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 NPhard 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. ZusammenfassungIn dieser Arbeit führen wir einen Ansatz in der robusten Optimierung ein, der auf einem binären Min-max-min Problem basiert. Die sogenannte Minmax-min robuste Optimierung erweitert die klassische robuste Optimierung, indem k verschiedene Lösungen anstatt einer Einzigen berechnet werden.Im Allgemeinen betrachtet man in der robusten Optimierung Probleme, deren Problemparameter unsicher sein können. Die Grundidee ist eine Unsicherheitsmenge U zu definieren, die alle relevanten Problemparameter enthält, die man auch Szenarien nennt. Das Ziel ist es eine Lösung zu berechnen, die für alle Szenarien in U zulässig ist und die den schlechtesten Zielfunktionswert bezüglich aller Szenarien in U optimiert.Als Spezialfall des K-adaptability Ansatzes für zweistufige robuste Proble...

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We consider robust combinatorial optimization problems where the decision maker can react to a scenario by choosing from a finite set of k solutions. This approach is appropriate for decision problems under uncertainty where the implementation of decisions requires preparing the ground. We focus on the case that the set of possible scenarios is described through a budgeted uncertainty set and provide three algorithms for the problem. The first algorithm solves heuristically the dualized problem, a nonconvex mixed-integer non-linear program (MINLP), via an alternating optimization approach. The second algorithm solves the MINLP exactly for k = 2 through a dedicated spatial branch-and-bound algorithm. The third approach enumerates k-tuples, relying on strong bounds to avoid a complete enumeration. We test our methods on shortest path instances that were used in the previous literature and on randomly generated knapsack instances, and find that our methods considerably outperform previous approaches. Many instances that were previously not solved within hours can now be solved within few minutes, often even faster.

In this work we study binary two-stage robust optimization problems with objective uncertainty. We present an algorithm to calculate efficiently lower bounds for the binary two-stage robust problem by solving alternately the underlying deterministic problem and an adversarial problem. For the deterministic problem any oracle can be used which returns an optimal solution for every possible scenario. We show that the latter lower bound can be implemented in a branch and bound procedure, where the branching is performed only over the first-stage decision variables. All results even hold for non-linear objective functions which are concave in the uncertain parameters. As an alternative solution method we apply a column-and-constraint generation algorithm to the binary two-stage robust problem with objective uncertainty. We test both algorithms on benchmark instances of the uncapacitated single-allocation hub-location problem and of the capital budgeting problem. Our results show that the branch and bound procedure outperforms the column-and-constraint generation algorithm.

We investigate a robust approach for solving the capacitated vehicle routing problem (CVRP) with uncertain travel times. It is based on the concept of K-adaptability, which allows one to calculate a set of k feasible solutions in a preprocessing phase before the scenario is revealed. Once a scenario occurs, the corresponding best solution may be picked out of the set of candidates. The aim is to determine the k candidates by hedging against the worst-case scenario, as is common in robust optimization. This idea leads to a min-max-min problem. In this paper, we propose an oracle-based algorithm for solving the resulting min-max-min CVRP, calling an exact algorithm for the deterministic problem in each iteration. Moreover, we adjust this approach such that heuristics for the CVRP can also be used. In this way, we derive a heuristic algorithm for the min-max-min problem, which turns out to yield good solutions in a short running time. All algorithms are tested on standard benchmark instances of the CVRP.