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Lagrange Multipliers in locally convex spaces

Mohammed Bachir, Joel Blot

Abstract

We give a general Lagrange multiplier rule for mathematical programming problems in a Hausdorff locally convex space. We consider infinitely many inequality and equality constraints. Our results gives in particular a generalisation of the result of J. Jahn in \cite{Ja}, replacing Fréchet-differentiability assumptions on the functions by the Gateaux-differentiability. Moreover, the closed convex cone with a nonempty interior in the constraints is replaced by a strictly general class of closed subsets introduced in the paper and called {\it ``admissible sets"}. Examples illustrating our results are given.

Lagrange Multipliers in locally convex spaces

Abstract

We give a general Lagrange multiplier rule for mathematical programming problems in a Hausdorff locally convex space. We consider infinitely many inequality and equality constraints. Our results gives in particular a generalisation of the result of J. Jahn in \cite{Ja}, replacing Fréchet-differentiability assumptions on the functions by the Gateaux-differentiability. Moreover, the closed convex cone with a nonempty interior in the constraints is replaced by a strictly general class of closed subsets introduced in the paper and called {\it ``admissible sets"}. Examples illustrating our results are given.
Paper Structure (11 sections, 16 theorems, 90 equations)

This paper contains 11 sections, 16 theorems, 90 equations.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. The dual convex cone in a Hausdorff locally convex space.
  4. Equi-Gateaux-differentiability and equi-semicontinuity.
  5. Admissible sets and the set of multipliers.
  6. Weak-admissible and admissible sets.
  7. The set of multipliers.
  8. Examples
  9. The main results.
  10. Inequalities constraints in a Hausdorff locally convex space.
  11. Optimization with inequality and equality constraints.

Key Result

Proposition 1

Let $K$ be a closed convex subset of $Y$. Then, we have In particular, $K=Y$ if and only if $\textnormal{bar}(K)=\lbrace 0 \rbrace$.

Theorems & Definitions (47)

  • Proposition 1
  • proof
  • Example 1
  • Proposition 2
  • proof
  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Example 2
  • ...and 37 more