Second Order Sufficient Conditions for Weak and Strict Constrained Minima

Abstract

Sufficient conditions for a weak (nonisolated) local constrained minimum are obtained for the problem: minimize $f( x )$ subject to $g_i \leqq 0,i = 1, \cdots ,m$, and $h_j ( x ) = 0,j = 1, \cdots ,p$, assuming twice differentiability of the problem functions. The key requirement is positive semidefiniteness of the Hessian of the associated Lagrangian in a suitable feasible neighborhood of a candidate point. This is an extension of certain results obtained by McCormick for characterizing a strict minimum.