Abstract
Recently Chen and Mangasarian proposed a class of smoothing functions for linear/nonlinear programs and complementarity problems that unifies many previous proposals. Here we study a non-interior continuation method based on these functions in which, like interior path-following methods, the iterates are maintained to lie in a neighborhood of some path and, at each iteration, one or two Newton-type steps are taken and then the smoothing parameter is decreased. We show that the method attains global convergence and linear convergence under conditions similar to those required for other methods. We also show that these conditions are in some sense necessary. By introducing an inexpensive active-set strategy in computing one of the Newton directions, we show that the method attains local superlinear convergence under conditions milder than those for other methods. The proof of this uses a local error bound on the distance from an iterate to a solution in terms of the smoothing parameter.
This research is supported by National Science Foundation Grant CCR-9311621.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Burke, J. and Xu, S., The global linear convergence of a non-interior path-following algorithm for linear complementarity problems, Preprint, Department of Mathematics, University of Washington, Seattle, Washington (December 1996); Math. Oper. Res.,to appear.
Chen, B. and Chen, X., A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints, Report, Department of Management and Systems, Washington State University, Pullman, Washington (March 1997).
Chen, B. and Chen, X., A global and local superlinear continuation-smoothing method for Po+Ro and monotone NCP, Report, Department of Management and Systems, Washington State University, Pullman, Washington (May 1997); SIAM J. Optim.,to appear.
Chen, B. and Harker, P. T., A non-interior-point continuation method for linear complementarity problems, SIAM J. Matrix Anal. Appl., 14 (1993), 1168–1190.
Chen, B. and Harker, P. T., A continuation method for monotone variational inequalities, Math. Programming, 69 (1995), 237–253.
Chen, B. and Harker, P. T., Smooth approximations to nonlinear complementarity problems, SIAM J. Optim., 7 (1997), 403–420.
Chen, B. and Xiu, N., A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing function, Report, Department of Management and Systems, Washington State University, Pullman, Washington (February 1997); SIAM J. Optim.,to appear.
Chen, C. and Mangasarian, O. L., Smoothing methods for convex inequalities and linear complementarity problems, Math. Programming, 71 (1995), 51–69.
Chen, C. and Mangasarian, O. L., A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), 97–138.
Chen, X., Qi, L., and Sun, D., Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comput., 67 (1998), 519–540.
Chen, X. and Ye, Y., On homotopy-smoothing methods for variational inequalities, Report, School of Mathematics, University of New South Wales, Sydney, Australia (December 1996); SIAM J. Control Optim.,to appear.
Cottle, R. W., Giannessi, F., and Lions, J.-L., Eds., Variational Inequalities and Complementarity Problems: Theory and Applications, John Wiley Sons, New York, New York, 1980.
Cottle, R. W., Pang, J.-S., and Stone, R. E., The Linear Complementarity Problem, Academic Press, New York, New York, 1992.
Ferris, M. C. and Pang, J.-S., editors, Complementarity and Variational Problems: State of the Art, SIAM Publishing, Philadelphia, Pennsylvania, 1997.
Gabriel, S. A. and Moré, J. J., Smoothing of mixed complementarity problems, in Complementarity and Variational Problems: State of the Art, edited by M. C. Ferris and J.-S. Pang, SIAM Publishing, Philadelphia, Pennsylvania, 1997, 105–116.
Harker, P., and Pang, J.-S., Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming, 48 (1990), 161–220.
Horn, R. A. and Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge, United Kingdom, 1985.
Hotta, K. and Yoshise, A., Global convergence of a class of non-interiorpoint algorithms using Chen-Harker-Kanzow functions for nonlinear cornplementarity problems, Discussion Paper 708, Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Japan (December 1996).
Jiang, H., Smoothed Fischer-Burmeister equation methods for the complementarity problem, Report, Department of Mathematics, University of Melbourne, Parkville, Australia (June 1997).
Kanzow, C., Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 17 (1996), 851–868.
Kanzow, C., A new approach to continuation methods for complementarity problems with uniform P-functions, Oper. Res. Letters, 20 (1997), 85–92.
Kanzow, C. and Jiang, H., A continuation method for (strongly) monotone variational inequalities, Preprint, Institute of Applied Mathematics, University of Hamburg, Hamburg, Germany (October 1994); Math. Programming,to appear.
Kanzow, C., Yamashita, N., and Fukushima, M., New NCP-functions and their properties, J. Optim. Theory Appl., 94 (1997), 115–135.
Kojima, M., Megiddo, N., and Mizuno, S., A general framework of continuation methods for complementarity problems, Math. Oper. Res., 18 (1993), 945–963.
Kojima, M., Megiddo, N., and Noma T., Homotopy continuation methods for nonlinear complementarity problems, Math. Oper. Res.,16 (1991), 754774.
Kojima, M., Megiddo, N., Noma, T. and Yoshise, A., A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science 538, Springer-Verlag, Berlin, 1991.
Kojima, M., Noma, T., and Yoshise, A., Global convergence in infeasibleinterior-point algorithms, Math. Programming, 65 (1994), 43–72.
Luo, Z.-Q. and Tseng, P., A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, edited by M. C. Ferris and J.-S. Pang, SIAM Publishing, Philadelphia, Pennsylvania, 1997, 204–225.
Moré, J. J. and Rheinboldt, W. C., On P- and S-functions and related classes of n-dimensional nonlinear mappings, Linear Algebra Appl., 6 (1973), 45–68.
Pang, J.-S., Complementarity problems, in Handbook on Global Optimization, edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Norwell, Massachusetts, 1995, 271–338.
Qi, L. and Sun, D., Improving the convergence of non-interior point algorithms for nonlinear complementarity problems, Preprint, School of Mathematics, University of New South Wales, Sydney, Australia (May 1997); Math. Comput.,to appear.
Qi, L., Sun, D. and Zhou, G., A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Preprint, School of Mathematics, University of New South Wales, Sydney, Australia (June 1997).
Smale, S., Algorithms for solving equations, in Proceedings of the International Congress of Mathematicians, edited by A. M. Gleason, American Mathematical Society, Providence, Rhode Island (1987), 172–195.
Tseng, P., Simplified analysis of an O(nL)-iteration infeasible predictor-corrector path following method for monotone LCP, in Recent Trends in Optimization Theory and Applications, edited by R. P. Agarwal, World Scientific, Singapore, 1995, 423–434.
Tseng, P., An infeasible path-following method for monotone complementarity problems, SIAM J. Optim., 7 (1997), 386–402.
Wright, S. J. and Ralph, D., A superlinear infeasible-interior-point algorithm for monotone complementarity problems, Math. Oper. Res., 21 (1996), 815–838.
Xu, S., The global linear convergence an infeasible non-interior path-following algorithm for complementarity problems with uniform P-functions, Preprint, Department of Mathematics, University of Washington, Seattle, Washington (December 1996).
Xu, S., The global linear convergence and complexity of a non-interior path-following algorithm for monotone LCP based on Chen-HarkerKanzow-Smale smoothing functions, Preprint, Department of Mathematics, University of Washington, Seattle, Washington (February 1997).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Tseng, P. (1998). Analysis of a Non-Interior Continuation Method Based on Chen-Mangasarian Smoothing Functions for Complementarity Problems. In: Fukushima, M., Qi, L. (eds) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods. Applied Optimization, vol 22. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6388-1_20
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6388-1_20
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4805-2
Online ISBN: 978-1-4757-6388-1
eBook Packages: Springer Book Archive