Near-optimal Approaches for Binary-Continuous Sum-of-ratios Optimization

In this paper, we investigate a class of non-convex sum-of-ratios programs relevant to decision-making in key areas such as product assortment and pricing, facility location and cost planning, and security games. These optimization problems, characterized by both continuous and binary decision variables, are highly non-convex and challenging to solve. To the best of our knowledge, no existing methods can efficiently solve these problems to near-optimality with arbitrary precision. To address this challenge, we explore a piecewise linear approximation approach that enables the approximation of complex nonlinear components of the objective function as linear functions. We then demonstrate that the approximated problem can be reformulated as a mixed-integer linear program, a second-order cone program, or a bilinear program, all of which can be solved to optimality using off-the-shelf solvers like CPLEX or GUROBI. Additionally, we provide theoretical bounds on the approximation errors associated with the solutions derived from the approximated problem. We illustrate the applicability of our approach to competitive joint facility location and cost optimization, as well as product assortment and pricing problems. Extensive experiments on instances of varying sizes are conducted to assess the efficiency of our method.