Min–max–min robust combinatorial optimization
- 913 Downloads
The idea of k-adaptability in two-stage robust optimization is to calculate a fixed number k of second-stage policies here-and-now. After the actual scenario is revealed, the best of these policies is selected. This idea leads to a min–max–min problem. In this paper, we consider the case where no first stage variables exist and propose to use this approach to solve combinatorial optimization problems with uncertainty in the objective function. We investigate the complexity of this special case for convex uncertainty sets. We first show that the min–max–min problem is as easy as the underlying certain problem if k is greater than the number of variables and if we can optimize a linear function over the uncertainty set in polynomial time. We also provide an exact and practical oracle-based algorithm to solve the latter problem for any underlying combinatorial problem. On the other hand, we prove that the min–max–min problem is NP-hard for every fixed number k, even when the uncertainty set is a polyhedron, given by an inner description. For the case that k is smaller or equal to the number of variables, we finally propose a fast heuristic algorithm and evaluate its performance.
KeywordsRobust optimization k-Adaptability Complexity
Mathematics Subject Classification90C27 90C57
We would like to thank the authors of  for providing us their instances and computational results for the uncertain shortest path problem.
- 1.Baumann, F., Buchheim, C., Ilyina, A.: Lagrangean decomposition for mean-variance combinatorial optimization. In: Combinatorial Optimization—Third International Symposium, ISCO 2014, Lecture Notes in Computer Science, vol. 8596, pp. 62–74. Springer, Berlin (2014)Google Scholar
- 7.Bertsimas, D., Sim, M.: Robust discrete optimization under ellipsoidal uncertainty sets (2004)Google Scholar
- 9.Buchheim, C., Kurtz, J.: Min-max-min robustness: a new approach to combinatorial optimization under uncertainty based on multiple solutions. In: International Network Optimization Conference—INOC 2015 (to appear)Google Scholar
- 12.Grötschel, M., Lovász, L., Schrijver, A.: Geometric methods in combinatorial optimization. In: Proceedings of Silver Jubilee Conference on Combinatorics, pp. 167–183 (1984)Google Scholar
- 14.Hanasusanto, G.A., Kuhn, D., Wiesemann, W.: K-adaptability in two-stage robust binary programming. Optim. Online (2015)Google Scholar
- 16.Liebchen, C., Lübbecke, M., Möhring, R., Stiller, S.: The concept of recoverable robustness, linear programming recovery, and railway applications. In: Robust and Online Large-scale Optimization, pp. 1–27. Springer, Berlin (2009)Google Scholar
- 17.Nikolova, E.: Approximation algorithms for reliable stochastic combinatorial optimization. In: Proceedings of APPROX ’10, Barcelona, Spain (2010)Google Scholar
- 18.Sim, M.: Robust optimization. Ph.D. thesis, Massachusetts Institute of Technology (2004)Google Scholar