Approaches for Biobjective Integer Linear Robust Optimization

Abstract

Real-world optimization problems often do not just involve multiple objectives but also uncertain parameters. In this case, the goal is to find Pareto-optimal solutions that are robust, i.e., reasonably good under all possible realizations of the uncertain data. Such solutions have been studied in many papers within the last ten years and are called robust efficient. However, solution methods for finding robust efficient solutions are scarce. In this paper, we develop three algorithms for determining robust efficient solutions to biobjective mixed-integer linear robust optimization problems. To this end, we draw from methods for both multiobjective optimization and robust optimization: dichotomic search for biobjective mixed-integer optimization problems and an optimization-pessimization approach from (single-objective) robust optimization, which iteratively adds scenarios and thereby increases the uncertainty set. We propose two algorithms that combine dichotomic search with the optimization-pessimization method as well as a dichotomic search method for biobjective linear robust optimization that exploits duality. On the way we derive some other results: We extend dichotomic search from biobjective linear problems to biobjective linear minmax problems and generalize the optimization-pessimization method from single-objective to multi-objective robust optimization problems. We implemented and tested the three algorithms on linear and integer linear instances and discuss their respective strengths and weaknesses.