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Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. (English) Zbl 1010.90085

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.

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.

90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
90C30Nonlinear programming
49J40Variational methods including variational inequalities
65H10Systems of nonlinear equations (numerical methods)
65K05Mathematical programming (numerical methods)
49M37Methods of nonlinear programming type in calculus of variations