Finite Codimensionality Method in Infinite-dimensional Optimization Problems

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.