Duality and Pricing in Multiple Right-Hand Choice Linear Programming Problems

Abstract

The main problem studied in this paper is a linear programming problem with multiple right-hand choice of the form Min{cx ∣ x ≥ 0, Ax ≥ b for some b ∈ B}, where B is a finite subset of Rm. The dual to the above problem is introduced to be of the form Max{Minb∈B π(b) ∣ π(A) ≤ c, π ∈ p}, where P is the collection of all subadditive, positively homogeneous and monotone functions from Rm to the extended real line. We prove that the above pair of problems satisfy both weak and strong duality, and provide necessary and sufficient conditions for strong duality to hold when the dual functions π are restricted to be linear. It is shown that the class of functions P arises naturally even in linear programming, i.e., when |B| = 1. Indeed, linear functions will not suffice for pricing functions in a degenerate linear programming problem. In fact, in this case the pricing function p is piecewise linear, superadditive and positively homogeneous and thus concave, i.e., −p ∈ P. On the other hand, we show that the pricing function of the multiple right-hand choice linear program is not necessarily concave. This indicates the difficulties of pricing in integer programming. Finally, a more symmetric pair of dual problems is introduced, and duality relationship between the two problems is studied.