Finite Codimensionality Method in Infinite-dimensional Optimization Problems

TL;DR

This paper introduces a finite codimensionality condition to establish first-order necessary optimality conditions in infinite-dimensional optimization, simplifying verification compared to traditional constraint qualifications.

Contribution

It develops an enhanced Fritz John type condition using finite codimensionality, applicable to both deterministic and stochastic control problems.

Findings

Provides a unified approach for optimal control problems.

Offers a more straightforward verification process.

Extends Fritz John conditions to infinite-dimensional settings.

Abstract

This paper is devoted to establishing an enhanced Fritz John type first-order necessary condition for a general constrained nonlinear infinite-dimensional optimization problem. Unlike traditional constraint qualifications in optimization theory, a condition of finite codimensionality is employed to ensure the existence of nontrivial Lagrange multipliers. As applications, first-order necessary conditions for optimal control problems of some deterministic/stochastic control systems are derived in a unified manner. Compared with the existing constraint qualifications, the finite codimensionality condition, which is equivalent to some suitable {\it a priori} estimates, can offer a more straightforward verification process in these applications.