Abstract
In this paper, we develop and analyze the convergence of a fairly general trust region method for solving a system of nonsmooth equations subject to some linear constraints. The method is based on the existence of an iteration function for the nonsmooth equations and involves the solution of a sequence of sub- problems defined by this function. A particular realization of the method leads to an arbitrary-norm trust region method. Applications of the latter method to the nonlinear complementarity and related problems are discussed. Sequential convergence of the method and its rate of convergence are established under certain regularity conditions similar to those used in the NE/SQP method [14] and its generalization [16]. Some computational results are reported.
This work was based on research supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739, and by the Office of Naval Research under project 4116687-01
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Aubin J. P. and Frankowska H. (1990), Set-Valued Analysis, Birkauser Boston.
Capuzzo Dolcetta I. and Mosco U. (1980), “Implicit Complementarity Problems and Quasi-Variational Inequalities,” in R. W. Cottle, F. Giannessi, and J. L. Lions, eds., Variational Inequality and Complementarity Problems, John Wiley (amp) Sons, Chichester, 75–87.
Dennis, J. E., Jr. and Schnabel, R. B. (1983), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs.
El Hallabi, M. and Tapia, R. A. (1987), “A Global Convergence Theory for Arbitrary Norm Trust Region Methods for Nonlinear Equations,” Mathematical Sciences Technical Report TR87-25, Rice University, Houston.
Ferris, M. C. and Lucidi, S. (1991), “Globally Convergent Methods for Nonlinear Equations,” Manuscript, Computer Science Department, University of Wisconsin, Madison.
Fletcher, R. (1987), Practical Methods of Optimization, John Wiley, New York.
Gill, P. E., Murray, W., Saunders, M. S., and Wright, M. H. (1984), “Users’s Guide for QPSOL (Version 3.2): A FORTRAN Package for Quadratic Programming,” Technical Report SOL 84 - 6, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford.
Han, S. P., Pang, J. S., and Rangaraj, N. (1992), “Globally Convergent Newton Methods for Nonsmooth Equations,” Mathematics of Operations Research, Vol 17, 586–607.
Hock, W. and Schittkowski, K. (1981), Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems, Springer- Verlag, Berlin.
Horn, R. A. and Johnson, C. A. (1985), Matrix Analysis, Cambridge University Press, London.
Pang, J. S. (1981), “The Implicit Complementarity Problem,” in O. L. Mangasarian, S. M. Robinson, and R. R. Meyer, eds., Nonlinear Programming, 4, Academic Press, New York, 487–518.
Pang, J. S. (1990), “Newton’s Method for B-Differentiable Equations,” Mathematics of Operations Research, Vol. 15, 311–341.
Pang, J. S. (1993), “Serial and Parallel Computations of Karush-Kuhn-Tucker Points via Nonsmooth Equations,” Manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.
Pang, J. S. and Gabriel, S. A. (1993), “NE/SQP: A Robust Algorithm for the Nonlinear Complementarity Problem,” Mathematical Programming, Vol. 58, forthcoming.
Pang, J. S., Han, S. P., and Rangaraj, N. (1991), “Minimization of Locally Lipschitzian Functions,” SIAM Journal on Optimization, Vol. 1 57–82.
Pang, J. S. and Qi, L. (1993), “Nonsmooth Equations: Motivation and Algorithms,” SIAM Journal on Optimization, Vol. 5, forthcoming.
Poliquin R. and Qi, L. (revised January 1993 ), “Iteration Functions in Some Nonsmooth Optimization Algorithms,” manuscript, Department of Applied Mathematics, The University of New South Wales, Kensington, New South Wales, Australia.
Qi, L. (1992), “Trust Region Algorithms for Solving Nonsmooth Equations,” Applied Mathematics Preprint AM 92/20, University of New South Wales, Kensington, New South Wales, Australia.
Qi, L. and Sun, J. (1992), “A General Globally Convergent Model of Trust Region Algorithms for Minimization of Locally Lipschitzian Functions,” Applied Mathematics Preprint AM 92/1, University of New South Wales, Kensington, New South Wales, Australia.
Ralph, D. (1993), “Global Convergence of Damped Newton’s Method for Non- smooth Equations via the Path Search,” Mathematics of Operations Research, Vol. 18, forthcoming.
Robinson, S. M. (1988), “Newton’s Method for a Class of Nonsmooth Functions,” Industrial Engineering Department, University of Wisconsin, Madison.
Robinson, S. M. (1991), “An Implicit Function Theorem for a Class of Nonsmooth Functions,” Mathematics of Operations Research, Vol. 16, 292–309.
Rockafellar, R. T. (1970), Convex Analysis, Princeton University Press, Princeton, New Jersey.
Shapiro, A. (1990), “On Concepts of Directional Differentiability,” Journal of Optimization Theory and Applications, Vol. 66, 477–487.
Tobin, R. L. (1988), “Variable Dimension Spatial Price Equilibrium Algorithm,” Mathematical Programming, Vol. 40, 33–51.
Yuan, Y. (1985), “Conditions for Convergence of Trust Region Algorithms for Nonsmooth Optimization,” Mathematical Programming, Vol. 31, 220–228.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Kluwer Academic Publishers
About this chapter
Cite this chapter
Gabriel, S.A., Pang, JS. (1994). A Trust Region Method for Constrained Nonsmooth Equations. In: Hager, W.W., Hearn, D.W., Pardalos, P.M. (eds) Large Scale Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3632-7_9
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3632-7_9
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3634-1
Online ISBN: 978-1-4613-3632-7
eBook Packages: Springer Book Archive