A globally convergent Newton method for convex SC1 minimization problems

  • J. S. Pang
  • L. Qi
Contributed Papers


This paper presents a globally convergent and locally superlinearly convergent method for solving a convex minimization problem whose objective function has a semismooth but nondifferentiable gradient. Applications to nonlinear minimax problems, stochastic programs with recourse, and their extensions are discussed.

Key Words

Nonsmooth optimization Newton method 


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  1. 1.
    Hiriart-Urruty, J. B., Strodiot, J. J., andNguyen, V. H.,Generalized Hessian Matrix and Second-Order Optimality Conditions for Problems with C 1,1 Data, Applied Mathematics and Optimization, Vol. 11, pp. 43–56, 1984.CrossRefGoogle Scholar
  2. 2.
    Klatte, D., andTammer, K.,On Second-Order Sufficient Optimality Conditions for C 1,1 Optimization, Optimization, Vol. 19, pp. 169–179, 1988.Google Scholar
  3. 3.
    Kummer, B.,Lipschitzian Inverse Functions, Directional Derivatives, and Application in C 1,1 Optimization, Journal of Optimization Theory and Applications, Vol. 70, pp 559–580, 1991.CrossRefGoogle Scholar
  4. 4.
    Han, S. P.,Superlinearly Convergent Variable-Metric Algorithms for General Nonlinear Programming Problems, Mathematical Programming, Vol. 11, pp. 263–282, 1976.CrossRefGoogle Scholar
  5. 5.
    Kojima, M., andShindo, S.,Extensions of Newton and Quasi-Newton Methods to Systems of PC 1 Equations, Journal of the Operations Research Society of Japan, Vol. 29, pp. 352–374, 1986.Google Scholar
  6. 6.
    Kuntz, L., andScholtes, S.,Structural Analysis of Nonsmooth Mappings, Inverse Functions, and Metric Projections, Manuscript, Institut für Statistik and Mathematische Wirtschafstheorie, Universität Karlsruhe, Karlsruhe, Germany, 1992.Google Scholar
  7. 7.
    Pang, J. S., andRalph, D.,Piecewise Smoothness, Local Invertibility, and Parametric Analysis of Normal Maps, Mathematics of Operations Research.Google Scholar
  8. 8.
    Qi, L.,Superlinearly Convergent Approximate Newton Methods for LC 1 Optimization Problems, Mathematical Programming, Vol. 64, pp. 277–294, 1994.CrossRefGoogle Scholar
  9. 9.
    Bertsekas, D. P.,Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, New York, 1982.Google Scholar
  10. 10.
    Qi, L., andSun, J.,A Nonsmooth Version of Newton's Method, Mathematical Programming, Vol. 58, pp. 353–367, 1993.CrossRefGoogle Scholar
  11. 11.
    Mifflin, R.,Semismooth and Semiconvex Functions in Constrained Optimization, SIAM Journal on Control and Optimization, Vol. 15, pp. 957–972, 1977.CrossRefGoogle Scholar
  12. 12.
    Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley, New York, New York, 1983.Google Scholar
  13. 13.
    Pang, J. S.,Newton Methods for B-Differentiable Equations, Mathematics of Operations Research, Vol. 15, pp. 311–341, 1990.Google Scholar
  14. 14.
    Robinson, S. M. Local Structure of Feasible Sets in Nonlinear Programming, Part 3: Stability and Sensitivity, Mathematical Programming Study, Vol. 30, pp. 45–66, 1987.Google Scholar
  15. 15.
    Shapiro, A.,On Concepts of Directional Differentiability, Journal of Optimization Theory and Applications, Vol. 66, pp. 477–487, 1990.CrossRefGoogle Scholar
  16. 16.
    Qi, L.,Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations, Mathematics of Operations Research, Vol. 18, pp. 227–244, 1993.Google Scholar
  17. 17.
    Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.Google Scholar
  18. 18.
    Jiang, H., andQi, L.,Local Uniqueness and Newton-Type Methods for Nonsmooth Variational Inequalities, Journal of Mathematical Analysis and Applications.Google Scholar
  19. 19.
    Chaney, R. W.,Piecewise C k Functions in Nonsmooth Analysis, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 15, pp. 649–660, 1990.Google Scholar
  20. 20.
    Pang, J. S., Han, S. P., andRangaraj, R.,Minimization of Locally Lipschitzian Functions, SIAM Journal on Optimization, Vol. 1, pp. 57–82, 1991.CrossRefGoogle Scholar
  21. 21.
    Gabriel, S. A., andPang, J. S.,A Trust-Region Method for Constrained Nonsmooth Equations, Large-Scale Optimization: State of the Art, Edited by W. W. Hager, D. W. Hearn, and P. Pardalos, Kluwer Academic Publishers, Boston, Massachusetts, pp. 159–186, 1993.Google Scholar
  22. 22.
    Pang, J. S., andGabriel, S. A.,NE/SQP: A Robust Algorithm for the Nonlinear Complementarity Problem, Mathematical Programming, Vol. 60, pp. 295–337, 1993.CrossRefGoogle Scholar
  23. 23.
    Pang, J. S., andQi, L.,Nonsmooth Equations: Motivation and Algorithms, SIAM Journal on Optimization, Vol. 3, pp. 443–465, 1993.CrossRefGoogle Scholar
  24. 24.
    Qi, L., andWomersley, R. S.,An SQP Algorithm for Extended Linear-Quadratic Problems in Stochastic Programming, Annals of Operations Research.Google Scholar
  25. 25.
    Danskin, J. M.,The Theory of Max-Min, Springer Verlag, New York, New York, 1967.Google Scholar
  26. 26.
    Ermoliev, Y., andWets, R. J. B.,Numerical Techniques in Stochastic Programming, Springer Verlag, Berlin, Germany, 1988.Google Scholar
  27. 27.
    Rockafellar, R. T., andWets, R. J. B.,A Lagrangian Finite-Generation Technique for Solving Linear-Quadratic Problems in Stochastic Programming, Mathematical Programming Study, Vol. 28, pp. 63–93, 1986.Google Scholar
  28. 28.
    Chen, X., Qi, L., andWomersley, R. S.,Newton's Method for Quadratic Stochastic Program with Recourse, Journal of Computational and Applied mathematics (to appear).Google Scholar
  29. 29.
    Attouch, H., andWets, R. J. B.,Epigraphical Analysis, in: Analyse Nonlinéaire, Edited by H. Attouch, J. P. Aubin, and R. J. B. Wets, Gauthier-Villars, Paris, France, pp. 73–100, 1989.Google Scholar
  30. 30.
    Bonnans, J. F., Gilbert, J. C., Lemarechal, C., andSagastizabal, C.,A Family of Variable-Metric Proximal Methods, Mathematical Programming, Vol. 68, pp. 15–48, 1995.Google Scholar
  31. 31.
    Teboulle, M.,Entropic Proximal Mappings with Applications to Nonlinear Programming, Mathematics of Operations Research, Vol. 17, pp. 670–690, 1992.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • J. S. Pang
    • 1
  • L. Qi
    • 2
  1. 1.Department of Mathematical SciencesThe Johns Hopkins UniversityBaltimore
  2. 2.Department of Applied MathematicsUniversity of New South WalesSydneyAustralia

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