A Survey of Derivative Characterization of Constrained Extrema

Abstract

In this chapter we review the theorems that characterize optimality by way of derivatives. First we formulate a very general optimization problem. Then we present characterization theorems for three types of problems: Finite dimensional, variational and problems in linear topological spaces. In each case we present theorems for equality — inequality constraints. The theorems in each case are: first order necessary conditions, first order sufficient conditions, second order necessary conditions and second order sufficient conditions. The scheme of representation is as follows: Statements of theorems are followed by remarks referring the reader to the earliest, known to us, proofs of the theorems. In some instances, slight generalizations of some theorems appear here for the first time, an indication of necessary modifications to existing proofs are presented. A case is “solved” if proofs for all the four types of characterization theorems exist. The only “unsolved” case is that of problems in linear topological spaces with inequality and with equality — inequality constraints. For this case, we present two conjectures about second order conditions that are analogous to the equality constraint case.