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Nonlinear robust optimization via sequential convex bilevel programming

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Abstract

In this paper, we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min–max problems which arise in the context of semi-infinite programming. Here, our main motivation are nonlinear inequality constrained robust optimization problems. In the first part of the paper, we propose a conservative approximation strategy for such nonlinear and non-convex robust optimization problems: under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given, we show how to formulate a lower-level concave min–max problem which approximates the robust counterpart in a conservative way. This approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties. In the second part of the paper, we review existing theory on optimality conditions for nonlinear lower-level concave min–max problems which arise in the context of semi-infinite programming. Regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem, we end up with a structured mathematical program with complementarity constraints (MPCC). The special hierarchical structure of this MPCC can be exploited in a novel sequential convex bilevel programming algorithm. We discuss the surprisingly strong global and locally quadratic convergence properties of this method, which can in this form neither be obtained with existing SQP methods nor with interior point relaxation techniques for general MPCCs. Finally, we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example.

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Acknowledgments

The authors would like to thank Quoc Tran Dinh for fruitful discussions on sequential convex programming [37]. Additionally, we thank two anonymous referees and the associate editor for substantial remarks which significantly improved this paper. This research was supported by the Research Council KUL via the Center of Excellence on Optimization in Engineering EF/05/006 (OPTEC, http://www.kuleuven.be/optec/), GOA AMBioRICS, IOF-SCORES4CHEM and PhD/postdoc/fellow grants, the Flemish Government via FWO (PhD/postdoc grants, projects G.0452.04, G.0499.04, G.0211.05,G.0226.06, G.0321.06, G.0302.07, G.0320.08, G.0558.08, G.0557.08, research communities ICCoS, ANMMM, MLDM) and via IWT (PhD Grants, McKnow-E, Eureka-Flite+), Helmholtz Gemeinschaft via vICeRP, the EU via ERNSI, Contract Research AMINAL, as well as the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, Dynamical systems, control and optimization, 2007–2011).

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  1. Optimization in Engineering Center (OPTEC), K. U. Leuven, Kasteelpark Arenberg 10, 3001 , Leuven-Heverlee, Belgium

    Boris Houska & Moritz Diehl

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Houska, B., Diehl, M. Nonlinear robust optimization via sequential convex bilevel programming. Math. Program. 142, 539–577 (2013). https://doi.org/10.1007/s10107-012-0591-2

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