In this example, we want to compute lower bounds for the problem of optimizing the linear function X+Y+Z
over the hyperbolicity cone of (X^2-Y^2-Z^2)*(4*X^2-Y^2-Z^2) in diretion [1,0,0].

We use the following formalization in a Maple worksheet

   > F := (X^2-Y^2-Z^2)*(4*X^2-Y^2-Z^2):
   > E := [1,0,0]:
   > lin := X+Y+Z:

We call the function RenegarRelaxations

   > RenegarRelaxations(F,E,lin);

The output of RenegarRelaxations is the following description of the lower bounds and corresponding minimizers :

    3-th DERIVATIVE
    Lower bound given by the 3-th derivative: []
    Computed in time 0.000

This is reasonable because the third derivative of F is a linear form, hence its hyperbolicity cone is a
hyperspace and X+Y+Z is unbounded on it. The other relaxations give

    2-th DERIVATIVE
    Lower bound given by the 2-th derivative: -2.098386677
    Computed in time 0.074
    Algebraic degree 2, multiplicity 1
    	      [[X = [1., 1.], Y = [-1.549193338, -1.549193338], Z = [-1.549193338, -1.549193338]]]
                               2
                          [5 _Z  - 12, 5 _Z, [12, 12]]

    1-th DERIVATIVE
    Lower bound given by the 1-th derivative: -.7888543820
    Computed in time 0.033
    Algebraic degree 2, multiplicity 1
    	      [[X = [1., 1.], Y = [-0.8944271910, -0.8944271910], Z = [-0.8944271910, -0.8944271910]]]
                                2
                           [5 _Z  - 4, 5 _Z, [4, 4]]


    0-th DERIVATIVE
    Lower bound given by the 0-th derivative: -.4142135624
    Computed in time 0.042
    Algebraic degree 2, multiplicity 1
    	      [[X = [1., 1.], Y = [-0.7071067812, -0.7071067812], Z = [-0.7071067812, -0.7071067812]]]
                  2            3              2          2
               [_Z  - 1/2, 4 _Z  - 5 _Z, [5 _Z  - 4, 5 _Z  - 4]]

We compute with the "0-th relaxation" the exact value of the minimizer with its rational parametrization.