Second-Order Necessary Conditions for Nonlinear Optimization Problems with Abstract Constraints: The Degenerate Case
Helmut Gfrerer · SIAM Journal on Optimization · 2007
In this paper we derive second-order necessary conditions for optimality for an optimization problem with abstract constraints in Banach spaces. Results for the nondegenerate case derived earlier [H. Gfrerer, SIAM J. Control Optim., 45 (2006), pp. 972–997] are extended to the degenerate case. For the mathematical programming problem, where the constraints are given by equality and finitely many inequality constraints, our approach applies to the degenerate case, when the equality constraints are not regular; our results appear to be new even in this special case. Our second-order necessary conditions are contained in the gap between the standard necessary and sufficient conditions, where the only difference is the change from a nonstrict to a strict inequality. Our results are formulated in such a way to be applicable also to vector optimization problems.