Higher-order necessary conditions in optimization theory: A systematic approach

Necessary conditions for constrained optimization problems are stated under weak assumptions. The presence of a generalized critical direction in these conditions is the basis for deriving necessary conditions of arbitrary order for various concrete problems. Two applications are considered. The first concerns first-and second-order necessary conditions in an infinite-dimensional vector space where the cost, equality and inequality functions possess finite-dimensional one-sided differentials. The second application involves first-, second- and third-order necessary conditions for a nonlinear programming problem in a Banach space with Frechet differentiability hypotheses. In both applications normality conditions are not required.