NDLI: Nonlinear robust optimization via sequential convex bilevel programming

Content Provider	Springer Nature Link
Author	Houska, Boris Diehl, Moritz
Copyright Year	2012
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.
Ending Page	577
Page Count	39
Starting Page	539
File Format	PDF
ISSN	00255610
e-ISSN	14364646
Journal	Mathematical Programming
Issue Number	1-2
Volume Number	142
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2012-09-15
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematical programming with complementarity constraints Semi-infinite optimization Theoretical, Mathematical and Computational Physics Semi-infinite programming Bilevel optimization Mathematical Methods in Physics Robust optimization Calculus of Variations and Optimal Control; Optimization Mathematics of Computing Numerical Analysis Methods of successive quadratic programming type Sequential convex programming Combinatorics Minimax problems
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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