Characterizing the solution set of convex optimization problems without convexity of constraints

Abstract

Some characterizations of solution sets of a convex optimization problem with a convex feasible set described by tangentially convex constraints are given. The results are expressed in terms of convex subdifferentials, tangential subdifferentials, and Lagrange multipliers. In order to characterize the solution set, we first introduce the so-called pseudo Lagrangian-type function and establish a constant pseudo Lagrangian-type property for the solution set. This property is still valid in the case of a pseudoconvex locally Lipschitz objective function, and then used to derive Lagrange multiplier-based characterizations of the solution set. Some examples are given to illustrate the significances of our theoretical results.