Abstract
In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set.
The extension of Robust Optimization methodology developed in this paper opens up new possibilities to solve efficiently multi-stage finite-horizon uncertain optimization problems, in particular, to analyze and to synthesize linear controllers for discrete time dynamical systems.
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References
Ben-Tal, A., Nemirovski, A.: Stable Truss Topology Design via Semidefinite Programming. SIAM J. Optimization 7, 991–1016 (1997)
Ben-Tal, A., Nemirovski, A.: Robust Convex Optimization. Math. Oper. Res. 23, 769–805 (1998)
Ben-Tal, A., Nemirovski, A.: Robust solutions to uncertain linear programs. OR Letters 25, 1–13 (1999)
Ben-Tal, A., El-Ghaoui, L., Nemirovski, A.: Robust semidefinite programming. In: Semidefinite Programming and Applications, R. Saigal, R., Vandenberghe, L., Wolkowicz, H. (eds), Kluwer Academic Publishers, 2000
Ben-Tal, A., Nemirovski, A.: Robust Optimization –- Methodology and Applications. Math. Program. Series B 92, 453–480 (2002)
Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable Robust Solutions of Uncertain Linear Programs: Math. Program. 99, 351–376 (2004)
Ben-Tal, A., Golany, B., Nemirovski, A., Vial, J.-Ph.: Supplier-Retailer Flexible Commitments Contracts: A Robust Optimization Approach. Accepted to Manufacturing & Service Operations Management
Bertsimas, D., Pachamanova, D., Sim, M.: Robust Linear Optimization under General Norms. Oper. Res. Lett. 32, 510–516 (2004)
Bertsimas, D., Sim, M.: The price of Robustness. Oper. Res. 52, 35–53 (2004)
Bertsimas, D., Sim, M.: Robust Discrete optimization and Network Flows. Math. Program. Series B 98, 49–71 (2003)
Bertsekas, D.P., Nedic, A., Ozdaglar, A.E.: Convex Analysis and Optimization, Athena Scientific, 2003
El-Ghaoui, L., Lebret, H.: Robust solutions to least-square problems with uncertain data matrices. SIAM J. Matrix Anal. Appl. 18, 1035–1064 (1997)
El-Ghaoui, L., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optimization 9, 33–52 (1998)
Grötschel, M., Lovasz, L., Schrijver, A.: The Ellipsoid Method and Combinatorial Optimization, Springer, Heidelberg, 1988
Soyster, A.L.: Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming. Oper. Res., 1973, pp. 1154–1157
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Research was supported by the Binational Science Foundation grant #2002038
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Ben-Tal, A., Boyd, S. & Nemirovski, A. Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems. Math. Program. 107, 63–89 (2006). https://doi.org/10.1007/s10107-005-0679-z
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DOI: https://doi.org/10.1007/s10107-005-0679-z