Necessary and sufficient KKT optimality conditions in non-convex optimization

Abstract

We study the optimization problems $$\min _\mathbf{x }\left\{ f(\mathbf{x }):g_j(\mathbf{x })\le 0, j=1,\ldots ,m\right\} $$ where Slater’s condition holds without the convexity of the feasible set and of the functions $$f,~ g_j$$ . At a feasible point $$\mathbf{x }$$ under question the functions $$g_j$$ are assumed to satisfy a non-degeneracy assumption, necessary and sufficient KKT optimality conditions are then considered in relation to the convexity of the level sets of f.