On Solving Discrete Fractional Programs and Its Applications to Assortment Optimization

Abstract

This work bridges the gap between the discrete and the continuous fractional programming literature by solving a class of 0-1 fractional programs as continuous fractional programs. Specifically, we consider 0-1 linear fractional programs under cardinality-type constraints and provide a continuous reformulation with integral maxima, albeit with a higher number of ratio terms. We, therefore, consider the direct relaxation and use the insights from the reformulation to show that the resultant fractional solution can be rounded off with a parametric guarantee. As applications, we develop a Lagrange relaxation-based upper bound solution for assortment optimization under the mixture-of-multinomial logit model and show that it improves upon the existing discretization-based approach. We then derive, as corollaries, tighter parametric bounds for a class of assortment optimization problems. Additionally, we illustrate that the reformulation can help improve the discrete local search heuristic solution by exploiting the continuous solution space. We substantiate this further numerically and show that the reformulation is quite effective and provides significant performance gains over the current approaches.

Keywords: Discrete Fractional Programming, Continuous Fractional Programming, Assortment Optimization