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

Fourier-Motzkin elimination is a projection algorithm for solving finite linear programs. Weextend Fourier-Motzkin elimination to semi-infinite linear programs which are linear programswith finitely many variables and infinitely many constraints. Applying projection leads to newcharacterizations of important properties for primal-dual pairs of semi-infinite programs suchas zero duality gap, feasibility, boundedness, and solvability. Extending the Fourier-Motzkinelimination procedure to semi-infinite linear programs yields a new classification of variablesthat is used to determine the existence of duality gaps. In particular, the existence of what theauthors term dirty variables can lead to duality gaps. Our approach has interesting applicationsin finite-dimensional convex optimization. For example, sufficient conditions for a zero dualitygap, such as the Slater constraint qualification, are reduced to guaranteeing that there are nodirty variables. This leads to completely new proofs of such sufficient conditions for zero duality.