A Maximization Algorithm of Pseudo-convex Quadratic Functions on Closed Convex Sets in Euclidean Spaces

Our aim in this paper is the search for the maxima of vectorial pseudo-convex quadratic functions. The motivation behind the choice of these functions is mainly computation. We shall give an algorithm to find such maxima and give some computational results at the end. Many papers were devoted to the numerical search of minima and maxima of convex functions, we cite for example [1,8]. In this paper, we show that some results of Enkhbat and Ibaraki [1,8] given in a context of convex functions can be carried on to pseudo-convex functions. We will give some necessary and sufficient conditions of optimality in the third section. We will also derive an algorithm to apply this program. We deal with the convergence of the algorithm in the fourth section. Some numerical results from problems that were treated in [1,2,7,8,9,10] are given at the end for illustration. Consider a quadratic and pseudo-convex f : R → R. The problem we are interested in is:


Introduction
Our aim in this paper is the search for the maxima of vectorial pseudo-convex quadratic functions. The motivation behind the choice of these functions is mainly computation. We shall give an algorithm to find such maxima and give some computational results at the end. Many papers were devoted to the numerical search of minima and maxima of convex functions, we cite for example [1,8]. In this paper, we show that some results of Enkhbat and Ibaraki [1,8] given in a context of convex functions can be carried on to pseudo-convex functions.
We will give some necessary and sufficient conditions of optimality in the third section. We will also derive an algorithm to apply this program. We deal with the convergence of the algorithm in the fourth section. Some numerical results from problems that were treated in [1,2,7,8,9,10] are given at the end for illustration.
Consider a quadratic and pseudo-convex f : R n → R. The problem we are interested in is: Recall that a differentiable function f is pseudo-convex (cf. [6]) if: Since f is quadratic, there is a square real matrix Q of order n and x, p ∈ R n such that: The derivative of f at x is ∇f (x) = Qx + p.
In this work, we do not suppose that Q is symmetric, nevertheless we can always transform our problem via a change of basis and a change of variable to the maximization of a function of the form f (x) = f (x 0 ) + 1 2 Dy, y where D is a diagonal matrix and y that will be clarified below.

Properties of the function under study
We begin by the following result.
where D is a diagonal matrix and x 0 is a vector such that ∇f (x 0 ) = 0.
Proof. Claim: Consider two matrices defined as follows: with a ii = q ii and ∀i = j, a ij = 1 2 (q ij + q ji ). Then, A is symmetric and Qx, x = Ax, x . Indeed, by construction of the matrix A, a ij = a ji . So A is symmetric. On the other hand, For the second part, we refer to Best [4] who shows that if f (x) = 1 2 Ax, x + p, x , where A is a symmetric matrix. We can make a change of variable and obtain where D is a diagonal matrix and x 0 is a vector such that ∇f (x 0 ) = 0. Indeed, A is real symmetric, so there is an orthogonal matrix S and a diagonal matrix D such that A = S.D.S T and S ⊤ = S −1 . Also, Ax 0 + p = 0 implies that x 0 is a solution of a linear system. Consider the vector y such that x = S.y + x 0 . Therefore, whatever the real matrix Q that defines the function f (x), we can always find a symmetric matrix D such that That way we have an equivalent formulation of f with terms of the form y 2 i only and no y i .y j .

Characterization of pseudoconvex quadratic functions
Denote by ν(D) the number of negative eigenvalues in D and π(D) the number of positive eigenvalues. Suppose in the sequel that D = (λ i ) i is a diagonal matrix and ν (D) = 1, π (D) = k − 1 and k ≤ n. We sort the eigenvalues of the matrix D so that λ 1 ≤ 0. Denote the sets: According to Greub [3], the following sets represent the solid cones: And according to Ferland,[2], if the real diagonal matrix D satisfies ν(D) = 1, then the quadratic form f (x) = Dx, x is pseudoconvex on either of the sets T + \ N and T − \ N defined by: x ∈ T + and Dx = 0 and T − \ N = x ∈ R n ; x ∈ T − and Dx = 0 If a function f is quadratic pseudoconvex, then we can easily check that: (iii) When f is pseudoconvex, it is quasiconvex.

Optimality Conditions and Algorithm
We define a level set as follows:
For the construction of the algorithm to solve our problem, we define the following functions: Ay, x − y and X(x) = max y∈C Y (y).
Proof. By definition of X(x) and Y (y), ∀y ∈ C x : By Theorem 3, we have that x * is a solution of (P ).

Algorithm
See Figure 1 below for our algorithm to find the maxima of f .

Lemma 3.3.
The sequence (f (x (k) ) k is strictly increasing.
Proof. For the general case, we suppose that x (k) is not a solution of the problem. First, let's prove that the sequence is increasing. Since x (k) is not a solution, X x (k) > 0. Hence So the sequence (f (x (k) ) k is increasing. Let's prove now that the sequence is strictly increasing.
A contradiction.
The sequence (f (x (k) ) k is strictly increasing and bounded from above by f (x * ), where x * is a solution of (P). So for all k = 1, 2, . . ., we have f x (0) < f (x (k) ) ≤ f (x * ), and the function f (x) is bounded. Since it's also continuous, we deduce that: There exists L ∈ R such that f (x (k) ) = L. Proof. We have (k) . which gives us lim k→∞ X(x (k) ) = 0.

Numerical Simulations
Now, that we have an algorithm to find solutions of (P ), we will use it on the following set of problems: The results of the numerical simulations are as follows (n is the number of variables):