Nonmonotone Path Following Methods for Nonsmooth Equations and Complementarity Problems
We present a homotopy path following method for nonsmooth equations, which is effective at solving highly nonlinear equations for which the norm of the residual has nonglobal local minima. The method is based on a class of homotopy mappings that are smooth in the interior of their domain. Sufficient conditions are presented that guarantee the existence of well-behaved zero curves of these homotopy mappings, which can be followed to a solution. These zero curves need not be monotonic in the homotopy parameter. The method is specialized to solve complementarity problems through the use of MCP functions and associated smoothers.
KeywordsNonsmooth equations complementarity problems homotopy methods smoothing path following
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- J. C. Alexander. The topological theory of an embedding method. In H. Wacker, editor, Continuation Methods, pages 37–68. Academic Press, New York, 1978.Google Scholar
- J. C. Alexander, R. B. Kellogg, T.-Y. Li, and J. A. Yorke. Piecewise smooth continuation, manuscript, 1979.Google Scholar
- S. C. Billups. A homotopy based algorithm for mixed complementarity problems. UCD/CCM Report No. 124, Department of Mathematics, University of Colorado at Denver, Denver, Colorado, 1998.Google Scholar
- M. C. Ferris and T. F. Rutherford. Accessing realistic complementarity problems within Matlab. In G. Di Pillo and F. Giannessi, editors, Nonlinear Optimization and Applications, pages 141–153. Plenum Press, New York, 1996.Google Scholar
- S. A. Gabriel and J. J. More. Smoothing of mixed complementarity problems. In M. C. Ferris and J. S. Pang, editors, Complementarity and Variational Problems: State of the Art, Philadelphia, Pennsylvania, 1997. SIAM Publications.Google Scholar
- T. S. Munson. Private communication, 1999.Google Scholar
- H. Sellami. A homotopy continuation method for normal maps. Mathematical Programming, pages 317–337, 1998.Google Scholar
- H. Sellami and S. M. Robinson. Homotopies based on nonsmooth equations for solving nonlinear variational inequalities. In G. Di Pillo and F. Giannessi, editors, Nonlinear Optimization and Applications, pages 327–343. Plenum Press, New York, 1996.Google Scholar
- H. Sellami and S. M. Robinson. Implementation of a continuation method for normal maps. Mathematical Programming, pages 563–578, 1997.Google Scholar