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

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