%0 Journal Article
%T Well-Posedness and Primal-Dual Analysis of Some Convex Separable Optimization Problems
%A Stefan M. Stefanov
%J Advances in Operations Research
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/279030
%X We focus on some convex separable optimization problems, considered by the author in previous papers, for which problems, necessary and sufficient conditions or sufficient conditions have been proved, and convergent algorithms of polynomial computational complexity have been proposed for solving these problems. The concepts of well-posedness of optimization problems in the sense of Tychonov, Hadamard, and in a generalized sense, as well as calmness in the sense of Clarke, are discussed. It is shown that the convex separable optimization problems under consideration are calm in the sense of Clarke. The concept of stability of the set of saddle points of the Lagrangian in the sense of Gol'shtein is also discussed, and it is shown that this set is not stable for the ¡°classical¡± Lagrangian. However, it turns out that despite this instability, due to the specificity of the approach, suggested by the author for solving problems under consideration, it is not necessary to use modified Lagrangians but only the ¡°classical¡± Lagrangians. Also, a primal-dual analysis for problems under consideration in view of methods for solving them is presented. 1. Introduction 1.1. Statement of Problems under Consideration: Preliminary Results In this paper, we study well-posedness and present primal-dual analysis of some convex separable optimization problems, considered by the author in previous papers. For the sake of convenience, in this subsection we recall main results of earlier papers that are used in this study. In paper [1], the following convex separable optimization problem was considered: where are twice differentiable strictly convex functions, and are twice differentiable convex functions, defined on the open convex sets in£¿£¿ , respectively, for every ,£¿£¿ , and . Assumptions for problem are as follows. for all . If for some , then the value is determined a priori.( ) . Otherwise, the constraints (2) and (3) are inconsistent and the feasible set , defined by (2)-(3), is empty. In addition to this assumption, we suppose that in some cases which are specified below.( ) (Slater¡¯s constraint qualification) There exists a point such that . Under these assumptions, the following characterization theorem (necessary and sufficient condition) for problem was proved in [1]. Denote by the values of , for which for the three problems under consideration in this paper, respectively. Theorem 1 (characterization of the optimal solution to problem ). Under the above assumptions, a feasible solution is an optimal solution to problem if and only if there exists a such that The
%U http://www.hindawi.com/journals/aor/2013/279030/