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Zbl 1010.90085
Ulbrich, Michael
Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. (English)
[J] SIAM J. Optim. 11, No.4, 889-917 (2001). ISSN 1052-6234; ISSN 1095-7189/e

Summary: We develop and analyze a class of trust-region methods for bound-constrained semismooth systems of equations. The algorithm is based on a simply constrained differentiable minimization reformulation. Our global convergence results are developed in a very general setting that allows for nonmonotonicity of the function values at subsequent iterates. We propose a way of computing trial steps by a semismooth Newton-like method that is augmented by a projection onto the feasible set. Under a Dennis-Moré-type condition we prove that close to a regular solution the trust-region algorithm turns into this projected Newton method, which is shown to converge locally $q$-superlinearly or quadratically, respectively, depending on the quality of the approximate subdifferentials used.\par As an important application we discuss how the developed algorithm can be used to solve nonlinear Mixed Complementarity Problems (MCPs). Hereby, the MCP is converted into a bound-constrained semismooth equation by means of an NCP-function. The efficiency of our algorithm is documented by numerical results for a subset of the MCPLIB problem collection.

MSC 2000:: *90C33 Complementarity problems
90C30 Nonlinear programming
49J40 Variational methods including variational inequalities
65H10 Systems of nonlinear equations (numerical methods)
65K05 Mathematical programming (numerical methods)
49M37 Methods of nonlinear programming type

Keywords: semismooth equation; nonmonotone trust region method; nonlinear mixed complementarity problem; nonsmooth Newton method; global convergence; superlinear and quadratic convergence

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