Two-Stage Robust Optimization Problems with Two-Stage Uncertainty

TL;DR

This paper extends two-stage robust optimization to include an additional adversary stage, analyzing complexity and solution methods for problems with budgeted uncertainty sets in multi-stage decision-making.

Contribution

It introduces a novel three-stage min-max-min-max framework and analyzes its computational complexity for continuous and discrete uncertainty sets.

Findings

Polynomial-time solutions for certain continuous cases

NP-hardness results for discrete cases

Special polynomial-time solvable case when adversarial costs are equal

Abstract

We consider two-stage robust optimization problems, which can be seen as games between a decision maker and an adversary. After the decision maker fixes part of the solution, the adversary chooses a scenario from a specified uncertainty set. Afterwards, the decision maker can react to this scenario by completing the partial first-stage solution to a full solution. We extend this classic setting by adding another adversary stage after the second decision-maker stage, which results in min-max-min-max problems, thus pushing two-stage settings further towards more general multi-stage problems. We focus on budgeted uncertainty sets and consider both the continuous and discrete case. For the former, we show that a wide range of robust combinatorial optimization problems can be decomposed into polynomially many subproblems, which can be solved in polynomial time for example in the case of (\textsc{representative}) \textsc{selection}. For the latter, we prove NP-hardness for a wide range of problems, but note that the special case where first- and second-stage adversarial costs are equal can remain solvable in polynomial time.