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Combinatorial Structures in Nonlinear Programming

Stefan Scholtes (s.scholtes***at***jims.cam.ac.uk)

Abstract: Non-smoothness and non-convexity in optimization problems often arise because a combinatorial structure is imposed on smooth or convex data. The combinatorial aspect can be explicit, e.g. through the use of ”max”, ”min”, or ”if” statements in a model, or implicit as in the case of bilevel optimization where the combinatorial structure arises from the possible choices of active constraints in the lower level problem. In analyzing such problems, it is desirable to decouple the combinatorial from the nonlinear aspect and deal with them separately. This paper suggests a problem formulation which explicitly decouples the two aspects. We show that such combinatorial nonlinear programs, despite their inherent non-convexity, allow for a convex first order local optimality condition which is generic and tight. The stationarity condition can be phrased in terms of Lagrange multipliers which allows an extension of the popular sequential quadratic programming (SQP) approach to solve these problems. We show that the favorable local convergence properties of SQP are retained in this setting. The computational effectiveness of the method depends on our ability to solve the subproblems efficiently which, in turn, depends on the representation of the governing combinatorial structure. We illustrate the potential of the approach by applying it to optimization problems with max-min constraints which arise, for example, in robust optimization.

Keywords: Nonlinear programming, piecewise smoothness, combinatorial structures, robust stochastic programming

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 3: Robust Optimization

Citation: Judge Institute of Management Research Papers Series No. WP 6/2002

Download: [Postscript][PDF]

Entry Submitted: 05/10/2002
Entry Accepted: 05/10/2002
Entry Last Modified: 05/18/2002

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