Constraint Qualifications for Uncertain Convex Optimization without Convexity of Constraint Data Uncertainty

Nithirat Sisarat, Rabian Wangkeeree

Abstract

Considering an uncertain convex optimization problem, within the present paper we study constraint qualifications as well as necessary and sufficient optimality conditions. Following the robust optimization approach, we present constraint qualifications for Lagrange multiplier characterizations of the robust constrained convex optimization with a robust convex feasible set described by lo- cally Lipschitz constraints which are satisfied for all possible uncertainties within the prescribed uncertainty sets and establish relations among various known con- straint qualifications. A new constraint qualification are described, and it is shown that this constraint qualification is the weakest constraint qualification for guar- anteeing the Lagrange multiplier conditions to be necessary for optimality of the robust constrained convex optimization problem. Consequently, we present how the robust best approximation that is immunized against data uncertainty can be obtained by characterizing the best approximation to any x from the robust coun- terpart of the robust feasible set without convexity of constraint data uncertainty, improving the corresponding results in the literature