstract:
In a Hilbert space setting, we study a class of firstorder algorithms which aim to solve structured monotone equations involving
the sum of potential and nonpotential {operators}. Precisely, we are looking for the zeros of an operator $A= \nabla f +B $
where $\nabla f$ is the gradient of a differentiable convex function $f$, and $B$ is a nonpotential monotone and cocoercive
operator. Our study is based on the inertial autonomous dynamic previously studied by the authors, which
involves dampings {controlled respectively} by the Hessian of $f$, and by a Newtontype correction term attached to $B$.
These geometric dampings attenuate the oscillations which occur with the inertial methods with viscous damping. Using Lyapunov
analysis, we study the convergence properties of the proximalgradient algorithms obtained by temporal discretization of this
dynamic. These results open the door to the design of firstorder accelerated algorithms in numerical optimization taking
into account the specific properties of potential and nonpotential terms.

[J102] 
S. Adly, L. Bourdin and G. Dhar.
The application of a universal separating vector lemma to optimal sampleddata control problems with nonsmooth Mayer cost function,
Mathematical Control and Related Fields, no. 4, 13181343 (2023).
Abstract
Abstract:
In this paper we provide a Pontryagin maximum principle for optimal sampleddata control problems with nonsmooth Mayer cost function. Our investigation leads us
to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of
a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set.
This socalled universal separating vector lemma is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle
for optimal sampleddata control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by
implementing an indirect numerical method.

[J101] 
S. Adly and Nguyen Nang Thieu.
Existence of solutions for a Lipschitzian vibroimpact problem with timedependent constraints,
Journal of Fixed Point Theory Algorithms Sci. Eng. (formerly Journal of Fixed Point Theory and Applications), Paper No. 3, 32 pp (2022).
Abstract
Abstract:
We study a mechanical system with a ﬁnite number of degrees of freedom, subjected to perfect timedependent frictionless unilateral (possibly nonconvex)
constraints with inelastic collisions on active constraints. The dynamic is described in the form of a secondorder measure diﬀerential inclusion.
Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed
to be uniformly proxregular. Our approach does not require any secondorder information or boundedness of the Hessians of the constraints involved in the problem
and are speciﬁc to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the timestepping algorithm and
construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem.
This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with timedependent nonconvex
constraints.

[J100] 
S. Adly, Huynh Van Ngai and Van Vu Nguyen. DennisMoré condition for setvalued vector fields and the superlinear convergence of Broyden updates in
Riemannian manifolds,
Journal of Convex Analysis 29, No. 3, (2022).
Abstract
Abstract:
This paper deals with the quasiNewton type scheme for solving generalized equations involving pointtoset vector fields on Riemannian manifolds. We establish some conditions
ensuring the superlinear convergence for the iterative sequence which approximates a solution of the generalized equations. Such conditions can be viewed as an extension of the classical
DennisMoré theorem in [1] as well as the Riemannian DennisMoré condition established in the work [2]. Furthermore, we also apply these results to consider the convergence of a
Broydentype update for the problem of solving generalized equations in Riemannian context. Our results are new even for classical equations defined by singlevalued vector fields.
[1] J. E. Dennis and J.J. Moré, A characterization of
superlinear convergence and its application to quasiNewton methods,
Mathematics of Computation, 28 (1974), pp. 549560.
[2] K.A. Gallivan, C.Qi, and P.A. Absil}, HighPerformance Scientific
Computing: Algorithms and Applications, Springer London, 2012, ch. A
Riemannian DennisMoré Condition, pp.281293.

[J99] 
S. Adly and H. Attouch. Firstorder inertial algorithms involving dry friction damping,
Mathematical Programming Series A, 193 (2022), no. 1, Ser. A, 405–445. Online here.
Abstract
Abstract:
In a Hilbert space $H$, based on inertial dynamics with dry friction damping, we introduce a new class of proximalgradient
algorithms
with finite convergence properties.
The function $f:H \to \R$ to minimize is supposed to be differentiable (not necessarily convex), and enters the algorithm
via its gradient.
The dry friction damping function
$\phi: H \to \R_+$ is convex with a sharp minimum at the origin, (typically $\phi(x) = r \x\$ with $r>0$). It enters the
algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities. This algorithm naturally occurs as
a discrete temporal version of an inertial differential inclusion involving viscous and dry friction together.
The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence to zero.
Then, replacing the potential function $f$ by its Moreau envelope, we extend the results to the case of a nonsmooth convex function $f$. In this case, the algorithm involves the proximal operators of $f$ and $\phi$ separately.
Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method. We then
consider the extension in the case of additive composite optimization, thus leading to new splitting methods. Numerical
experiments are given for Lassotype problems. The performance profiles, as a comparison tool, highlight the effectiveness of
two variants of the Nesterov accelerated method with dry friction.

2021
[J98]

S. Adly and R.T. Rockafellar.
Sensitivity analysis of maximal monotone inclusions via the protodifferentiability of the resolvent operator,
Mathematical Programming no. 12, Ser. B, 37–54 (2021).
Abstract
Abstract:
This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximal
monotone operators in a Hilbert space. The perturbation of all the data involved in the problem is taken into account.
Using the concept of protodifferentiability of a multifunction and the notion of semidifferentiability of a singlevalued map,
we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the
protoderivative of the resolvent operator associated with the maximal monotone parameterized variational inclusion.
This shows that the derivative of the solution of the parametrized variational inclusion obeys the same pattern by being
itself a solution of a variational inclusion involving the semiderivative and the protoderivative of the associated maps.
An application to the study of the sensitivity analysis of a parametrized primaldual composite monotone inclusion is given.
Under some sufficient conditions on the data, it is shown that the primal and the dual solutions are differentiable and their
derivatives belong to the derivative of the associated KuhnTucker set.

[J97]

S. Adly, H. Attouch, and Van Nam Vo.
Asymptotic behavior of Newtonlike inertial dynamics involving the sum of potential and nonpotential terms,
Journal of Fixed Point Theory and Applications, Paper No. 17, 30 pp (2021).
Abstract
Abstract:
In a Hilbert space $\HH$, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations
involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator $A= \nabla f + B$
where $\nabla f$ is the gradient of a continuously differentiable convex function $f$, and $B$ is a nonpotential monotone and
cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled
respectively by the Hessian of the potential $f$ and by a Newtontype correction term attached to $B$. Based on a fixedpoint
argument, we show the wellposedness of the Cauchy problem. Then we show the weak convergence as $t\to+\infty$ of the
generated trajectories towards the zeros of $\nabla f + B$. The convergence analysis is based on the appropriate setting of
the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and
attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the secondorder evolution equation as
a firstorder dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open
the door to the design of new firstorder accelerated algorithms in optimization, taking into account the specific properties of
potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of
the nonpotential term.

[J96]

S. Adly, F. Nacry, and L. Thibault.
New metric properties for proxregular sets,
Mathematical Programming 189 (2021), no. 12, Ser. B, 7–36.
Abstract
Abstract:
In this paper, we present diverse new metric properties that proxregular sets share with convex ones. At the heart of our work
lie the LegendreFenchel transform and complements of balls. First, we show that a connected proxregular set is completely
determined by the LegendreFenchel transform of a suitable perturbation of its indicator function. Then, we prove that such a
function is also the right tool to extend, to the context of proxregular sets, the famous connection between the distance
function and the support function of a convex set. On the other hand, given a proxregular set, we examine the intersection of
complements of open balls containing the set.
We establish that the distance of a point to a proxregular set is the maximum of the distances of the point from boundaries of
all such complements separating the set and the point. This is in line with the known result expressing the distance from a
convex set in terms of separating hyperplanes. To the best of our knowledge, these results are new in the literature and show
that the class of proxregular sets has good properties known in convex analysis.

[J95]

S. Adly and Ba Khiet Le.
DouglasRachford splitting algorithm for solving statedependent maximal monotone inclusions,
Optimization Letters, 15, no. 8, 28612878 (2021).
Abstract
Abstract:
In this paper, we provide a new application of the DouglasRachford splitting method for finding a zero of two maximal monotone
operators where one of the operators depends on the state. Our proposed algorithms are simple with a significant rate of
convergence and can be implemented under general conditions. Applications to generalized Nash games and quasivariational
inequalities are provided to illustrate the obtained results.

[BC]

S. Adly, D. Goeleven, and R. Oujja.
Friction models in the framework of setvalued and convex analysis,
Book Chapter, pp 122 (2021): Nonlinear Analysis and Global Optimization
Abstract
Abstract:
Most models of friction are nonsmooth in the sense that the functions involved in the models are discontinuous. The pioneering works of JeanJacques Moreau (19232014) and Panagiotis D. Panagiotopoulos (19501998)
catalyzed the development of a mathematical framework applicable to the study of nonsmooth mechanical problems using advanced
results of modern convex analysis and setvalued analysis. The approach of Moreau and Panagiotopoulos can, in particular, be used to write a precise and rigorous mathematical model describing the friction force and the stickslip phenomena. This approach using setvalued functions left aside the complicated transition processes between "stick" and "slip" but led to rigorous mathematical models like differential inclusions and variational inequalities.
In this expository paper, we summarize the approaches dealing with friction through different models.

[J94]

S. Adly and H. Attouch.
Finite time stabilization of continuous inertial dynamics combining dry friction with Hessiandriven damping,
Journal of Convex Analysis, Vol. 28, no. 2, 281310, (2021).
Abstract
Abstract:
In a Hilbert space $\H$, we study the stabilization in finite time of the trajectories generated by a continuous (in time $t$)
damped inertial dynamic system. The potential function $f:\cH \to \R$ to be minimized is supposed to be differentiable, not necessarily convex.
It enters the dynamic via its gradient. The damping results from the joint action of dry friction, viscous friction, and a geometric damping driven by
the Hessian of $f$.
The dry friction damping function $\phi:\H \to \R_+$, which is convex with a sharp minimum at the origin (typically $\phi(x) = r \x\$ with $r>0$),
enters the dynamic via its subdifferential.
It acts as a soft threshold operator on the velocities and is at the origin of the stabilization property in finite time.
The Hessiandriven damping, which enters the dynamics in the form $\nabla^2 f(x(t)) \dot{x}(t)$,
permits to control and attenuate the oscillations which occur naturally with the inertial effect.
We give two different proofs, in a finitedimensional setting, of the existence of strong solutions
of this secondorder differential inclusion. One is based on a fixedpoint argument and the use of LeraySchauder theorem; the other one is
based on the Yosida approximation technique and the Mosco convergence. We also give an existence and uniqueness result in a general Hilbert
framework by assuming that the Hessian of the function $f$ is Lipschitz continuous on the bounded sets of $\cH$.
Then, we study the convergence properties of the trajectories as $t \to +\infty$ and show their stabilization property in finite time.
The convergence results tolerate the presence of perturbations, errors, under the sole assumption of their asymptotic convergence to zero.
Then, we extend the study to the case of a nonsmooth convex function $f$.

[BC] 
S. Adly, D. Goeleven and R. Oujja.
Wellposedness of nonsmooth Lurie dynamical systems involving maximal monotone operators,
In: Parasidis, I.N., Providas, E., Rassias, T.M. (eds)
Mathematical Analysis in Interdisciplinary Research. Springer Optimization and Its Applications,
vol 179. Springer, Cham (2021). https://doi.org/10.1007/9783030847210_4.
Abstract
Abstract:
Many physical phenomena can be modeled as a feedback connection of a linear dynamical systems combined with a nonlinear function which satisfies a
sector condition. The concept of absolute stability, proposed by Lurie and Postinikov \cite{lur'e} in the early 1940s, constitutes an important tool in
the theory of control systems. Lurie dynamical systems have been studied extensively in the literature with nonlinear (but smooth) feedback functions, that can be
formulated as an ordinary differential equation. Many concrete applications in engineering can be modeled by a setvalued feedback law in order to take into account
the nonsmooth relation between the output and the state variables. In this paper, we show the wellposedness of nonsmmoth Lurie dynamical systems involving maximal
monotone operators. This includes the case where the setvalued law is given by the subdifferential of a convex, proper and lower semicontinuous function.
Some existence and uniqueness results are given depending on the data of the problem and particularly the interplay between the matrix $D$ and the setvalued
map $\F$. We will also give some conditions ensuring that the the extended resolvent $(D+\F)^{1}$ is single valued and Lipschitz continuous. The main tools used
are derived from convex analysis and maximal monotone theory.

2020
[J93]

S. Adly, F. Nacry, and L. Thibault.
Proxregular sets and LegendreFenchel transform related to separation properties,
Accepted for publication in Optimization.
Abstract
Abstract:
This paper is devoted to nonconvex/proxregular separations of sets in Hilbert spaces.
We introduce the LegendreFenchel rconjugate of a prescribed function and rquadratic support functionals and points of a
given set, all associated with a positive constant r. By means of these concepts, we obtain nonlinear functional separations for points and proxregular sets.
In addition to such functional separations, we also establish geometric separation results with balls for a proxregular set and a strongly convex set.

[J92]

S. Adly and H. Attouch
Finite convergence of proximalgradient inertial algorithms combining dry friction with Hessiandriven damping,
SIAM Journal on Optimization (SIOPT), Vol. 30, No. 3, pp. 21342162 (2020).
Abstract
Abstract:
In a Hilbert space $H$, we introduce a new class of proximalgradient algorithms with finite convergence properties.
These algorithms naturally occur as discrete temporal versions of an inertial differential inclusion which is damped under the joint action of three dampings: a viscous damping, a geometric damping driven by the Hessian, and a dry friction damping.
The function $f:H \to R$ to be minimized is supposed to be differentiable (not necessarily convex) and enters the algorithm via its gradient.
The dry friction damping function $\phi:H \to \R_+$ is convex with a sharp minimum at the origin (typically $\phi(x) = r \x\$ with $r>0$).
It enters the algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities.
The geometric damping driven by the Hessian intervenes in the dynamics in the form $\nabla^2 f(x(t)) \dot{x}(t)$.
By treating this term as the time derivative of $\nabla f(x(t))$, this gives, in discretized form, firstorder algorithms.
The Hessiandriven damping allows controlling and attenuating the oscillations. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence to zero.
Replacing the potential function $f$ by its Moreau envelope, we extend the results to the case of a nonsmooth convex function $f$.
In this case, the algorithm involves the proximal operators of $f$ and $\phi$ separately.
Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method.
We then consider the extension in the case of additive composite optimization, thus leading to splitting methods.
Numerical experiments are given for Lassotype problems. The performance profiles, as a comparison tool, highlight the effectiveness of a variant of the Nesterov accelerated method with dry friction and Hessiandriven damping.

[J91]

S. Adly and T. Haddad,
WellPosedness of nonconvex degenerate sweeping process via unconstrained evolution problems,
Nonlinear Analysis: Hybrid Systems, 36 (2020).
Abstract
Abstract:
The main concern of this paper is the study of the degenerate sweeping process involving uniform proxregular sets via an unconstrained differential inclusion by showing that the sets of solutions of the two problems coincide. This principle of reduction to an unconstrained evolution problem can be seen as a penalization of the subdifferential of the distance function. Using this reduction technique, an existence and uniqueness result of a Lipschitz perturbed version of the degenerate sweeping process is proved in the finitedimensional setting. An application is given to quasistatic unilateral dynamics in nonsmooth mechanics where the moving set is described by a finite number of inequalities. We provide sufficient verifiable conditions ensuring both the proxregularity and the Lipschitz continuity with respect to the Hausdorff distance of the moving set.

[J90]

S. Adly and D. Goeleven,
A nonsmooth approach for the modelling of a mechanical rotary drilling system with friction,
Evolution Equations and Control Theory, no. 4, 915–934. (2020).
Abstract
Abstract:
In this paper, we show how the approach of nonsmooth dynamical systems can be used to develop a suitable method for the modelling of a rotary oil drilling system with friction. We study different kinds of frictions and analyze the mathematical properties of the involved dynamical systems. We show that using a general Stribeck model for the frictional contact, we can formulate the rotary drilling system as a wellposed evolution variational inequality. Several numerical simulations are also given to illustrate both the model and the theoretical results.

[J89]

S. Adly and T. Haddad,
On Evolution quasiVariational Inequalities and Implicit statedependent Sweeping Processes,
Discrete and Continuous Dynamical SystemsS, 13, no. 6, 17911801, (2020).
Abstract
Abstract:
In this paper, we study a variant of the statedependent sweeping process with velocity constraint. The constraint C(·,u) depends upon the unknown state u, which causes one of the main difficulties in the mathematical treatment of quasivariational inequalities. Our aim is to show how a fixedpoint approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixedpoint theorem combined with a recent existence and uniqueness theorem in the case where the moving set C does not depend explicitly on the state u (i.e., C:=C(t)) given in [AHT], we prove a new existence result of solutions of the quasivariational sweeping process in the infinitedimensional Hilbert spaces with a velocity constraint. Contrary to the classical statedependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.

2019
[J88]

S. Adly,
A coderivative approach to the robust stability of composite parametric variational systems. Applications in nonsmooth mechanics,
Journal of Optimization Theory and Applications, 180, no. 1, pp. 6290, (2019).
Abstract
Abstract:
The main concern of this paper is to investigate the Lipschitzianlike stability property (namely Aubin property) of the solution map of possibly nonmonotone variational systems with composite superpotentials. Using Mordukhovich coderivative criterion and a secondorder subdifferential analysis, we provide simple and verifiable characterizations of this property in terms of the data involved in the problem. Applications are given in nonsmooth mechanics.

[J87]

S. Adly, A. Hantoute and B. Tran Nguyen.
Lyapunov stability of differential inclusions with Lipschitz Cusco perturbations of maximal monotone operators,
SetValued and Variational Analysis, 28, no. 2, 345368, (2020).
Abstract
Abstract:
We give criteria for weak and strong invariant closed sets for differential inclusions given in $\mathbb{R}^{n}$ and governed by Lipschitz Cusco perturbations of maximal monotone operators. Correspondingly, we provide different characterizations for the associated strong Lyapunov functions. The resulting conditions only depend on the data of the system.

[J86]

A. Aboussoror, S. Adly and S. Salim.
An extended conjugate duality for generalized semiinfinite programming problems via a convex decomposition,
Optimization, 69, no. 78, 16351654, (2020).
Abstract
Abstract:
We present an extended conjugate duality for a generalized semiinfinite programming problem $({\mathcal P})$. The extended duality is defined in the context of the absence of convexity of problem $({\mathcal P})$, by means of a decomposition into a family of convex subproblems and a conjugate dualization of the subproblems. Under appropriate assumptions, we establish strong extended duality and provide necessary and sufficient optimality conditions for problem $({\mathcal P})$. These extended conjugate duality and optimality conditions are new in the literature of generalized semiinfinite programming.

[J85]

S. Adly and L. Bourdin.
On a decomposition formula for the resolvent operator of the sum of two setvalued maps with monotonicity assumptions,
Applied Mathematics and Optimization, 80, no. 3, 715–732, (2019).
Abstract
Abstract:
The aim of the present work is to provide an explicit decomposition formula for the resolvent operator $\mathcal{J}_{A+B}$ of the sum of two setvalued maps $A$ and $B$ in a Hilbert space. For this purpose, we introduce a new operator called the $A$resolvent operator of $B$, denoted by $\mathcal{J}_{A,B}$, which generalizes the usual notion. Then, our main result lies in the decomposition formula $\mathcal{J}_{A+B} = \mathcal{J}_A \circ \mathcal{J}_{A,B}$ holding true when $A$ is monotone. Several properties of $\mathcal{J}_{A,B}$ are deeply investigated in this paper. In particular, the relationship between $\mathcal{J}_{A,B}$ and an extension of the classical DouglasRachford operator is established, which allows us to propose a weakly convergent algorithm that computes numerically $\mathcal{J}_{A,B}$ (and thus $\mathcal{J}_{A+B}$ from the decomposition formula) when $A$ and $B$ are maximal monotone. In order to illustrate our theoretical results, we give an application in elliptic PDEs, specifically the decomposition formula is used to point out the relationship between the classical obstacle problem and a new nonlinear PDE involving a partially blinded elliptic operator. Some numerical experiments using the finite element method are carried out in order to support our approach.

[J84]

A. Aboussoror, S. Adly and F.E. Saissi .
Optimality Conditions for Strong Semivectorial Bilevel Programming Problems via a Conjugate Duality,
Journal of Pure and Applied Functional Analysis, 4, no. 2, 151–176, (2019).
Abstract
Abstract:
We are concerned with a strong semivectorial nonlinear bilevel programming problem where the upper and lower levels are vectorial and scalar, respectively. For such a problem, we give a duality approach via scalarization, regularization, and a conjugate duality. Then, via this duality approach, we provide necessary and sufficient optimality conditions for the initial semivectorial bilevel programming problem. This duality approach extends the one given in [1] from the scalar case to the semivectorial one.
[1] A. Aboussoror, S. Adly, and F. E. Saissi, An extended FenchelLagrange duality approach and optimality conditions for strong bilevel programming problems, SIAM J. Optim., Vol. 27, No. 2, pp. 12301255, 2017.

[J83]

S. Adly and F. Nacry.
Wellposedness of discontinuous secondorder nonconvex statedependent sweeping processes,
Applied Mathematics and Optimization, 79 (2019), no. 2, 515546.
Abstract
Abstract:
In this paper, we study the existence of solutions for a time and statedependent discontinuous nonconvex secondorder sweeping process with a multivalued perturbation. The moving set is assumed to be proxregular, relatively ballcompact with bounded variation. The perturbation of the normal cone is a scalarly upper semicontinuous convexvalued multimapping satisfying a linear growth condition possibly timedependent. As an application of the theoretical results, we investigate the theory of evolution quasivariational inequalities.

[J82]

S. Adly and M. Sofonea.
Timedependent Inclusions and Sweeping Processes in Contact Mechanics,
Zeitschrift für angewandte Mathematik und Physik ZAMP: 70 (2019), no. 2, Art. 39, 19 pp.
Abstract
Abstract:
We consider a class of timedependent inclusions in Hilbert spaces for which we state and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities, convex analysis, and fixedpoint theory. Then we use this result to prove the unique weak solvability of a new class of Moreau's sweeping processes with constraints in velocity. Our results are useful in the study of mathematical models that describe the quasistatic evolution of deformable bodies in contact with an obstacle. To provide some examples, we consider three viscoelastic contact problems which lead to timedependent inclusions and sweeping processes in which the unknowns are the displacement and velocity fields, respectively. Then we apply our abstract results in order to prove the unique weak solvability of the corresponding contact problems.

[J81]

S. Adly and T. Zakaryan.
Sensitivity properties of parametric nonconvex evolution inclusions with application to optimal control problems,
Setvalued and Variational Analysis, 27 (2019), no. 2, pp 549568.
Abstract
Abstract:
The main concern of this paper is to investigate sensitivity properties of parametric evolution systems of first and second order involving a general class of nonconvex functions. Using recent results on the stability of the subdifferentials, with respect to the Gamma convergence, of the associated sequence of pln (primal lower nice) and semiconvex functions, we give some continuity properties of the solution set associated with these problems. The particular case of the parametric sweeping process involving proxregular sets is studied in detail. As an application, we study the sensitivity analysis of the generalized Bolza/Mayer problem governed by a nonsmooth dynamic of a sweeping process type.

[J80]

S. Adly, L. Bourdin, and F. Caubet.
On a decomposition formula for the proximal operator of the sum of two convex functions,
Journal of Convex Analysis 26 (2019), No. 3
Abstract
Abstract:
The main result of this theoretical paper is an original decomposition formula for the proximal operator of the sum of two proper, lower semicontinuous, and convex functions $f$ and $g$. We introduce a new operator, called the $f$proximal operator of $g$ and denoted by $\prox^f_g$, which generalizes the classical notion. We prove the decomposition formula $\prox_{f+g} = \prox_f \circ \prox^f_g$ and present several properties and characterizations of $\prox^f_g$. We also establish its relationship with a generalized version of the classical DouglasRachford operator and provide a weakly convergent algorithm for numerical computation. While the algorithm has been previously considered in the literature, our work explicitly expresses and studies the $\prox^f_g$ operator, contributing to its theoretical understanding. Finally, we demonstrate the usefulness of the decomposition formula in the context of sensitivity analysis of linear variational inequalities of the second kind in a Hilbert space.

[J79]

S. Adly, T. Haddad, and B.K. Le.
Statedependent implicit sweeping process in the framework of quasistatic evolution quasivariational inequalities,
Journal of Optimization Theory and Applications, 182, no. 2, 473493, (2019).
Abstract
Abstract:
This paper addresses the existence and uniqueness of solutions for a class of statedependent sweeping processes with constrained velocity in Hilbert spaces, without assuming any compactness condition. We introduce a new notion called hypomonotonicitylike of the normal cone to the moving set, which holds in many important cases. By combining this notion with an implicit time discretization and a Cauchy technique, we establish the strong convergence of approximate solutions to the unique solution. Using tools from convex analysis, we demonstrate the equivalence between this implicit statedependent sweeping process and quasistatic evolution quasivariational inequalities. We further apply the results to study the statedependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.

[J78]

S. Adly, A. Hantoute, and Bao Tran Nguyen.
Lyapunov Stability of Differential Inclusions Involving ProxRegular Sets via Maximal Monotone Operators,
Journal of Optimization Theory and Applications, 182, no. 3, 906934, (2019).
Abstract
Abstract:
This paper studies the existence and Lyapunov stability of solutions for differential inclusions governed by the normal cone to a proxregular set and subject to a Lipschitzian perturbation. We show that these seemingly more general nonsmooth dynamics can be reformulated into the classical theory of differential inclusions involving maximal monotone operators. This result is novel in the literature and allows us to leverage the extensive achievements in this class of monotone operators to establish the desired existence result, stability analysis, and continuity and differentiability properties of the solutions. The connection between these two models of differential inclusions is facilitated by a viability result for maximal monotone operators. As an application, we investigate a Luenbergerlike observer, which is shown to exponentially converge to the actual state when the initial estimation remains in a neighborhood of the original system's initial value.

[J77]

S. Adly, A. Hantoute, and Bao Tran Nguyen.
Weak Lyapunov functions and differential inclusions involving proxregular sets,
Journal of Convex and Nonlinear Analysis, Volume 20, Number 1, (2019).
Abstract
Abstract:
We provide criteria for weak Lyapunov functions associated with differential inclusions given in $\mathbb{R}^{n}$ and governed by Lipschitz Cusco perturbations of maximal monotone operators. Additionally, we apply these results to study the existence and stability of differential inclusions involving normal cones to proxregular sets.

2018
[J76]

S. Adly and L. Bourdin.
Sensitivity analysis of variational inequalities via twice epidifferentiability and protodifferentiability of the proximity operator,
SIAM Journal on Optimization (SIOPT), 28 (2018), no. 2, 16991725.
Abstract
Abstract:
In this paper, we investigate the sensitivity analysis of parameterized nonlinear variational inequalities of the second kind in a Hilbert space.
We extend the notions of twice epidifferentiability and protodifferentiability to the case of a parameterized lower semicontinuous convex function and its subdifferential,
respectively. By establishing the link between these notions and introducing the concept of convergent supporting hyperplanes, we derive an exact formula for the protoderivative of
the generalized proximity operator associated with a parameterized variational inequality. This allows us to show the differentiability of the associated solution with respect to
the parameter and derive a new variational inequality involving semi and twice epiderivatives of the data. We provide an application to parameterized convex optimization problems
and discuss the case of smooth convex optimization problems with inequality constraints. Our approach offers new perspectives for theoretical and computational issues in nonlinear optimization.

[J75]

S. Adly and T. Haddad.
An implicit sweeping process approach to quasistatic evolution variational inequalities,
SIAM Journal on Mathematical Analysis, 50 (2018), no. 1, 761–778.
Abstract
Abstract:
In this paper, we study a new variant of Moreau's sweeping process with a velocity constraint. Based on an adapted version of Moreau's catchingup algorithm, we establish the wellposedness (existence and uniqueness) of this problem in a general framework. We show the equivalence between this implicit sweeping process and a quasistatic evolution variational inequality, which is a common formulation for mechanical problems involving unilateral contact and friction. We provide an application by reformulating the quasistatic antiplane frictional contact problem for linear elastic materials with short memory as an implicit sweeping process with a velocity constraint. This connection between the implicit sweeping process and the quasistatic evolution variational inequality is facilitated by standard tools from convex analysis and presents a new contribution to the literature.

[J74]

S. Adly and B.K. Le.
On semicoercive sweeping process with velocity constraint,
Optimization letters, 12 (2018), no. 4, 831843. DOI: 10.1007/s1159001711492.
Abstract
Abstract:
In this note, we study the existence of solutions for a class of sweeping processes with velocity in the moving set. We solve a variational inequality at each iteration and aim to improve Theorem 5.1 in a recent paper [1] to allow for possibly unbounded moving sets. The theoretical result is supported by examples in nonregular electrical circuits.
[1] S. Adly, T. Haddad, and L. Thibault, "Convex Sweeping Process in the framework of Measure Differential Inclusions and Evolution Variational Inequalities," Math. Program., Ser. B (2014) 148 (1), 547.

[J73]

S. Adly and A. Aboussoror.
New Necessary and Sufficient Optimality Conditions for Strong Bilevel Programming Problems,
Journal of Global Optimization, 70 (2018), no. 2, 309–327.
Abstract
Abstract:
In this paper, we investigate a strong bilevel programming problem and establish necessary and sufficient optimality conditions. To analyze this problem, we use a regularization technique and global optimization tools. The derived optimality conditions, expressed in terms of maxmin conditions with linked constraints, are novel in the literature.

[J72]

S. Adly, F. Nacry, and L. Thibault.
Proxregularity approach to generalized equations and image projection,
ESAIM: Control, Optimisation and Calculus of Variations 24, pp 677708 (2018). DOI: cocv170112.
Abstract
Abstract:
In this paper, we first investigate the proxregularity behavior of solution mappings to generalized equations. This study is realized through a nonconvex uniform RobinsonUrsescu type theorem. Then, we derive new significant results for the preservation of proxregularity under various and usual set operations. The role and applications of proxregularity of solution sets of generalized equations are illustrated with dynamical systems with constraints.

[J71]

S. Adly and B.K. Le.
Secondorder State Dependent Sweeping Process with Unbounded and Nonconvex Constraints,
Pure and Applied Functional Analysis, no. 2, 271285 (2018).
Abstract
Abstract:
In this paper, using an implicit discrete scheme, we prove the existence of solutions for a class of secondorder sweeping processes with possibly unbounded and proxregular moving sets depending on both time and state in a Hilbert space. We use the local excess (local onesided Lipschitz continuity property) instead of the global Hausdorff distance to describe the way the set of constraints is moving.

[J70]

S. Adly, H. V. Ngai, and N. Van Vu.
Newtontype method for solving generalized equation on Riemannian manifolds,
Journal of Convex Analysis, Volume 25, No. 2, pp 341370 (2018).
Abstract
Abstract:
This paper is devoted to the study of Newtontype algorithm for solving inclusions involving setvalued maps defined on Riemannian manifolds. We provide some sufficient conditions ensuring the existence as well as the quadratic convergence of the Newton sequence. The material studied in this paper is based on Riemannian geometry as well as variational analysis, where metric regularity property is a key point.

[J69]

S. Adly, A. Hantoute, and Bao Tran Nguyen.
Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators,
Journal of Mathematical Analysis and Applications, Volume 457, Issue 2, 15 January 2018, pages 10171037.
Abstract
Abstract:
We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed, while the invariance criteria are still written by using only the data of the system. So, no need for the explicit knowledge of neither the solution of this differential inclusion nor the semigroup generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for $a$Lyapunov pairs of lower semicontinuous (not necessarily weaklylsc) functions associated with these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out using techniques from nonsmooth analysis.

[J68]

A. Aboussoror, S. Adly, and F.E. Saissi.
Duality and Optimality Conditions for a Class of Semivectorial Bilevel Programming Problems,
Vietnam Journal of Mathematics, 46 (2018), no. 1, 197–214.
Abstract
Abstract:
In this paper, we are concerned with a semivectorial bilevel programming problem $(S)$ where the upper and lower levels are vectorial and scalar, respectively. Our aim is to give a duality approach for problem $(S)$ via scalarization. However, the considered scalarized problem of $(S)$ does not satisfy a constraint qualification that we need for our duality approach. To overcome this obstacle, we first give a regularized problem $(S_\epsilon)$ of $(S)$ ($\epsilon>0$) whose scalarized problem $(S_\epsilon^s)$ satisfies this condition. Afterwards, we consider the FenchelLagrange dual to $(S_\epsilon^s)$. Using a stability result and under appropriate assumptions, we establish strong duality and provide optimality conditions for $(S_\epsilon^s)$ and its dual. Then, via the duality given for the previous regularizedscalarized case, we provide necessary optimality conditions for a class of properly efficient solutions of $(S)$. Finally, sufficient optimality conditions are given for $(S)$ without using duality. Examples are given for illustration.

[J67]

S. Adly and H.V. Ngai.
QuasiNewton methods for solving nonsmooth equations: Generalized DennisMoré theorem and Broyden's update,
Journal of Convex Analysis, 25 (2018), No. 4, pp 10751104.
Abstract
Abstract:
In this paper, we study the quasiNewton method by using setvalued approximations for solving generalized equations without smoothness assumptions. The setvalued approximations appear naturally when dealing with nonsmooth problems or even in smooth cases, where data in almost concrete applications are not exact. We present a generalization of the classical DennisMoré theorem, which gives a characterization of the $q$superlinear convergence of the quasiNewton iterates. The local linear and superlinear convergences of the method, especially a modification of the Broyden update method, are investigated. We present an example showing that the classical Broyden update method is no longer linearly convergent when the function involved in the nonlinear equation is not smooth. A modified version of the Broyden update is proposed, and its convergence is proved. These results are new and can be considered as both an improvement and an extension of some results that have appeared recently in the literature on this subject.

[J66]

S. Adly and B.K. Le.
Unbounded StateDependent Sweeping Processes with Perturbations in Uniformly Convex and $q$Uniformly Smooth Banach Spaces,
Numerical Algebra, Control and Optimization (NACO AIMS), Volume 8, Number 1, pp. 8195 (2018). DOI: 10.3934/naco.2018005.
Abstract
Abstract:
In this paper, the existence of solutions for a class of first and second order unbounded statedependent sweeping processes with perturbation in uniformly convex and $q$uniformly smooth Banach spaces is analyzed by using a discretization method. The sweeping process is a particular differential inclusion with a normal cone to a moving set and is of great interest in many concrete applications. The boundedness of the moving set, which plays a crucial role in
the existence of solutions in many works in the literature, is not necessary in the present paper. The compactness assumption on the moving set is also improved.

2017
[J65]

S. Adly, A. Aboussoror, and F.E. Saissi.
An Extended FenchelLagrange Duality Approach and Optimality Conditions for Strong Bilevel Programming Problems,
SIAM Journal On Optimization (SIOPT), no. 2, 1230–1255 (2017).
Abstract
Abstract:
In this paper, we give a conjugate duality approach for a strong bilevel programming problem $(S)$. The approach is based on the use of a regularization of problem $(S)$ and the socalled FenchelLagrange duality. We first show that a regularized problem of $(S)$ admits solutions and any accumulation point of a sequence of regularized solutions solves $(S)$. Then, via this duality approach, we provide necessary and sufficient optimality conditions for the regularized problem. Finally, necessary and sufficient optimality conditions are given for $(S)$. We note that such an approach, which allows us the application of the FenchelLagrange duality to the class of strong bilevel programming problems, is new in the literature.

[J64]

S. Adly, H.V. Ngai, and N. Van Vu.
Stability of metric regularity with setvalued perturbations and application to Newton's method for solving generalized equations,
Journal SetValued and Variational Analysis, September 2017, Volume 25, Issue 3, pp 543–567.
Abstract
Abstract:
In this paper, we deal firstly with the question of the stability of the metric regularity under setvalued perturbation. By adopting the measure of closeness between two multifunctions, we establish some stability results on the semilocal/local metric regularity. These results are applied to study the convergence of iterative schemes of Newtontype methods for solving generalized equations in which the setvalued part is approximated. Some examples illustrating the applicability of the proposed method are discussed.

[J63]

S. Adly and B.K. Le.
Nonconvex Sweeping Processes Involving Maximal Monotone Operators,
Optimization 66, no. 9, 1465–1486 (2017).
Abstract
Abstract:
By using a regularization method, we study in this paper the global existence and uniqueness property of a new variant of nonconvex sweeping processes involving maximal monotone operators. The system can be considered as a maximal monotone differential inclusion under a control term of normal cone type forcing the trajectory to be always contained in the desired moving set. When the set is fixed, one can show that the unique solution is rightdifferentiable everywhere and its rightderivative is rightcontinuous. Nonsmooth Lyapunov pairs for this system are also analyzed.

[J62]

S. Adly, A. Hantoute, and B.K. Le.
Maximal Monotonicity and CyclicMonotonicity Arising in Nonsmooth Lur'e Dynamical Systems,
Journal of Mathematical Analysis and Applications, 448 (2017), no. 1, 691–706
Abstract
Abstract:
We study a precomposition of a maximal monotone operator with linear mappings, which preserves the maximal monotonicity in the setting of reflexive Banach spaces. Instead of using the adjoint of such linear operators, as in the usual precomposition, we consider a more general situation involving operators which satisfy the socalled passivity condition. We also provide similar analysis for the preservation of the maximal cyclic monotonicity. These results are applied to derive existence results for nonsmooth Lur'e dynamical systems.

[J61]

A. Aboussoror, S. Adly and F.E. Saissi.
StrongWeak Nonlinear Bilevel Problems: Existence of Solutions in a Sequential Setting,
Journal Setvalued and Variational Analysis, 25 (2017), no. 1, 113–132. DOI 10.1007/s1122801603694
Abstract
Abstract:
The paper deals with a strongweak nonlinear bilevel problem $({S})$ which generalizes the wellknown weak and strong ones. For such a problem, we first give a regularization based on the use of strict $\epsilon$solutions of the lower level problem. Under appropriate assumptions, we prove the existence of solutions to the regularized problem $(S_{\epsilon})$. Then, we show that any accumulation point of a sequence of regularized solutions solves the original problem $(S)$. The obtained result is an extension of the one given in \cite{[3]} for weak bilevel programming problems. It also generalizes the result on the existence of solutions given in \cite{[26]} for the linear finite dimensional case.

[J60]

A. Aboussoror and S. Adly.
A duality approach and optimality conditions for simple convex bilevel programming problems,
Pacific Journal of Optimization, Volume 13, Number 1, pp 123135 (2017).
Abstract
Abstract:
The paper deals with a convex bilevel programming problem $({S})$ which never satisfies the Slater's condition. Using $\epsilon$approximate solutions of the lower level problem, we introduce a regularized bilevel problem $(S_{\epsilon})$ of $(S)$ that satisfies this condition. We show that $\inf S_{\epsilon}\to\inf S$ when $\epsilon$ goes to zero and that any accumulation point of a sequence of regularized solutions solves the original problem $(S)$. We provide optimality conditions for the regularized problem via the FenchelLagrange duality. Then, necessary optimality conditions are established for the solutions of $(S)$ that are accumulation points of sequences of regularized solutions. Finally, sufficient optimality conditions are established for $(S)$.

[J59]

S. Adly, F. Nacry, and L. Thibault.
Discontinuous Sweeping Process with ProxRegular Sets,
ESAIM: Control, Optimisation and Calculus of Variations, Vol. 23, No 4 pp 12931329 (2017):
Abstract
Abstract:
In this paper, we study the wellposedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving proxregular sets in Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure, and the perturbation is assumed to satisfy a Lipschitz property. The existence of a solution with bounded variation is achieved thanks to the Moreau's catchingup algorithm adapted to this kind of problem. Various properties and estimates of jumps of the solution are also provided. We give sufficient conditions to ensure the uniform proxregularity when the moving set is described by inequality constraints. As an application, we consider a nonlinear differential complementarity system which is a combination of an ordinary differential equation with a nonlinear complementarity condition. Such problems appear in many areas such as nonsmooth mechanics, nonregular electrical circuits, and control systems.

2016
[J58]

S. Adly, F. Nacry, and L. Thibault.
Preservation of proxregularity of sets and application to constrained optimization,
SIAM Journal on Optimization SIOPT, Vol. 26, No 1, pp 448473 (2016)
Abstract
Abstract:
In this paper, we first provide counterexamples showing that sublevels of proxregular functions and levels of differentiable mappings with Lipschitz derivatives may fail to be proxregular. Then, we prove the uniform proxregularity of such sets under usual verifiable qualification conditions. The preservation of uniform proxregularity of intersection and inverse image under usual qualification conditions is also established. Applications to constrained optimization problems are given.

[J57]

S. Adly, B. Brogliato, and B.K. Le.
Implicit Euler TimeDiscretization of a Class of Lagrangian Systems with SetValued Robust Controller,
Journal of Convex Analysis 23 (2016), No. 1,
pp 2352.
Abstract
Abstract:
A class of Lagrangian continuous dynamical systems with setvalued controller and subjected to a perturbation force has been thoroughly studied in [S. Adly, B. Brogliato, B. K. Le, Wellposedness, robustness and stability analysis of a setvalued controller for Lagrangian systems, SIAM J. Control Optim., 51(2), 15921614, 2013]. In this paper, we study the time discretization of these setvalued systems with an implicit Euler scheme. Under some mild conditions, the wellposedness (existence and uniqueness of solutions) of the discretetime scheme, as well as the convergence of the sequences of discrete positions and velocities in finite steps are assured. Furthermore, the approximate piecewise linear function generated by these discrete sequences is shown to converge to the solution of the continuous time differential inclusion with order $\frac{1}{2}$. Some numerical simulations on a twodegree of freedom example illustrate the theoretical developments.

[J56]

S. Adly and Ba Khiet Le.
Unbounded Second Order State Dependent Moreau's Sweeping Processes in Hilbert Spaces,
Journal of Optimization Theory and Applications, (2016) 169 pp 407–423.
Abstract
Abstract:
In this paper, an existence and uniqueness result of a second order sweeping process with velocity in the moving set under perturbation in infinitedimensional Hilbert spaces is studied by using an implicit discretization scheme. It is assumed that the moving set depends on the time, the state, and is allowed to be unbounded. The compactness assumption on the moving set is improved compared to previous works. Our methodology is based on convex and variational analysis.

[J55]

S. Adly, H. V. Ngai, and N. Van Vu.
Newton's method for solving generalized equations: Kantorovich's and Smale's approaches,
Journal of Mathematical Analysis and Applications, 439 (2016), no. 1, 396–418
Abstract
Abstract:
In this paper, we study the Newtontype method for solving generalized equations involving setvalued maps in Banach spaces. Kantorovichtype theorems (both local and global versions) are proved, as well as the quadratic convergence of the Newton sequence. We also extend both Smale's classical $(\alpha, \gamma)$theory to generalized equations. These results are new and can be considered as an extension of many known ones in the literature for the classical nonlinear case. Our approach is based on tools from variational analysis, where the metric regularity concept plays an important role in our analysis.

[J54]

S. Adly, A. Hantoute, and M. Théra.
Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain,
Mathematical Programming Ser. B. 157, No 2, pp 349–374 (2016)
Abstract
Abstract:
The general theory of Lyapunov's stability of firstorder differential inclusions in Hilbert spaces has been studied by the authors in a previous work. This new contribution focuses on the natural case when the maximally monotone operator governing the given inclusion has a domain with nonempty interior. This setting permits having nonincreasing Lyapunov functions on the whole trajectory of the solution to the given differential inclusion. It also allows some more explicit criteria for Lyapunov's pairs. Some consequences to the viability of closed sets are given, as well as some useful cases relying on the continuity or/and convexity of the involved functions. Our analysis makes use of standard tools from convex and variational analysis.

[J53]

S. Adly, Ta T.H. Trang, and Vu N. Phat.
Finitetime stabilization and $H_\infty$ Control of nonlinear timevarying delay systems via output feedback,
Journal of Industrial and Management Optimization, Vol. 12, Issue 1, 2016, pp 303315.
Abstract
Abstract:
This paper studies the robust finitetime H∞ control for a class of nonlinear systems with timevarying delay and disturbances via output feedback. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delaydependent conditions for the existence of output feedback controllers are established in terms of linear matrix inequalities (LMIs). The proposed conditions allow us to design the output feedback controllers which robustly stabilize the closedloop system in the finitetime sense. An application to H∞ control of uncertain linear systems with interval timevarying delay is also given. A numerical example is given to illustrate the efficiency of the proposed method.

[J52]

S. Adly, T.T. Trang, and V.N. Phat.
Optimal Guaranteed Cost Control of Nonlinear TimeVarying Delay Systems via Static Output Feedback,
Pacific Journal of Optimization (2016), Vol. 12, N° 3, pp. 649667
Abstract
Abstract:
The optimal guaranteed cost control problem via static output feedback controller is addressed in this paper for a class of dynamical systems with interval timevarying delays and nonlinear perturbations. By introducing a set of improved LyapunovKrasovskii functionals, a novel delaydependent condition for output feedback guaranteed cost control with guaranteed exponential stability is derived in terms of linear matrix inequalities (LMIs). Then, a design method of robust guaranteed cost control via output feedback controller is applied for uncertain linear systems. The design of output feedback controllers can be carried out in a systematic and computationally efficient manner via the use of LMIbased algorithms. Numerical examples are included to illustrate the effectiveness of the obtained result.

[J51]

S. Adly, A. Hantoute, and B.K. Le.
Nonsmooth Lur'e Dynamical Systems in Hilbert Spaces,
SetValued and Variational Analysis,Vol. 24, Issue 1, pp 1335.
Abstract
Abstract:
In this paper, we study the wellposedness and stability analysis of setvalued Lur'e dynamical systems in infinitedimensional Hilbert spaces. The existence and uniqueness results are established under the socalled passivity condition. Our approach uses a regularization procedure for the term involving the maximal monotone operator. The Lyapunov stability as well as the invariance properties are considered in detail. In addition, we give some sufficient conditions ensuring the robust stability of the system in finitedimensional spaces. The theoretical developments are illustrated by means of some examples dealing with nonregular electrical circuits. This work extends and improves some of the recent results given in [bg2, bg1, cs]. Our methodology is based on tools from setvalued and variational analysis.

2015
[J50]

S. Adly, R. Cibulka, and H.V. Ngai.
Newton's method for solving inclusions using setvalued approximations,
SIAM Journal on Optimization, 25 (2015), no. 1, 159–184
Abstract
Abstract:
Results on stability of both local and global metric regularity under setvalued perturbations are presented. As an application, we study (super)linear convergence of the Newtontype iterative process for solving generalized equations. The possibility to choose setvalued approximations allows us to describe several iterative schemes in a unified way (such as inexact Newton method, nonsmooth Newton method for semismooth functions, inexact proximal point algorithm, etc.). Moreover, it also covers a forwardbackward splitting algorithm for finding a common zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton's method is discussed.

[J49]

S. Adly and H. Rammal.
A new method for solving SecondOrder Cone Eigenvalue Complementarity Problem,
Journal of Optimization Theory and Applications, 165 (2015), no. 2, 563–585
Abstract
Abstract:
Eigenvalue complementarity problem EiCP with nonnegativity constraints has become a fruitful discipline within mathematical programming. In this paper, we extend EiCP to problem where the nonnegative orthant, i.e., the Pareto cone is replaced by the product of second order cones SOC. We reformulate such problem to find the roots of a semismooth function. Furthermore, we generalize the Lattice Projection Method LPM proposed first in [AR] to solve the second order cone eigenvalue complementarity problem SOCEiCP. The originality of this work, in comparison with [AR], is that we use a globalization of the semismooth Newton method SNM to approximate the Lorentz eigenvalues. Surprisingly, this kind of subject has never been studied before due to the difficulty of this problem in the sense that the Lorentz spectrum is not always finite. Finally, LPM is then compared to the semismooth Newton methods with line search: SNMmin and SNMFB, by using the performance profiles [Cops, perf] as a comparison tool. The numerical experiments highlight that the LPM solver is efficient and robust for solving SOCEiCP.

2014
[J48]

S. Adly, A.L. Dontchev, and M. Théra.
On onesided Lipschitz stability of setvalued contractions,
Numerical Functional Analysis and Optimization, 35 (2014), no. 79, 837–850
Abstract
Abstract:
We show that a result by T. C. Lim [On fixedpoint stability for setvalued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985) 436441] can be sharpened significantly by using a generalization of a theorem by Arutyunov regarding fixed points of composition of mappings. A global version of the LyusternikGraves theorem is a corollary of this estimate as well. We apply the generalization of Lim's result to derive onesided Lipschitz properties of the solution mapping of a differential inclusion with a parameter.

[J47]

S. Adly and R. Cibulka.
Quantitative stability of a generalized equation. Application to nonregular electrical circuits,
Journal of Optimization Theory and Applications JOTA, 160 (2014), no. 1, 90–110
Abstract
Abstract:
The paper is devoted to the study of several stability properties (such as Aubin property, calmness, and isolated calmness) of a special nonmonotone generalized equation. The theoretical results are applied in the theory of nonregular electrical circuits involving electronic devices like ideal diode, practical diode, and DIACs (DIode Alternating Current).

[J46]

S. Adly, T. Haddad, and L. Thibault.
Convex Sweeping Process in the framework of Measure Differential Inclusions and Evolution Variational Inequalities,
Mathematical Programming, 148 (2014), no. 12, Ser. B, pages: 5–47
Abstract
Abstract:
In this paper, we analyze and discuss the wellposedness of two new variants of the socalled sweeping process, introduced by J.J. Moreau in the early 70's [More71] with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset C(t), supposed to have a bounded retraction, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a firstorder sweeping process with velocity in the moving set C(t). This class of problems subsumes as a particular case, the evolution variational inequalities (widely used in applied mathematics and unilateral mechanics [DL]). Assuming that the moving subset C(t) has continuous variation for every t in [0,T] with C(0) bounded, we show that the problem has at least a Lipschitz continuous solution. The wellposedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process for the study of vector hysteresis operators in the elastoplastic model [Krej91], the planning procedure in mathematical economy [Henr], and to nonregular electrical circuits containing nonsmooth electronic devices like diodes [abb]. The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like the other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau's seminal work.

[J45]

S. Adly and B. K. Le.
Stability and invariance results for a class of nonmonotone setvalued Lur’e dynamical systems,
Applicable Analysis, 93 (2014), no. 5, 1087–1105.
Abstract
Abstract:
In this paper, we analyze the wellposedness, stability, and invariance results for a class of nonmonotone setvalued Lur'e dynamical system, which has been widely studied in control and applied mathematics [l]. Many recent researches deal with the case when the setvalued part is the subdifferential of some proper, convex, lower semicontinuous function to use the nice properties of maximally monotone operators. But in practice, particularly in electronics, there are some devices such as diac, silicon controller rectifier (SCR)... of which the voltagecurrent characteristics are not monotone but only locally hypomonotone. This fact motivates us to write the paper, which is organized as follows: firstly, the existence and uniqueness of solutions are proved by using Filippov's method and local hypomonotonicity; then, the stability analysis and generalized LaSalle's invariance principle are presented. The theoretical results are supported by numerical simulations for some examples in electronics. Our methodology is based on nonsmooth and variational analysis.

[J44]

A. Aboussoror and S. Adly
Generalized SemiInfinite Programming: Optimality Conditions Involving Reverse Convex Problems,
Numerical Functional Analysis and Optimization, 35 (2014), no. 79, 816–836
Abstract
Abstract:
The paper deals with a generalized semiinfinite programming problem (S). Under appropriate assumptions, for such a problem, we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a minmax problem constrained with compact convex linked constraints.

2013
[J43]

S. Adly, B. Brogliato, and B. K. Le.
Wellposedness, Robustness and Stability Analysis of a SetValued Controller for Lagrangian Systems,
SIAM Journal on Control and Optimization 51 (2013), no. 2, 1592–1614
Abstract
Abstract:
This paper deals with the analysis of a class of nonsmooth robust controllers for Lagrangian systems with nontrivial mass matrix. First, the existence and uniqueness of solutions are analyzed, then the Lyapunov stability, the KrasovskiiLaSalle invariance principle, and finitetime convergence properties are studied.

[J42]

S. Adly and H. Rammal.
A New Method for Solving Pareto Eigenvalue Complementarity Problems,
Computational Optimization and Applications 55, No 3, pp 703731 (2013).
Abstract
Abstract:
In this paper, we introduce a new method called the Lattice Projection Method (LPM) for solving eigenvalue complementarity problems. The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues and eigenvectors of the complementarity problems. The LPM is compared to SNM$_{\rm min}$ and SNM$_{\rm FB}$, two methods widely discussed in the literature for solving nonlinear complementarity problems, by using performance profiles as a comparison tool. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving eigenvalue complementarity problems. Finally, Pareto bieigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method.

[J41]

S. Adly, R. Cibulka, and H. Massias.
Variational Analysis and Generalized Equations in Electronics. Stability and Simulation Issues,
SetValued and Variational Analysis 21 (2013), no. 2, 333–358.
Abstract
Abstract:
The paper is devoted to the study of the Aubin/Lipschitzlike property and the isolated calmness of a particular nonmonotone generalized equation arising in electronics. The variational and nonsmooth analysis is applied in the theory of nonregular electrical circuits involving electronic devices like ideal diodes, practical diodes, DIACs, silicon controlled rectifiers (SCR), and transistors. We also discuss the relationship of our results to the ones using classical techniques from (smooth) analysis and provide a simulation for several simple electrical circuits which are chosen to cover the most common nonsmooth elements in electronics. The simulations of the electrical circuits discussed in this paper are performed by using Xcos (a component of Scilab).

[J40]

S. Adly and O. Chau.
On some dynamic thermal nonclamped contact problems,
Mathematical Programming serie B, No 12, pp 526 (2013)
Abstract
Abstract:
We study a class of dynamic thermal subdifferential contact problems with friction, for long memory viscoelastic materials, without the clamped condition, which can be put into a general model of system defined by a secondorder evolution inequality, coupled with a firstorder evolution equation. We present and establish an existence and uniqueness result, by using general results on firstorder evolution inequality, with monotone operators and fixedpoint methods. Finally, a fully discrete scheme for numerical approximations is provided, and corresponding various numerical computations in dimension two will be given.

[J39]

S. Adly and J.V. Outrata.
Qualitative Stability of a Class of NonMonotone Variational Inclusions. Application in Electronics,
Journal of Convex Analysis 20, 1 (2013)
Abstract
Abstract:
The main concern of this paper is to investigate some stability properties (namely Aubin property and isolated calmness) of a special nonmonotone variational inclusion. We provide a characterization of these properties in terms of the problem data and show their importance for the design of electrical circuits involving nonsmooth and nonmonotone electronic devices like DIAC (DIode Alternating Current). Circuits with other devices like SCR (Silicon Controlled Rectifiers), Zener diodes, thyristors, varactors, and transistors can be analyzed in the same way.

2012
[J38]

S. Adly, D. Goeleven, and B. K. Le.
Stability Analysis and Attractivity Results of a DCDC Buck Converter,
SetValued and Variational Analysis (2012) 20:331353
Abstract
Abstract:
Using tools from setvalued and variational analysis, we propose a mathematical formulation for a power DCDC Buck converter. We prove the existence of trajectories for the model. Stability and asymptotic stability results are established. The theoretical results are supported by some numerical simulations with a discussion about explicit and implicit schemes.

[J37]

S. Adly, A. Hantoute, and M. Thera.
Nonsmooth Lyapunov pairs for infinitedimensional firstorder differential inclusions,
Nonlinear Analysis: Theory, Methods and Applications 75, 3 (2012) 9851008
Abstract
Abstract:
The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated with firstorder differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described in terms of the contingent directional derivative. We also propose a unifying review of many other criteria given in the literature. Our approach is based on advanced tools of variational analysis and generalized differentiation.

[J36]

S. Adly, O. Chau, and M. Rochdi.
Solvability of a class of thermal dynamical contact problems with subdifferential conditions,
Numerical Algebra, Control and Optimization 2, 1, pp 91104 (2012).
Abstract
Abstract:
We study a class of dynamic thermal subdifferential contact problems with friction, for long memory viscoelastic materials, which can be put into a general model defined by a secondorder evolution inequality, coupled with a firstorder evolution equation. We present and establish an existence and uniqueness result, using general results on firstorder evolution inequality with monotone operators and fixedpoint methods.

2011
[J35]

A. Aboussoror and S. Adly.
A FenchelLagrange Duality Approach for a Bilevel Programming Problem with ExtremalValue Function,
Journal of Optimization Theory and Applications 149, 2 (2011) 254268
Abstract
Abstract:
In this paper, for a bilevel programming problem (S) with an extremalvalue function, we first give its FenchelLagrange dual problem. Under appropriate assumptions, we show that a strong duality holds between them. Then, we provide optimality conditions for (S) and its dual. Finally, we show that the resolution of the dual problem is equivalent to the resolution of a onelevel convex minimization problem.

[J34]

K. Addi, S. Adly, and H. Saoud.
Finitetime Lyapunov stability analysis of evolution variational inequalities,
Discrete and Continuous Dynamical Systems  Series A 31, 4 (2011) 10231038
Abstract
Abstract:
Using Lyapunov's stability and LaSalle's invariance principle for nonsmooth dynamical systems, we establish conditions for finitetime stability of evolution variational inequalities. The theoretical results are illustrated by examples drawn from electrical circuits involving nonsmooth elements like diodes.

[J33]

S. Adly and A. Seeger.
A nonsmooth algorithm for coneconstrained eigenvalue problems,
Computational Optimization and Applications 49, 2 (2011) 299318
Abstract
Abstract:
We study several variants of a nonsmooth Newtontype algorithm for solving an eigenvalue problem of the form Kx(Ax−Bx)K+. Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics. The symbol K refers to a closed convex cone in the Euclidean space R^n and (A,B) is a pair of possibly asymmetric matrices of order n. Special attention is paid to the case in which K is the nonnegative orthant of R^n. The more general case of a possibly unpointed polyhedral convex cone is also discussed in detail.

[J32]

A. Aboussoror, S. Adly, and V. Jalby.
Weak Nonlinear Bilevel Problems: Existence of Solutions via Reverse Convex and Convex Maximization Problems,
Journal of Industrial and Management Optimization 7, 3 (2011) 559571
Abstract
Abstract:
In this paper, for a class of weak bilevel programming problems, we provide sufficient conditions guaranteeing the existence of global solutions. These conditions are based on the use of reverse convex and convex maximization problems.

2010
[J31]

S. Adly, M. AitMansour, and M. Bergounioux.
Optimal control of a quasivariational obstacle problem,
Journal of Global Optimization Vol. 47, Number 3, pp 421435 (2010)
Abstract
Abstract:
We consider an optimal control problem where the statecontrol relation is given by a quasivariational inequality, namely a generalized obstacle problem. We provide an existence result for solutions to such a problem. The main tool is a stability result, based on the Moscoconvergence theory, that establishes the weak closeness of the controltostate operator. We conclude the paper with some examples.

2009
[J30]

S. Adly, D. Goeleven, and M. Théra.
Periodic solutions of Evolution Variational Inequalities: a method of guiding functions,
Chinese Annals of Mathematics, Series B, no. 3, pp 261272 (2009)

2008
[J29]

K. Addi, S. Adly, D. Goeleven, and H. Saoud.
A sensitivity analysis of a class of semicoercive variational inequalities using recession tools,
Journal of Global Optimization, vol. 40, no. 1, pp. 727 (2008)

2007
[J28]

S. Adly, K. Addi, B. Brogliato, and D. Goeleven.
A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of nonregular circuits in electronics,
Nonlinear Analysis: Hybrid Systems and Applications, N° 1, pp. 315324 (2007).

[J27]

S. Adly, D. Goeleven, and M. Théra.
A continuation method for a class of periodic evolution variational inequalities,
Some Topics in Industrial and Applied Mathematics. Series in Contemporary Applied Mathematics CAM 8, pp. 128 (2007).

[CP]

A. Ahmad, S. Adly, D. Ghazanfarpour, and O. Terraz.
Stability analysis of filtered massspring systems,
Proceedings of theory and practice of computer graphics, 2007, pp. 4552 (Bangor, England).

[J26]

S. Adly, D. Goeleven, and M. Théra.
Existence Results for a Class of Periodic Evolution Variational Inequalities,
Chinese Annals of Mathematics  Series B, Volume 28, Number 6, pp. 629650 (2007)

2006
[BC]

S. Adly, H. Attouch, and A. Cabot.
Finite time stabilization of nonlinear oscillators subject to dry friction,
Progresses in Nonsmooth Mechanics and Analysis (edited by P. Alart, O. Maisonneuve and R.T. Rockafellar),
Advances in Mathematics and Mechanics, Kluwer, pp. 289304, (2006).

[J25]

S. Adly.
Attractivity Theory for Second Order Nonsmooth Dynamical Systems with application to dry friction,
Journal of Mathematical Analysis and Applications, 322, pp. 10551070 (2006).

[J24]

S. Adly, M. AitMansour, and L. Scrimali.
Sensitivity analysis of solutions to a class of quasivariational inequalities,
Bulletino Della Unione Mathematica Italiana (BUMI), pp. 767772 (2006).

[CP]

S. Adly, E. Ernst, and M. Théra.
Stability in frictional unilateral elasticity revisited: an application of the theory of semicoercive variational inequalities,
Proceeding of American Institute of Physics, Vol. 835, pp. 111 (2006).

[CP]

K. Addi, S. Adly, B. Brogliato, and D. Goeleven.
The approach of Moreau and Panagiotopoulos: use it in electronics,
accepté dans Proceeding of the International Conference
on Nonsmooth/Nonconvex Mechanics with Applications in Engineering, pp. 471478 (2006).

[J23]

S. Adly.
Stability of linear semicoercive variational inequalities in Hilbert spaces: application to the SignoriniFichera Problem,
Journal of Nonlinear and Convex Analysis, N° 3, pp. 325334 (2006).

[J22]

S. Adly, K. Addi, D. Goeleven, and M. Théra.
A nonsymmetric linear complementarity problem to solve a quasistatic rolling frictional contact problem,
Journal of Nonlinear and Convex Analysis, N°3, pp. 315324 (2006).

[CP]

A. Ahmad, S. Adly, D. Ghazanfarpour, and O. Terraz.
Stabilisation par filtrage de méthodes d'intégration explicite,
Journées AFIG (Association Française d'Informatique Graphique), pp. 7380 (2006).

2004
[J21]

S. Adly, E. Ernst, and M. Théra.
Norm Closure of the Barrier cone in normed linear spaces,
Proceeding of the American Mathematical Society, 132, no. 10, pp. 29112915 (2004)

[J20]

S. Adly, E. Ernst, and M. Théra.
Wellpositioned closed convex sets and wellpositioned closed convex functions,
Journal of Global Optimization, 29 (4), pp. 337351 (2004).

[J19]

S. Adly and D. Goeleven.
A stability Theory for secondorder nonsmooth dynamical systems with application to friction problems,
Journal de Mathématiques Pures et Appliquées, Vol. 83, pp. 1751 (2004).

2003
[J18]

S. Adly, E. Ernst, and M. Théra.
On the closedness of the algebraic difference of closed convex sets,
Journal de Mathématiques Pures et Appliquées, Vol. 82, 9, pp. 12191249 (2003).

2002
[J17]

S. Adly, E. Ernst, and M. Théra.
Stability of Noncoercive Variational Inequalities,
Communication in Contemporary Mathematics, 4 No 1, pp. 145160 (2002).

[J16]

S. Adly, E. Ernst, and M. Théra.
On the converse of the Dieudonné theorem in reflexive Banach spaces,

[J15]

S. Adly, E. Ernst, and M. Théra.
A Characterization of Convex and Semicoercive Functionals,
Journal of Convex Analysis, Volume 8, No 1, pp. 127148 (2001).

[J14]

S. Adly, E. Ernst, and M. Théra.
Stabilité de l'ensemble des solutions d'une inéquation variationnelle non coercive,
Compte Rendu de l'Académie des Sciences (CRAS), T. 333 Série I, pp. 409414 (2001).

2000
[J13] 
S. Adly and D. Goeleven.
A Discretization Theory for a Class Semicoercive Unilateral Problems,
Numerishe Mathematik,
87, pp 134 (2000).
Abstract
Abstract:
In this paper, we present a convergence analysis applicable to the approximation of a large class of semicoercive variational inequalities.
The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finiteelement approximations of semicoercive unilateral
problems in mechanics are discussed. In particular, a SignoriniFichera unilateral contact model and some obstacle problem with frictions are studied.
The theoretical conditions proved are in good agreement with the numerical ones.

[J12] 
S. Adly and D. Motreanu.
Periodic solutions for second order differential equations involving nonconvex superpotentials,
Journal of Global Optimization,
17, pp 917 (2000).
Abstract
Abstract:
The paper establishes the existence of a nonconstant periodic solution of a general second order nonautonomous
Hamiltonian system with discontinuous nonlinearities. The multiplicity of solutions is also studied.

[J11] 
S. Adly and D. Motreanu.
Location of eigensolutions to variationalhemivariational inequalities,
Journal of Nonlinear and Convex analysis,
Vol. 1, No 3, pp 255270, (2000).
Abstract
Abstract:
In this paper, we develop a method to deal with the existence and the
location of eigensolutions of a general class of variational problems, namely
variationalhemivariational inequalities. We discuss a general example of
nonlinear and nonsmooth eigenvalue problem with a constraint on a convex
set.

1999
[J10] 
S. Adly and W. Oettli.
Solvability of generalized nonlinear symmetric variational inequalities,
Bullettin of the Austrelian Mathematical Society,
Ser. B (40), pp 289300 (1999).
Abstract
Abstract:
This paper deals with the study of a general class of nonlinear variational inequalities. An existence result is given, and a perturbed iterative
scheme is analyzed for solving such problems.

1998
[J9] 
S. Adly, G. Buttazzo and M. Théra.
Critical points for nonsmooth energy functions and applications,
Nonlinear Analysis, Theory Methods and Applications,
Vol. 32, No 6, pp. 711718 (1998).
Abstract
Abstract:
This paper contains some existence results of critical points of nonsmooth and nonconvex energy functionals under suitable assumptions. Some applications are given to
semicoercive nonlinear PDE's and to differential equations involving discontinuous nonlinearities.

[J8] 
S. Adly, D. Goeleven and D. Motreanu.
Unilateral Hamiltonian Systems: a survey on the inequality approach,
Advances in Nonlinear Variational Inequalities,
1 , no. 1, pp 1126 (1998).
Abstract
Abstract:
This paper contains some existence and multiplicity results
for periodic solutions of second order nonautonomous and
nonsmooth Hamiltonian systems involving nonconvex
superpotentials. This study is achieved by proving the
existence of homoclinic solutions. These solutions are
constructed as critical points of the corresponding
nonconvex and nonsmooth energy functional.

1997
[J7] 
S. Adly, D. Goeleven and D. Motreanu.
Periodic and homoclinic solutions for a class of unilateral problems,
Discrete and Continuous Dynamical Systems,
vol. 3, Number 4, October (1997).
Abstract
Abstract:
This paper contains some existence and multiplicity results
for periodic solutions of second order nonautonomous and
nonsmooth Hamiltonian systems involving nonconvex
superpotentials. This study is achieved by proving the
existence of homoclinic solutions. These solutions are
constructed as critical points of the corresponding
nonconvex and nonsmooth energy functional.

1996
[J6] 
S. Adly, D. Goeleven and M. Théra.
Recession mappings and noncoercive variational inequalities,
Journal of Nonlinear Analysis, Theory Methods and Applications,
Vol. 26, No. 9, pp 15731603 (1996).
Abstract
Abstract:
In this paper, we present a quite general existence theory
applicable to several class of noncoercive variational inequalities and
related variational problems. This result is based on the
asymptotic behavior of the sets and the operators which are
involved in the variational inequality. Applications are then
given to several strongly nonlinear boundaries problems at
resonance, to the NavierStockes equations, to an obstacle
problem, to the unilateral buckling of a simply supported
beam and to the unilateral buckling of a thin elastic plate.

[J5] 
S. Adly.
A perturbed iterative method for a general class of variational inequalities,
Serdica Mathematical Journal ,
Vol. 22, pp 6982 (1996).
Abstract
Abstract:
The generalized WienerHopf equation and the approximation methods are
used to propose a perturbed iterative method to compute the solutions of
a general class of nonlinear variational inequalities.

[J4] 
S. Adly.
Iterative algorithms and sensivity analysis for a general class of variational inclusions,
Journal of Mathematical Analysis and Applications ,
201, pp 609630 (1996).
Abstract
Abstract:
In the present paper, we study a perturbed iterative method for solving
a general class of variational inclusions. An existence result which
generalizes some known results in this field, a convergence result and
a new iterative method are given. We also prove the continuity of the
perturbed solution to a parametric variational inclusion problem.
Several special cases are discussed.

1995
[J3] 
S. Adly and D. Goeleven.
Periodic solutions for a class of hemivariational inequalities,
Communications on Applied Nonlinear Analysis ,
2, No. 2, pp 4758 (1995).
Abstract
Abstract:
The field of inequality problems has seen a considerable
development in mathematics and unilateral mechanics.
Particularly, the theory of variational inequalities is now a
welldeveloped theory in mathematics.
The mechanical meaning of a variational inequality is given by
the formulation of the principle of virtual work when a
monotone stressstrain or reactiondisplacement condition hold.
In the case of lack of monotonicity of these underlying
stressstrain or reactiondisplacement conditions, the
variational expressions of the principle of virtual work are
no longer variational inequalities. Another type of inequality
expression arises as variational formulation of the problem.
These inequalities are called hemivariational inequalities and
were born only 12 years ago. The idea of these new inequality
methods are due to Panagiotopoulos. For more details concerning
these approach, we refer the reader to the book of P.D.
Panagiotopoulos and of Naniewicz & Panagiotopoulos.
It is clear that much could be done to extend the existence
theory of variational inequality to this case.
This paper starts with the study of a system of differential
equations involving discontinuous nonlinearities. The method
employed is based on nonsmooth critical point theory of
Chang.
We shall show that our approach contributes to the mathematical
theory of hemivariational inequalities.

[J2] 
S. Adly and D. Goeleven.
Homoclinic orbits for a class of hemivariational inequalities,
Journal of Applicable Analysis ,
Vol. 58, pp 229240 (1995).
Abstract
Abstract:
The hemivariational inequalities approach has been now proved to be
very efficient to describe the behavior of several complex structures
as multilayered plates, adhesive joints, composite structures, etc.
For details concerning the physical problems, we refer the reader to
the works of P. D. Panagiotopoulos. As a consequence of the contribution
s of P. D. Panagiotopoulos, the study of hemivariational inequalities
has emerged as an interesting branch of applied mathematics and this topi
c is now the object of the attention of several engineers and
mathematicians as P. D. Panagiotopoulos, Z. Naniewicz, D. Goeleven &
M. Théra, S. Adly & D. Goeleven.
Using a new compact imbedding theorem of C. De Coster and M. Willem,
we prove the existence of homoclinic orbits for a class of
hemivariational inequalities.

[J1] 
S. Adly, D. Goeleven and M. Théra.
Recession methods in monotone variational hemivariational inequalities,
Journal of Topological Methods in Nonlinear Analysis ,
Vol. 5, No. 2, pp. 397409, (1995).
Abstract
Abstract:
In this note, using the well known Ky Fan minimax principle,
we prove an existence theorem applicable to a large class of
variational hemivariational inequalities involving
hemicontinuous monotone operators.
