A SYSTEMATIC APPROACH TO HIGHER-ORDER NECESSARY CONDITIONS IN OPTIMIZATION THEORY*

Abstract. Necessary conditions for an abstract optimization problem are derived under weak assump-tions. 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 in detail. The first concerns first- and second-order necessary conditions for a constrained optimization problem in an infinite-dimensional vector space where the cost, equality and inequality functions possess differentials of a finite-dimensional one-sided character. The second application concerns first-, second- and third-order necessary conditions for a constrained optimization problem in a Banach space with Fr6chet differentiability hypotheses. In both applications normality conditions are not required. Several well-known results are generalized. Key words, higher-order necessary conditions, normality condition, convex constraint set, mathemati-cal programmin