Abstract: This brief examines mathematical models in nonsmooth mechanics and nonregular electrical circuits, including evolution variational inequalities, complementarity systems, differential inclusions, second-order 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.

Abstract: In this paper, we introduce a new approximate sliding mode observer for Lur'e set-valued dynamical systems, particularly addressing challenges posed by uncertainties not within the standard range of observation. Traditionally, Luenberger-like 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

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 single-valued 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: well-posedness 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.

Abstract: In this paper, we investigate the asymptotic properties of a particular class of state-dependent 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 quasi-variational inequalities. As a result, we introduce a new derivative-free 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 second-order differentiability conditions of the data.

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 Hessian-driven damping to attenuate inherent oscillations in inertial methods. By analyzing the high-resolution ODEs of order $p=0,1,2$ for these algorithms, we gain insights into their similarities and differences. All three algorithms share the same low-resolution 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 high-resolution ODEs of order $2$ in $h$ (termed super-resolution). 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 CPU-time, both (RAG) and (NAG) outperform (IGAHD). All three algorithms exhibit similar behavior when evaluating based on gradient norms.

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 time-scale 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 Hessian-driven damping. These dynamics are derived as high-resolution 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 Levenberg-Marquardt 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 high-dimensional problems. The interplay between numerical algorithms and continuous-time 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.

Abstract: In a Hilbert setting, our study focuses on the dynamical system introduced by Su-Boyd-Cand\`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 second-order evolution equation to a first-order 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.

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 first-order 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 first-order differential equation, followed by the application of the method of averaging}. By applying this methodology, we derive fast convergence results for second-order time-evolution systems with dry friction, asymptotically vanishing viscous damping, and implicit Hessian-driven damping.

Abstract: In this paper, we study} a sliding mode observer for a class of set-valued Lur'e systems subject to uncertainties. We show that our approach has obvious advantages than the existing Luenberger-like observers. {Furthermore, we provide an effective continuous approximation to eliminate the chattering effect in the sliding mode technique.

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 Kurdyka-Lojasiewicz 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.

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 epi-differentiability, 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 Signorini-type 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.

Abstract: In a Hilbert framework, for convex differentiable optimization, we analyze the long-time 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 first-order 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 sub-differential. 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 second-order time-evolution systems involving dry friction, asymptotically vanishing viscous damping, and Hessian-driven damping in the implicit form.

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 Lipschitz-likeness of the constraint set-valued mapping. The obtained results are compared with the existing ones and analyzed by several examples.

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 Newton-type damping. This contrasts with the authors' previous study where explicit Newton-type damping was considered, which, for the potential term, corresponds to Hessian-driven 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 second-order 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 first-order 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.

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 so-called 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 Fischer-Burmeister 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.

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\'e-Kuratowski 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\'e-Kuratowski 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.

Abstract: In this paper, we propose to solve cone-constrained eigenvalue problems by using interior-point methods. Precisely, we focus the study on an adaptation of the Mehrotra Predictor Corrector Method (MPCM) and a Non-Parametric 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.

Abstract: The present paper investigates the sensitivity analysis, with respect to right-hand 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 epi-differentiability, 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 epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.

Abstract: In a Hilbert space setting, we study a class of first-order 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 Newton-type 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 proximal-gradient algorithms obtained by temporal discretization of this dynamic. These results open the door to the design of first-order accelerated algorithms in numerical optimization taking into account the specific properties of potential and nonpotential terms.

Abstract: In this paper we provide a Pontryagin maximum principle for optimal sampled-data 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 so-called universal separating vector lemma is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data 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.

Abstract: We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order 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 prox-regular. Our approach does not require any second-order 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 time-stepping 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 time-dependent nonconvex constraints.

Abstract: This paper deals with the quasi-Newton type scheme for solving generalized equations involving point-to-set 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 Dennis-Moré theorem in [1] as well as the Riemannian Dennis-Moré condition established in the work [2]. Furthermore, we also apply these results to consider the convergence of a Broyden-type update for the problem of solving generalized equations in Riemannian context. Our results are new even for classical equations defined by single-valued vector fields.

[1] J. E. Dennis and J.J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Mathematics of Computation, 28 (1974), pp. 549--560.
[2] K.A. Gallivan, C.Qi, and P.-A. Absil}, High-Performance Scientific Computing: Algorithms and Applications, Springer London, 2012, ch. A Riemannian Dennis-Moré Condition, pp.281--293.

Abstract: In a Hilbert space $H$, based on inertial dynamics with dry friction damping, we introduce a new class of proximal-gradient 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 Lasso-type problems. The performance profiles, as a comparison tool, highlight the effectiveness of two variants of the Nesterov accelerated method with dry friction.

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 proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the proto-derivative 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 semi-derivative and the proto-derivative of the associated maps. An application to the study of the sensitivity analysis of a parametrized primal-dual 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 Kuhn-Tucker set.

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 Newton-type correction term attached to $B$. Based on a fixed-point argument, we show the well-posedness 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 second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order 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.

Abstract: In this paper, we present diverse new metric properties that prox-regular sets share with convex ones. At the heart of our work lie the Legendre-Fenchel transform and complements of balls. First, we show that a connected prox-regular set is completely determined by the Legendre-Fenchel 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 prox-regular sets, the famous connection between the distance function and the support function of a convex set. On the other hand, given a prox-regular set, we examine the intersection of complements of open balls containing the set. We establish that the distance of a point to a prox-regular 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 prox-regular sets has good properties known in convex analysis.

Abstract: In this paper, we provide a new application of the Douglas-Rachford 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.

Abstract: Most models of friction are nonsmooth in the sense that the functions involved in the models are discontinuous. The pioneering works of Jean-Jacques Moreau (1923-2014) and Panagiotis D. Panagiotopoulos (1950-1998) catalyzed the development of a mathematical framework applicable to the study of nonsmooth mechanical problems using advanced results of modern convex analysis and set-valued 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 stick-slip phenomena. This approach using set-valued 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.

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 Hessian-driven 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 finite-dimensional setting, of the existence of strong solutions of this second-order differential inclusion. One is based on a fixed-point argument and the use of Leray-Schauder 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 $f$.

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 set-valued feedback law in order to take into account the nonsmooth relation between the output and the state variables. In this paper, we show the well-posedness of nonsmmoth Lurie dynamical systems involving maximal monotone operators. This includes the case where the set-valued 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 set-valued 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.

Abstract: This paper is devoted to nonconvex/prox-regular separations of sets in Hilbert spaces. We introduce the Legendre-Fenchel r-conjugate of a prescribed function and r-quadratic 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 prox-regular sets. In addition to such functional separations, we also establish geometric separation results with balls for a prox-regular set and a strongly convex set.

Abstract: In a Hilbert space $H$, we introduce a new class of proximal-gradient 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, first-order algorithms. The Hessian-driven 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 Lasso-type problems. The performance profiles, as a comparison tool, highlight the effectiveness of a variant of the Nesterov accelerated method with dry friction and Hessian-driven damping.

Abstract: The main concern of this paper is the study of the degenerate sweeping process involving uniform prox-regular 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 finite-dimensional 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 prox-regularity and the Lipschitz continuity with respect to the Hausdorff distance of the moving set.

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 well-posed evolution variational inequality. Several numerical simulations are also given to illustrate both the model and the theoretical results.

Abstract: In this paper, we study a variant of the state-dependent 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 quasi-variational inequalities. Our aim is to show how a fixed-point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed-point 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 quasi-variational sweeping process in the infinite-dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.

Abstract: The main concern of this paper is to investigate the Lipschitzian-like stability property (namely Aubin property) of the solution map of possibly nonmonotone variational systems with composite superpotentials. Using Mordukhovich coderivative criterion and a second-order 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.

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.

Abstract: We present an extended conjugate duality for a generalized semi-infinite 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 semi-infinite programming.

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 set-valued 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 Douglas-Rachford 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.

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 Fenchel-Lagrange duality approach and optimality conditions for strong bilevel programming problems, SIAM J. Optim., Vol. 27, No. 2, pp. 1230-1255, 2017.

Abstract: In this paper, we study the existence of solutions for a time and state-dependent discontinuous nonconvex second-order sweeping process with a multivalued perturbation. The moving set is assumed to be prox-regular, relatively ball-compact with bounded variation. The perturbation of the normal cone is a scalarly upper semicontinuous convex-valued multimapping satisfying a linear growth condition possibly time-dependent. As an application of the theoretical results, we investigate the theory of evolution quasi-variational inequalities.

Abstract: We consider a class of time-dependent 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 fixed-point 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 time-dependent 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.

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 semi-convex functions, we give some continuity properties of the solution set associated with these problems. The particular case of the parametric sweeping process involving prox-regular 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.

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 Douglas-Rachford 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.

Abstract: This paper addresses the existence and uniqueness of solutions for a class of state-dependent sweeping processes with constrained velocity in Hilbert spaces, without assuming any compactness condition. We introduce a new notion called hypomonotonicity-like 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 state-dependent sweeping process and quasistatic evolution quasi-variational inequalities. We further apply the results to study the state-dependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.

Abstract: This paper studies the existence and Lyapunov stability of solutions for differential inclusions governed by the normal cone to a prox-regular 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 Luenberger-like 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.

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 prox-regular sets.

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 epi-differentiability and proto-differentiability to the case of a parameterized lower semi-continuous 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 proto-derivative 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 epi-derivatives 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.

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 catching-up algorithm, we establish the well-posedness (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.

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), 5--47.

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 max-min conditions with linked constraints, are novel in the literature.

Abstract: In this paper, we first investigate the prox-regularity behavior of solution mappings to generalized equations. This study is realized through a nonconvex uniform Robinson-Ursescu type theorem. Then, we derive new significant results for the preservation of prox-regularity under various and usual set operations. The role and applications of prox-regularity of solution sets of generalized equations are illustrated with dynamical systems with constraints.

Abstract: In this paper, using an implicit discrete scheme, we prove the existence of solutions for a class of second-order sweeping processes with possibly unbounded and prox-regular moving sets depending on both time and state in a Hilbert space. We use the local excess (local one-sided Lipschitz continuity property) instead of the global Hausdorff distance to describe the way the set of constraints is moving.

Abstract: This paper is devoted to the study of Newton-type algorithm for solving inclusions involving set-valued 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.

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 semi-continuous (not necessarily weakly-lsc) 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.

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 Fenchel-Lagrange 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 regularized-scalarized 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.

Abstract: In this paper, we study the quasi-Newton method by using set-valued approximations for solving generalized equations without smoothness assumptions. The set-valued 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 Dennis-Moré theorem, which gives a characterization of the $q$-superlinear convergence of the quasi-Newton 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.

Abstract: In this paper, the existence of solutions for a class of first and second order unbounded state-dependent 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.

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 so-called Fenchel-Lagrange 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 Fenchel-Lagrange duality to the class of strong bilevel programming problems, is new in the literature.

Abstract: In this paper, we deal firstly with the question of the stability of the metric regularity under set-valued perturbation. By adopting the measure of closeness between two multifunctions, we establish some stability results on the semi-local/local metric regularity. These results are applied to study the convergence of iterative schemes of Newton-type methods for solving generalized equations in which the set-valued part is approximated. Some examples illustrating the applicability of the proposed method are discussed.

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 right-differentiable everywhere and its right-derivative is right-continuous. Nonsmooth Lyapunov pairs for this system are also analyzed.

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 so-called 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.

Abstract: The paper deals with a strong-weak nonlinear bilevel problem $({S})$ which generalizes the well-known 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.

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 Fenchel-Lagrange 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 $(S)$.

Abstract: In this paper, we study the well-posedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving prox-regular 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 catching-up 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 prox-regularity 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.

Abstract: In this paper, we first provide counterexamples showing that sublevels of prox-regular functions and levels of differentiable mappings with Lipschitz derivatives may fail to be prox-regular. Then, we prove the uniform prox-regularity of such sets under usual verifiable qualification conditions. The preservation of uniform prox-regularity of intersection and inverse image under usual qualification conditions is also established. Applications to constrained optimization problems are given.

Abstract: A class of Lagrangian continuous dynamical systems with set-valued controller and subjected to a perturbation force has been thoroughly studied in [S. Adly, B. Brogliato, B. K. Le, Well-posedness, robustness and stability analysis of a set-valued controller for Lagrangian systems, SIAM J. Control Optim., 51(2), 1592--1614, 2013]. In this paper, we study the time discretization of these set-valued systems with an implicit Euler scheme. Under some mild conditions, the well-posedness (existence and uniqueness of solutions) of the discrete-time 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 two-degree of freedom example illustrate the theoretical developments.

Abstract: In this paper, an existence and uniqueness result of a second order sweeping process with velocity in the moving set under perturbation in infinite-dimensional 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.

Abstract: In this paper, we study the Newton-type method for solving generalized equations involving set-valued maps in Banach spaces. Kantorovich-type 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.

Abstract: The general theory of Lyapunov's stability of first-order 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.

Abstract: This paper studies the robust finite-time H∞ control for a class of nonlinear systems with time-varying delay and disturbances via output feedback. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delay-dependent 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 closed-loop system in the finite-time sense. An application to H∞ control of uncertain linear systems with interval time-varying delay is also given. A numerical example is given to illustrate the efficiency of the proposed method.

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 time-varying delays and nonlinear perturbations. By introducing a set of improved Lyapunov-Krasovskii functionals, a novel delay-dependent 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 LMI-based algorithms. Numerical examples are included to illustrate the effectiveness of the obtained result.

Abstract: In this paper, we study the well-posedness and stability analysis of set-valued Lur'e dynamical systems in infinite-dimensional Hilbert spaces. The existence and uniqueness results are established under the so-called 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 finite-dimensional 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 set-valued and variational analysis.

Abstract: Results on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super-)linear convergence of the Newton-type iterative process for solving generalized equations. The possibility to choose set-valued approximations allows us to describe several iterative schemes in a unified way (such as inexact Newton method, non-smooth Newton method for semi-smooth functions, inexact proximal point algorithm, etc.). Moreover, it also covers a forward-backward 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.

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.

Abstract: We show that a result by T. -C. Lim [On fixed-point stability for set-valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985) 436--441] 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 Lyusternik-Graves theorem is a corollary of this estimate as well. We apply the generalization of Lim's result to derive one-sided Lipschitz properties of the solution mapping of a differential inclusion with a parameter.

Abstract: The paper is devoted to the study of several stability properties (such as Aubin property, calmness, and isolated calmness) of a special non-monotone generalized equation. The theoretical results are applied in the theory of non-regular electrical circuits involving electronic devices like ideal diode, practical diode, and DIACs (DIode Alternating Current).

Abstract: In this paper, we analyze and discuss the well-posedness of two new variants of the so-called 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 first-order 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 well-posedness 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.

Abstract: In this paper, we analyze the well-posedness, stability, and invariance results for a class of non-monotone set-valued Lur'e dynamical system, which has been widely studied in control and applied mathematics [l]. Many recent researches deal with the case when the set-valued 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 voltage-current characteristics are not monotone but only locally hypo-monotone. 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 hypo-monotonicity; 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.

Abstract: The paper deals with a generalized semi-infinite 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 min-max problem constrained with compact convex linked constraints.

Abstract: This paper deals with the analysis of a class of nonsmooth robust controllers for Lagrangian systems with non-trivial mass matrix. First, the existence and uniqueness of solutions are analyzed, then the Lyapunov stability, the Krasovskii-LaSalle invariance principle, and finite-time convergence properties are studied.

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 bi-eigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method.

Abstract: The paper is devoted to the study of the Aubin/Lipschitz-like property and the isolated calmness of a particular non-monotone generalized equation arising in electronics. The variational and non-smooth analysis is applied in the theory of non-regular 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 non-smooth elements in electronics. The simulations of the electrical circuits discussed in this paper are performed by using Xcos (a component of Scilab).

Abstract: We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, without the clamped condition, which can be put into a general model of system defined by a second-order evolution inequality, coupled with a first-order evolution equation. We present and establish an existence and uniqueness result, by using general results on first-order evolution inequality, with monotone operators and fixed-point methods. Finally, a fully discrete scheme for numerical approximations is provided, and corresponding various numerical computations in dimension two will be given.

Abstract: The main concern of this paper is to investigate some stability properties (namely Aubin property and isolated calmness) of a special non-monotone 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 non-monotone 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.

Abstract: Using tools from set-valued and variational analysis, we propose a mathematical formulation for a power DC-DC 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.

Abstract: The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated with first-order 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.

Abstract: We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, which can be put into a general model defined by a second-order evolution inequality, coupled with a first-order evolution equation. We present and establish an existence and uniqueness result, using general results on first-order evolution inequality with monotone operators and fixed-point methods.

Abstract: In this paper, for a bilevel programming problem (S) with an extremal-value function, we first give its Fenchel-Lagrange 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 one-level convex minimization problem.

Abstract: Using Lyapunov's stability and LaSalle's invariance principle for nonsmooth dynamical systems, we establish conditions for finite-time stability of evolution variational inequalities. The theoretical results are illustrated by examples drawn from electrical circuits involving nonsmooth elements like diodes.

Abstract: We study several variants of a nonsmooth Newton-type 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.

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.

Abstract: We consider an optimal control problem where the state-control relation is given by a quasi-variational 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 Mosco-convergence theory, that establishes the weak closeness of the control-to-state operator. We conclude the paper with some examples.

Abstract: In this paper, we present a convergence analysis applicable to the approximation of a large class of semi-coercive variational inequalities. The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finite-element approximations of semi-coercive unilateral problems in mechanics are discussed. In particular, a Signorini-Fichera unilateral contact model and some obstacle problem with frictions are studied. The theoretical conditions proved are in good agreement with the numerical ones.

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.

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 variational-hemivariational inequalities. We discuss a general example of nonlinear and nonsmooth eigenvalue problem with a constraint on a convex set.

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.

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.

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.

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.

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 Navier-Stockes equations, to an obstacle problem, to the unilateral buckling of a simply supported beam and to the unilateral buckling of a thin elastic plate.

Abstract: The generalized Wiener-Hopf 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.

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.

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 well-developed theory in mathematics. The mechanical meaning of a variational inequality is given by the formulation of the principle of virtual work when a monotone stress-strain or reaction-displacement condition hold. In the case of lack of monotonicity of these underlying stress-strain or reaction-displacement 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.

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.

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.