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

TL;DR

This paper establishes conditions under which Lagrange multipliers exist for optimization problems in Lebesgue spaces with pointwise bounds and finitely many constraints, even without interior points, and extends to nonlinear constraints.

Contribution

It proves that a Slater point is both sufficient and necessary for Lagrange multipliers in these spaces, broadening classical constraint qualification results.

Findings

Slater point existence guarantees Lagrange multipliers.

Necessary and sufficient conditions are identified for multiplier existence.

Extension to nonlinear constraints is demonstrated.

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.