Kuhn–Tucker sufficiency for global minimum of multi-extremal mathematical programming problems

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 new Kuhn–Tucker sufficiency conditions for possibly multi-extremal nonconvex mathematical programming problems which may have many local minimizers that are not global. We derive the sufficiency conditions by first constructing weighted sum of square underestimators of the objective function and then by characterizing the global optimality of the underestimators. As a consequence, we derive easily verifiable Kuhn–Tucker sufficient conditions for general quadratic programming problems with equality and inequality constraints. Numerical examples are given to illustrate the significance of our criteria for multi-extremal problems.