Inverse Optimization in Countably Infinite Linear Programs

Abstract

Given the objective coefficients and a feasible solution for a linear program, inverse optimization involves finding a new vector of objective coefficients that (i) is as close as possible to the original vector; and (ii) would make the given feasible solution optimal. This problem is well-studied for finite-dimensional linear programs. We develop a duality-based inverse optimization framework for countably infinite linear programs (CILPs) -- problems that include a countably infinite number of variables and constraints. Using the standard weighted absolute sum metric to quantify distance between cost vectors, we provide conditions under which constraints in the inverse optimization problem can be reformulated as a countably infinite set of linear constraints. We propose a convergent algorithm to solve the resulting infinite-dimensional mathematical program. This algorithm involves solving a sequence of finite-dimensional linear programs. We apply these results to inverse optimization in infinite-horizon non-stationary Markov decision processes.