Faster Algorithms for Min-max-min Robustness for Combinatorial Problems with Budgeted Uncertainty

TL;DR

This paper introduces three new algorithms for solving robust combinatorial optimization problems with budgeted uncertainty, significantly improving solution times for complex instances compared to previous methods.

Contribution

The paper develops three algorithms for min-max-min robustness problems under budgeted uncertainty, including a heuristic, an exact method for two solutions, and a tuple enumeration approach.

Findings

Algorithms outperform previous methods on shortest path and knapsack instances.

Many previously unsolvable instances are now solved within minutes.

The approaches significantly reduce computational time for complex robust optimization problems.

Abstract

We consider robust combinatorial optimization problems where the decision maker can react to a scenario by choosing from a finite set of $k$ solutions. This approach is appropriate for decision problems under uncertainty where the implementation of decisions requires preparing the ground. We focus on the case that the set of possible scenarios is described through a budgeted uncertainty set and provide three algorithms for the problem. The first algorithm solves heuristically the dualized problem, a non-convex mixed-integer non-linear program (MINLP), via an alternating optimization approach. The second algorithm solves the MINLP exactly for $k=2$ through a dedicated spatial branch-and-bound algorithm. The third approach enumerates $k$-tuples, relying on strong bounds to avoid a complete enumeration. We test our methods on shortest path instances that were used in the previous literature and on randomly generated knapsack instances, and find that our methods considerably outperform previous approaches. Many instances that were previously not solved within hours can now be solved within few minutes, often even faster.