A Systematic Approach to Higher-Order Necessary Conditions in Optimization Theory
Dennis S. Bernstein · SIAM Journal on Control and Optimization · 1984
Necessary conditions for an abstract optimization problem are derived 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 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 Fréchet differentiability hypotheses. In both applications normality conditions are not required. Several well-known results are generalized.