An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles

TL;DR

This paper introduces an active set algorithm integrated into a branch-and-bound framework to efficiently solve robust combinatorial optimization problems with ellipsoidal uncertainty, outperforming existing solvers.

Contribution

It presents a novel active set method for dual bounds in robust combinatorial optimization, leveraging separation oracles and closed-form solutions for improved efficiency.

Findings

Outperforms Gurobi's mixed-integer SOCP solver on various instances.

Efficiently solves problems with uncertain coefficients in ellipsoidal sets.

Demonstrates effectiveness on shortest path and traveling salesman problems.

Abstract

We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurobi.