Journal of Convex Analysis 19 (2012), No. 4, 927--953
Copyright Heldermann Verlag 2012
On the Stability of the Optimal Value and the Optimal Set in Optimization Problems
N. Dinh
Dep. of Mathematics, International University, Vietnam National University, Ho Chi Minh City, Vietnam
ndinh02@yahoo.fr
Miguel A. Goberna
Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain
mgoberna@ua.es
Marco Antonio López
Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain
marco.antonio@ua.es
The paper develops a stability theory for the optimal value and the optimal set mapping of optimization problems posed in a Banach space. The problems considered in this paper have an arbitrary number of inequality constraints involving lower semicontinuous (not necessarily convex) functions and one closed abstract constraint set. The considered perturbations lead to problems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the abstract constraint set can also be perturbed. The spaces of functions involved in the problems (objective and constraints) are equipped with the metric of the uniform convergence on the bounded sets, meanwhile in the space of closed sets we consider, coherently, the Attouch-Wets topology. The paper examines, in a unified way, the lower and upper semicontinuity of the optimal value function, and the closedness, lower and upper semicontinuity (in the sense of Berge) of the optimal set mapping. This paper can be seen as a second part of the stability theory presented in a previous paper of the authors ["On the stability of the feasible set in optimization problems", SIAM J. Optim. 20 (2010) 2254-2280], where we studied the stability of the feasible set mapping (completed here with the analysis of the Lipschitz-like property).
Keywords: Stability, infinite dimensional optimization, optimal value function, optimal set mapping.
MSC: 90C31, 90C48; 90C34, 49K40
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