Generalizing convexity for second order optimality conditions

Abstract

Usually local or global convexity properties of the Lagrange function are employed in second order conditions for some point \( \overline{x} \) to be a local or global solution for a constrained minimization problem. In this paper we present, in section 1, an appropriate generalization of local and global convexity, which takes into account the structure of the feasible set and thus enables us to narrow the usual gap between necessary and sufficient optimality conditions. In section 2 we deal with quadratic problems for which we specify similar global optimality conditions.