Hindawi Publishing Corporation Journal of Optimization Volume 2013, Article ID 346131, 6 pages http://dx.doi.org/10.1155/2013/346131 Research Article Quasiconvex Semidefinite Minimization Problem R. Enkhbat 1 and T. Bayartugs 2 1 National University of Mongolia, Mongolia 2 Mongolian University of Science and Technology, Mongolia Correspondence should be addressed to R. Enkhbat; renkhbat46@yahoo.com Received 23 May 2013; Accepted 7 November 2013 Academic Editor: Jein-Shan Chen Copyright © 2013 R. Enkhbat and T. Bayartugs. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce so-called semidefnite 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 Semidefnite linear programming can be regarded as an extension of linear programming and solves the following problem: min⟨, ⟩  , ⟨  ,⟩  ⩽  , =1,2,...,,  ≽ 0, (1) where ∈ R × is a matrix of variables and   ∈ R × ,= 1,2,...,. ≽0 is notation for “ is positive semidefnite”. ⟨⋅, ⋅⟩  denotes Frobenius norm and ‖‖  =√ ⟨, ⟩  . Semidefnite programming fnds many applications in engineering and optimization [1]. Most interior-point meth- ods for linear programming have been generalized to semidefnite convex programming [1–3]. Tere are many works devoted to the semidefnite convex programming problem but less attention so for has been paid to quasiconvex programming semidefnite quasiconvex minimization prob- lem. Te aim of this paper is to develop theory and algorithms for the semidefnite quasiconvex programming. Te paper is organized as follows. Section 2 is devoted to formulation of semidefnite quasiconvex programming and its global optimality conditions. In Section 3, we consider an approx- imation of the level set of the objective function and its properties. 2. Problem Definition and Optimality Conditions Let  be matrices in R × , and defne a scalar matrix function as follows: : R × → R. (2) Defnition 1. Let () be a diferentiable function of the matrix . Ten   () = (  ()   ) × . (3) Introduce the Frobenius scalar product as follows: ⟨, ⟩  =  ∑ =1  ∑ =1     , ∀,  ∈ R × . (4) If (⋅) is diferentiable, then it can be checked that  ( + ) −  () = ⟨  (),⟩  +  (‖‖  ). (5) Defnition 2. A set D ⊂ R × is convex if  + (1 − ) ∈ D for all ,  ∈ D and  ∈ [0, 1].