Quasiconvex Semidefinite Minimization Problem

⟨⋅, ⋅⟩ F denotes Frobenius norm and ‖X‖ F = √⟨A,A⟩ F . 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 problembut 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.

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][2][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.

Problem Definition and Optimality Conditions
Let  be matrices in R × , 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 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  ∈ .
Denote by D a constraint set of the problem as follows: Then problems (12)-( 14) reduce to min In general, the set D is nonconvex.Problem ( 16) is nonconvex and belongs to a class of global optimization problems in Banach space.We formulate a new global optimality condition for problem (16) in the following.For this purpose,we introduce the level set  () () of the function  : R × → R at a point  ∈ R × : Then global optimality conditions for problem (16) can be formulated as follows.
Theorem 6.Let  be a solution of problem (16).Then where holds for all  ∈  and  ≥ 0, then condition (18) becomes sufficient.
Example 7. Consider the following problem: where  1 is convex and differentiable on  × and  2 is concave and differentiable on  × .Suppose that  1 and  2 are defined positively on a ball  containing a subset  ⊂  × ; that is, We will call this problem as the mixed fractional minimization problem.By Lemma 4, we can easily show that () is quasiconvex.Hence, the optimality condition (13) at a solution  of ( 27) is as follows:

An Algorithm for the Convex Minimization Problem
We consider the quasiconvex minimization problem as a special case of problem ( 16): where  :  × →  is strongly convex and continuously differentiable and  is an arbitrary compact set in  × .In this case, then we can weaken condition (19) as shown in the next theorem.
Clearly,   () ̸ = 0 by assumption (32).Now define   as follows for  > 0: Then, by the convexity of , we have that is, Thus we get Define a function ℎ :  + →  as where Note that ℎ() ⩾ 0 and ℎ(0) < 0. There are two cases with respect to the values of ℎ() which we should consider.
Proof.This is an obvious consequence of the following relations: which are fulfilled for all  ∈  () () and  ∈ .
Now we are ready to present an algorithm for solving problem (30).We also suppose that one can efficiently solve the problem of computing min ∈ ⟨  (),  − ⟩  for any given  ∈  × .

Algorithm MIN
Input.A strongly quasiconvex function  and a compact set .
Output.A solution  to the minimization problem (30).
Step  If there exists a  such that (  ) = 0 then, by Theorem 11,   is a solution to problem (30) and in this case the proof is complete.Therefore, without loss of generality, we can assume that ignored (  ) < 0 for all  and prove the theorem by contradiction.If the assertion is false; that is,   is not a minimizing sequence for problem (30), the following inequality holds: lim  → ∞ inf  (  ) >  * . (49) subject to D = { ∈ R × |  =