Projection: A Unified Approach to Semi-Infinite Linear Programs and Duality in Convex Programming

TL;DR

This paper extends Fourier-Motzkin elimination to semi-infinite linear programs, providing new characterizations of duality properties and conditions for zero duality gap, with applications in convex optimization.

Contribution

It introduces a unified projection approach for semi-infinite linear programs, revealing new classifications of variables and conditions for duality gaps and solvability.

Findings

Extended Fourier-Motzkin elimination to semi-infinite LPs

Characterized duality gap conditions using dirty variables

Provided new proofs for zero duality conditions in convex optimization

Abstract

Fourier-Motzkin elimination is a projection algorithm for solving finite linear programs. We extend Fourier-Motzkin elimination to semi-infinite linear programs which are linear programs with finitely many variables and infinitely many constraints. Applying projection leads to new characterizations of important properties for primal-dual pairs of semi-infinite programs such as zero duality gap, feasibility, boundedness, and solvability. Extending the Fourier-Motzkin elimination procedure to semi-infinite linear programs yields a new classification of variables that is used to determine the existence of duality gaps. In particular, the existence of what the authors term dirty variables can lead to duality gaps. Our approach has interesting applications in finite-dimensional convex optimization. For example, sufficient conditions for a zero duality gap, such as existence of a Slater point, are reduced to guaranteeing that there are no dirty variables. This leads to completely new proofs of such sufficient conditions for zero duality.