Global Optimality Conditions for Nonlinear Programming Problems with Linear Equality Constraints

Some necessary global optimality conditions and sufficient global optimality conditions for nonconvex minimization problems with a quadratic inequality constraint and a linear equality constraint are derived. In particular, global optimality conditions for nonconvex minimization over a quadratic inequality constraint which extend some known global optimality conditions in the existing literature are presented. Some numerical examples are also given to illustrate that a global minimizer satisfies the necessary global optimality conditions but a local minimizer which is not global may fail to satisfy them.