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

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 the Slater constraint qualification, are reduced to guaranteeing that there are no dirty variables. This leads to completely new proofs of such sufficient conditions for zero duality.

Recent contributions to linear semi-infinite optimization

“…To the best of our knowledge, the …rst semi-in…nite version of that method, which provides linear representations of the projections of closed convex sets on the coordinate hyperplanes, was introduced in [77] to characterize the socalled Motzkin decomposable sets (i.e., those sets which can be expressed as sums of polyhedral convex sets with closed convex cones, as the optimal set S of P when c 2 rint M ), see also [81] and [77]. The second and third semi-in…nite versions of the Fourier elimination method are due to A. Basu, K. Martin, and C. Ryan ( [12], [13], [14]) and to K. Kortanek and Q. Zhang [127], respectively, these four papers dealing with LSIO duality theory.…”

“…The primal problem P is reformulated in [13] by aggregating an additional variable z representing upper bounds for hc; i to be minimized on the feasible set, contained in R n+1 ; in order to get known and new duality results for the pair P D. This formulation of P ; called standard by the authors, has been reformulated again in [127] as a conic linear program from which the so-called classi…cation duality theorems for semi-in…nite linear programs [125] have been recovered.…”

The Slater Conundrum: Duality and Pricing in Infinite-Dimensional Optimization

“…Following a similar approach, Shapiro [44] considers conditions that imply zero duality gap for the algebraic dual of infinite conic programs and only later introduces topologies. Basu, Martin and Ryan [10] provide an algebraic condition for zero duality gap by extending Fourier-Motzkin elimination to semi-infinite linear programming. Other work involving the order-algebraic approach include Barvinok [8], Nash [34] and Pintér [35].…”

Strong duality and sensitivity analysis in semi-infinite linear programming

“…al. [4] and also state and prove several additional lemmas that elucidate further insights into non-finite-support duals. Here we define the Fourier-Motzkin operator, which plays a key role in our theory.…”

“…Third, the analytical approach to sensitivity analysis in Goberna and López [13] is grounded in convex-analytic methods that focus on topological properties of cones and epigraphs, whereas our approach uses Fourier-Motzkin elimination, an algebraic tool that appeared in the study of semiinfinite linear programming duality in Basu et al [4]. Earlier work by Goberna et al [11] explored extensions of Fourier-Motzkin elimination to semi-infinite linear systems but did not explore its implications for duality.…”