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On Lagrange multipliers of constrained optimization in Hilbert spaces

Zhiyu Tan

Abstract

In this paper we introduce the essential Lagrange multiplier and establish the solid mathematical foundation of constrained optimization in Hilbert spaces with sharp results on the mathematical foundation of quadratic-programming based methods such as the SQP method, the necessary and sufficient conditions for the existence and uniqueness of Lagrange multipliers, the essential difference of the theory of Lagrange multipliers in finite and infinite-dimensional spaces and an essential characterization of the convergence of the classical augmented Lagrangian method. They are achieved by a newly developed decomposition framework for Lagrange multipliers of the Karush-Kuhn-Tucker system of constrained optimization problems in Hilbert spaces, which is totally different from the existing theories based on separation theorems.

On Lagrange multipliers of constrained optimization in Hilbert spaces

Abstract

In this paper we introduce the essential Lagrange multiplier and establish the solid mathematical foundation of constrained optimization in Hilbert spaces with sharp results on the mathematical foundation of quadratic-programming based methods such as the SQP method, the necessary and sufficient conditions for the existence and uniqueness of Lagrange multipliers, the essential difference of the theory of Lagrange multipliers in finite and infinite-dimensional spaces and an essential characterization of the convergence of the classical augmented Lagrangian method. They are achieved by a newly developed decomposition framework for Lagrange multipliers of the Karush-Kuhn-Tucker system of constrained optimization problems in Hilbert spaces, which is totally different from the existing theories based on separation theorems.
Paper Structure (19 sections, 18 theorems, 152 equations)

This paper contains 19 sections, 18 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. A general optimization problem and the surrogate model
  3. Some preliminary results
  4. A first order necessary condition
  5. The surrogate model
  6. Proof of Theorem \ref{['thm:sm_e']}
  7. Nonsmooth cases
  8. The essential Lagrange multiplier
  9. The essential Lagrange multiplier
  10. The proper Lagrange multiplier
  11. The convergence of multipliers of the classical ALM
  12. Revisiting the weak form asymptotic KKT system
  13. Conclusions
  14. The proof of Theorem \ref{['thm:w-akkt']}
  15. The classical augmented Lagrangian method
  16. ...and 4 more sections

Key Result

Theorem 1.1

For a given minimizer $u^*$ of the constrained optimization problem (eq:GO), the surrogate model exists for all $f\in \mathcal{F}(u^*)$ at the minimizer if and only if Guignard's condition (eq:GCQ) holds, where $\mathcal{F}(u^*)$ is the set of all Fréchet differentiable objective functions which hav

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.1
  • Definition 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.1
  • Remark 2.2
  • ...and 34 more