Optimality Conditions for Variational Problems in Incomplete Functional Spaces

TL;DR

This paper introduces a new method for deriving necessary optimality conditions in variational problems within incomplete function spaces, utilizing generalized differentiation and constraint qualifications to handle nonconvex problems.

Contribution

It presents a novel approach that simplifies the derivation of optimality conditions in incomplete spaces, extending results to nonconvex variational problems with velocity constraints.

Findings

Derived necessary optimality conditions for nonconvex variational problems

Provided a simplified proof of first-order optimality conditions

Extended results to broad classes of function spaces including C^k

Abstract

This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach involves reducing a variational problem to a (nondynamic) problem of constrained optimization in a normed space and then applying the results recently obtained for the latter class using generalized differentiation. In this way, we derive necessary optimality conditions for nonconvex problems of the calculus of variations with velocity constraints under the weakest metric subregularity-type constraint qualification. The developed approach leads us to a short and simple proof of the First-order necessary optimality conditions for such and related problems in broad spaces of functions, including those of class C^k.