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 nonlinear infinite-dimensional optimization problem with constraints. Different from the classical constraint qualifications in optimization theory, a finite codimensionality condition is used to guarantee the existence of nontrivial Lagrange multipliers. Further, we show that, such a condition is equivalent to a suitable {\it a priori} estimate, which can be easily verified for some interesting examples. As applications, first order necessary conditions for optimal control problems of some deterministic/stochastic control systems are derived in a unified way.