Second Order Conditions for Kuhn-Tucker Sufficient Optimality for Optimization Problems with Linear Matrix Inequality Constraints

Abstract

The Kuhn-Tucker Sufficiency Theorem states that a feasible point that satisfies the Kuhn-Tucker conditions is a global minimizer for a convex programming problem for which a local minimizer is global. In this paper, we present a new second order conditions for Kuhn-Tucker sufficiency for minimizing a smooth function with linear matrix inequality constraints and bounds on the variables. In particular, we provide a necessary and sufficient conditions for a local minimizer to be a global minimizer over a box when the objective function is weighted sum of squares and linear functions. Numerical examples are given to illustrate the significance of sufficiency criteria.