Robust optimization with incremental recourse

TL;DR

This paper introduces a robust incremental optimization framework allowing decision adjustments after observing uncertain costs, demonstrating polynomial solvability for linear programs with polyhedral uncertainty and NP-hardness in discrete cases.

Contribution

It develops a new robust incremental optimization model, analyzes its computational complexity, and provides polynomial-time solutions for certain cases while proving NP-hardness for others.

Findings

Robust incremental linear programs with polyhedral uncertainty are solvable in polynomial time.

NP-hardness is established for robust incremental linear programming with discrete uncertainty.

The robust incremental shortest path problem is NP-complete even with limited modifications.

Abstract

In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem.