Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty

Abstract

We introduce robust weak sharp and robust sharp solution to a convex programming with the objective and constraint functions involved uncertainty. The characterizations of the sets of all the robust weak sharp solutions are obtained by means of subdiferentials of convex functions, DC fuctions, Fermat rule and the robust-type subdifferential constraint qualification, which was introduced in X.K. Sun, Z.Y. Peng and X. Le Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optim Lett. 10. (2016), 1463-1478. In addition, some applications to the multi-objective case are presented.

Optimality conditions for robust weak sharp efficient solutions of nonsmooth uncertain multiobjective optimization problems

“…It is obvious that the functions f i (•, u i ), i = 1, 2 and g(•, v) are not convex. Therefore, [35,Theorem 4.2] is not applicable for this example.…”

“…Very recently, with the intention to answer the question "How about optimality condition for weak sharp solutions, particularly, in a robust optimization? ", Kerdkaew and Wangkeeree [35] introduced robust weak sharp and robust sharp solution to a convex cone-constrained optimization problem with data uncertainty and some optimality conditions for the robust weak sharp solutions problem were established. Moreover, as an application, the authors presented the characterization of the robust weak sharp weakly efficient solution sets for convex uncertain multiobjective optimization problems.…”

“…Motivated by above-mentioned works, especially [34][35][36], we aim to establish necessary and sufficient optimality conditions for the robust weak sharp efficient solutions of an uncertain multiobjective optimization problem with data incertainty in both objective and constraints functions. Our obtained optimality conditions are presented in terms of multipliers and limiting/Mordukhovich subdifferential of the related functions.…”