Equivalence of Suboptimality Loss and Prediction Loss in Inverse Problems of Mixed Integer Linear Programming

Abstract

This paper deals with an inverse optimization problem (IOP) for mixed integer linear programming (MILP), which is, for example, to find which aspects are important in shift scheduling. Solutions to the IOP for convex programming exist in known methods. However, in the inverse problem of MILP, there was no efficient way to reduce the prediction loss to zero. To effectively minimize the prediction loss in MILP, this paper attributes it to the problem of minimizing the suboptimality loss with equivalence of suboptimality and prediction loss. In MILP, there exists γ,ε>0 such that for almost every true weight, we estimate the error between the true and learning weights as O(exp(-γk^1/2 +(1/ε²)log k )), where k is a number of updates in projected subgradient method, and we reduce the prediction loss to zero in a finite number of updates. We confirm these in numerical experiments.