Optimality conditions for nonconvex problems over nearly convex feasible sets

Abstract

In this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson’s constraint qualification, which reduces to the combination of generalized Slater’s condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson’s constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P).

Optimality and Duality for Robust Optimization Problems Involving Intersection of Closed Sets

“…For instance, when m := 1, the nonempty subset C 1 is a closed convex set in R n , f is a continuous convex function, and g 1 , ..., g p , are continuously differentiable functions, Chieu et al in [6] examined links among various constraint qualifications including Karush-Kuhn-Tucker conditions for an optimization problem without uncertainties. In the case of m := 1 and the constraints related to a convex cone, Ghafari and Mohebi in [12] provided a new characterization of the Robinson constraint qualification, which collapses to the validation of a generalized Slater constraint qualification and a sharpened nondegeneracy condition for a (no uncertainty) nonconvex optimization problem involving nearly convex feasible sets.…”