Robust solutions of uncertain mixed-integer linear programs using decomposition techniques

TL;DR

This paper introduces a decomposition-based approach to solve robust mixed-integer linear programs with uncertain data, improving scalability and providing explicit constraint violation probabilities.

Contribution

It offers a novel interpretation of the robust counterpart using decomposition techniques, enabling efficient solutions and probabilistic insights for large-scale problems.

Findings

Decomposes second-order cone problems into linear and quadratic subproblems.

Improves tractability for large-scale robust MILPs.

Provides exact probability of constraint violation with known distribution moments.

Abstract

Robust optimization is a framework for modeling optimization problems involving data uncertainty and during the last decades has been an area of active research. If we focus on linear programming (LP) problems with i) uncertain data, ii) binary decisions and iii) hard constraints within an ellipsoidal uncertainty set, this paper provides a different interpretation of their robust counterpart (RC) inspired from decomposition techniques. This new interpretation allows the proposal of an ad-hoc decomposition technique to solve the RC problem with the following advantages: i) it improves tractability, specially for large-scale problems, and ii) it provides the exact probability of constraint violation in case the probability distribution of uncertain parameters are completely defined by using first and second-order probability moments. An attractive aspect of our method is that it decomposes the second-order cone programming problem, associated with the robust counterpart, into a linear master problem and different quadratically constrained problems (QCP) of considerable lower size. The optimal solution is achieved through the solution of these master and subproblems within an iterative scheme based on cutting plane approximations of the second-order cone constraints. In addition, proof of convergence of the iterative method is given.