Kipp Martin

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

Duality theory is pervasive in finite-dimensional optimization. There is growing interest in solving infinite-dimensional optimization problems and hence a corresponding interest in duality theory in infinite dimensions. Unfortunately, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. In finite dimensions, a dual solution is represented by a vector of “dual prices” that index the primal constraints and have a natural economic interpretation. In infinite dimensions, we show that this simple dual structure, and its associated economic interpretation, may fail to hold for a broad class of problems with constraint vector spaces that are Riesz spaces (ordered vector spaces with a lattice structure) that either are

Strong duality and sensitivity analysis in semi-infinite linear programming

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The demand weighted vehicle routing problem

-order complete or satisfy the projection property. In these spaces we show that the existence of interior points required by common constraint qualifications for zero duality gap (such as Slaters condition) implies the existence of ...

Optimizing multinomial logit profit functions

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

A Modeling System for Mixed Integer Linear Programming Using XML Technologies

In this paper, we present an exact method for the demand weighted vehicle routing problem. This problem arises when the objective is to minimize the distance traveled by the vehicles weighted by the number of passengers on the routes. The resulting model is a non-convex, mixed-integer quadratic program and we show how to reformulate this problem as a linear integer program with auxiliary variables. The reformulation provides very tight lower bounds on the optimal solution value and little enumeration is required. However, the reformulated model is very large and therefore, it is solved with a branch, price and cut algorithm. Using COIN-OR software, we provide computational results with actual data from a large urban university.