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Finite Codimensionality Method in Infinite-dimensional Optimization Problems

Xu Liu, Qi Lü, Haisen Zhang, Xu Zhang

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.

Finite Codimensionality Method in Infinite-dimensional Optimization Problems

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.

Paper Structure

This paper contains 15 sections, 13 theorems, 322 equations.

Table of Contents

  1. Introduction
  2. First-order necessary condition in optimization problems
  3. Some preliminaries
  4. First-order necessary conditions for the problem (P)
  5. Application to an optimal control problem in infinite dimension
  6. Verification of finite codimensionality condition in optimal control problems
  7. An optimal control problem for an evolution equation with end-point state constraint
  8. An optimal control problem for an elliptic equation with pointwise state constraint
  9. An optimal control problem for stochastic differential equations with endpoint state constraint
  10. Formulation of the problem and a necessary condition for optimal controls
  11. The finite codimensionality of $\hbox{Var}_{L^2_\mathbb{F}(0, T; \mathbb{R}^m)}f(\bar{u})$
  12. Further comments
  13. Proofs of (\ref{['appendix']}) and (\ref{['appendix_add']})
  14. Proof of Proposition \ref{['add**']}
  15. Proof of Lemma \ref{['codim converg']}

Key Result

Lemma 2.1

2 For any $x\in X$, the set $\partial \hbox{\rm dist}( x,E)$ is a nonempty convex weak$^*$-compact subset of $X^{\prime}$ and $\partial \hbox{\rm dist}( x,E)\subseteq B_{X^{\prime}}(0,1)$. Moreover, $\hbox{\rm dist}( \cdot,E)$ is directional differentiable, and for any $x,v\in X$, where $\hbox{\rm dist}^{\prime}(x, E; v)$ is the directional derivative of $\hbox{\rm dist}( \cdot,E)$ at $x$ along t

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Definition 2.4
  • Example 2.1
  • Definition 2.5
  • Remark 2.1
  • Definition 2.6
  • Definition 2.7
  • ...and 25 more