An oracle-based framework for robust combinatorial optimization

TL;DR

This paper introduces an oracle-based framework for solving min-max robust combinatorial optimization problems with uncertain linear objectives, leveraging convex relaxations and branch-and-bound methods to improve solution bounds and computational efficiency.

Contribution

The paper presents a novel oracle-based approach that efficiently solves robust combinatorial problems under uncertainty, adaptable to various uncertainty sets and capable of handling complex problems.

Findings

Outperforms linearization methods on robust minimum spanning tree problems.

Provides better dual bounds for robust traveling salesman problems.

Efficiently solves problems using existing solvers as oracles.

Abstract

We propose a general solution approach for min-max-robust counterparts of combinatorial optimization problems with uncertain linear objectives. We focus on the discrete scenario case, but our approach can be extended to other types of uncertainty sets such as polytopes or ellipsoids. Concerning the underlying certain problem, the algorithm is entirely oracle-based, i.e., our approach only requires a (primal) algorithm for solving the certain problem. It is thus particularly useful in case the underlying problem is hard to solve, or only defined implicitly by a given software addressing the certain case. The idea of our algorithm is to solve the convex relaxation of the robust problem by a simplicial decomposition approach, the main challenge being the non-differentiability of the objective function in the case of discrete or polytopal uncertainty. The resulting dual bounds are then used within a tailored branch-and-bound framework for solving the robust problem to optimality. By a computational evaluation, we show that our method outperforms straightforward linearization approaches on the robust minimum spanning tree problem. Moreover, using the Concorde solver for the certain oracle, our approach computes much better dual bounds for the robust traveling salesman problem in the same amount of time.