A biobjective approach to robustness based on location planning

TL;DR

This paper introduces a biobjective approach to robustness in optimization, balancing recovery costs and solution quality, modeled as a location planning problem, with methods for computing efficient solutions and reducing uncertainty sets.

Contribution

It formulates a novel biobjective robustness model linking recovery costs and solution quality as a location problem, with solution methods for linear and quasiconvex cases.

Findings

Weakly Pareto efficient solutions can be computed via cost minimization for fixed worst-case objectives.

Approaches are developed for linear and quasiconvex problems with finite uncertainty sets.

Conditions are identified under which the uncertainty set size can be reduced without affecting solutions.

Abstract

Finding robust solutions of an optimization problem is an important issue in practice, and various concepts on how to define the robustness of a solution have been suggested. The idea of recoverable robustness requires that a solution can be recovered to a feasible one as soon as the realized scenario becomes known. The usual approach in the literature is to minimize the objective function value of the recovered solution in the nominal or in the worst case. As the recovery itself is also costly, there is a trade-off between the recovery costs and the solution value obtained; we study both, the recovery costs and the solution value in the worst case in a biobjective setting. To this end, we assume that the recovery costs can be described by a metric. We demonstrate that this leads to a location planning problem, bringing together two fields of research which have been considered separate so far. We show how weakly Pareto efficient solutions to this biobjective problem can be computed by minimizing the recovery costs for a fixed worst-case objective function value and present approaches for the case of linear and quasiconvex problems for finite uncertainty sets. We furthermore derive cases in which the size of the uncertainty set can be reduced without changing the set of Pareto efficient solutions.