Second Order Conditions for Constrained Minima

This paper establishes two sets of "second order" conditions-one which is necessary, the other which is sufficient-in order that a vector x* be a local minimum to the constrained optimization problem: minimize f(x) subject to the constraints $$ g_{i}(x)\geqq 0,i=1,\cdots ,m,\; \rm{and} \; h_{i}(x)=0,j=1,\cdots,p, $$ where the problem functions are twice continuously differentiable. The necessary conditions extend the well-known results, obtained with Lagrange multipliers, which apply to equality constrained optimization problems, and the Kuhn-Tucker conditions, which apply to mixed inequality and equality problems when the problem functions are required only to have continuous first derivatives. The sufficient conditions extend similar conditions which have been developed only for equality constrained problems. Examples of the applications of these sets of conditions are given.