Min-max-min robust combinatorial optimization

Abstract

In this thesis we introduce a robust optimization approach which is based on a binary min-max-min problem. The so called Min-max-min Robust Optimization extends the classical min-max approach by calculating k different solutions instead of one. Usually in robust optimization we consider problems whose problem parameters can be uncertain. The basic idea is to define an uncertainty set U which contains all relevant problem parameters, called scenarios. The objective is then to calculate a solution which is feasible for every scenario in U and which optimizes the worst objective value over all scenarios in U. As a special case of the K-adaptability approach for robust two-stage problems, the min-max-min robust optimization approach aims to calculate k different solutions for the underlying combinatorial problem, such that, considering the best of these solutions in each scenario, the worst objective value over all scenarios is optimized. This idea can be modeled as a min-max-min problem. In this thesis we analyze the complexity of the afore mentioned problem for convex and for discrete uncertainty sets U. We will show that under further assumptions the problem is as easy as the underlying combinatorial problem for convex uncertainty sets if the number of calculated solutions is greater than the dimension of the problem. Additionally we present a practical exact algorithm to solve the min-max-min problem for any combinatorial problem, given by a deterministic oracle. On the other hand we prove that if we fix the number of solutions k, then the problem is NP-hard even for polyhedral uncertainty sets and the unconstrained binary problem. For the case when the number of calculated solutions is lower or equal to the dimension we present a heuristic algorithm which is based on the exact algorithm above. Both algorithms are tested and analyzed on random instances of the knapsack problem, the vehicle routing problem and the shortest path problem. For discrete uncertainty sets we show that the min-max-min problem is NP-hard for a selection of combinatorial problems. Nevertheless we prove that it can be solved in pseudopolynomial time or admits an FPTAS if the min-max problem can be solved in pseudopolynomial or admits an FPTAS respectively.