Quasiconvex Semidefinite Minimization Problem

We introduce so-called semidefinite quasiconvex minimization problem. We derive new global optimality conditions for the above problem. Based on the global optimality conditions, we construct an algorithm which generates a sequence of local minimizers which converge to a global solution. 1. Introduction Semidefinite linear programming can be regarded as an extension of linear programming and solves the following problem: where is a matrix of variables and . is notation for “ is positive semidefinite”. denotes Frobenius norm and . Semidefinite programming finds many applications in engineering and optimization [1]. Most interior-point methods for linear programming have been generalized to semidefinite convex programming [1–3]. There are many works devoted to the semidefinite convex programming problem but less attention so for has been paid to quasiconvex programming semidefinite quasiconvex minimization problem. The aim of this paper is to develop theory and algorithms for the semidefinite quasiconvex programming. The paper is organized as follows. Section 2 is devoted to formulation of semidefinite quasiconvex programming and its global optimality conditions. In Section 3, we consider an approximation of the level set of the objective function and its properties. 2. Problem Definition and Optimality Conditions Let be matrices in , and define a scalar matrix function as follows: Definition 1. Let be a differentiable function of the matrix . Then Introduce the Frobenius scalar product as follows: If is differentiable, then it can be checked that Definition 2. A set is convex if for all and . Definition 3. The function is said to be quasiconvex on if The well-known property of a convex function [3] can be easily generalized as follows. Lemma 4. A function is quasiconvex if and only if the set is convex for all . Proof Necessity. Suppose that is an arbitrary number and . By the definition of quasiconvexity, we have which means that the set is convex. Sufficiency. Let be a convex set for all . For arbitrary , define . Then and . Consequently, , for any . This completes the proof. Lemma 5. Let be a quasiconvex and differentiable function. Then the inequality for implies that where and denotes the Frobenius scalar product of two matrices. Proof. Since is quasiconvex, for all and such that . By Taylor's formula, there is a neighborhood of the point on which: From the fact that , we obtain which completes the proof. Consider the problem of minimizing a differentiable quasiconvex matrix function subject to constraints where are scalar functions and are positive