Strong duality and sensitivity analysis in semi-infinite linear programming

Finite-dimensional linear programs satisfy strong duality (SD) and have the “dual 8 pricing” (DP) property. The (DP) property ensures that, given a sufficiently small perturbation of 9 the right-hand-side vector, there exists a dual solution that correctly “prices” the perturbation by 10 computing the exact change in the optimal objective function value. These properties may fail in 11 semi-infinite linear programming where the constraint vector space is infinite dimensional. Unlike 12 the finite-dimensional case, in semi-infinite linear programs the constraint vector space is a modeling 13 choice. We show that, for a sufficiently restricted vector space, both (SD) and (DP) always hold, 14 at the cost of restricting the perturbations to that space. The main goal of the paper is to extend 15 this restricted space to the largest possible constraint space where (SD) and (DP) hold. Once (SD) 16 or (DP) fail for a given constraint space, then these conditions fail for all larger constraint spaces. 17 We give sufficient conditions for when (SD) and (DP) hold in an extended constraint space. Our 18 results require the use of linear functionals that are singular or purely finitely additive and thus 19 not representable as finite support vectors. The key to understanding these linear functionals is 20 the extension of the Fourier-Motzkin elimination procedure to semi-infinite linear programs. 21