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Slater conditions without interior points for programs in Lebesgue spaces with pointwise bounds and finitely many constraints

Gerd Wachsmuth

Abstract

We consider optimization problems in Lebesgue spaces with pointwise box constraints and finitely many additional linear constraints. We prove that the existence of a Slater point which lies strictly between the pointwise bounds and which satisfies the linear constraints is sufficient for the existence of Lagrange multipliers. Surprisingly, the Slater point is also necessary for the existence of Lagrange multipliers in a certain sense. We also demonstrate how to handle additional finitely many nonlinear constraints.

Slater conditions without interior points for programs in Lebesgue spaces with pointwise bounds and finitely many constraints

Abstract

We consider optimization problems in Lebesgue spaces with pointwise box constraints and finitely many additional linear constraints. We prove that the existence of a Slater point which lies strictly between the pointwise bounds and which satisfies the linear constraints is sufficient for the existence of Lagrange multipliers. Surprisingly, the Slater point is also necessary for the existence of Lagrange multipliers in a certain sense. We also demonstrate how to handle additional finitely many nonlinear constraints.
Paper Structure (8 sections, 13 theorems, 74 equations)

This paper contains 8 sections, 13 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tangent cones and normal cones
  4. Polyhedric sets
  5. Linear Constraints
  6. Necessity of Slater points
  7. Nonlinear Constraints
  8. A counterexample

Key Result

Lemma 4

Let measurable functions $x_a, x_b \colon \Omega \to [-\infty,\infty]$ be given such that is nonempty and we fix $x \in K$. The estimates hold for all $\varepsilon > 0$. For $p \in [1,\infty)$, we even have

Theorems & Definitions (34)

  • Definition 3
  • Lemma 4
  • proof
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 24 more