Mathematical Programming

, Volume 58, Issue 1–3, pp 353–367

A nonsmooth version of Newton's method

  • Liqun Qi
  • Jie Sun
Article

Abstract

Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth. Thus, the extended Newton's method can be used in the augmented Lagrangian method for solving nonlinear programs.

Key words

Newton's methods generalized Jacobian semismoothness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J.V. Burke and L. Qi, “Weak directional closedness and generalized subdifferentials,”Journal of Mathematical Analysis and Applications 159 (1991) 485–499.Google Scholar
  2. [2]
    R.W. Chaney, “Second-order necessary conditions in constrained semismooth optimization,”SIAM Journal on Control and Optimization 25 (1987) 1072–1081.Google Scholar
  3. [3]
    R. W. Chaney, “Second-order necessary conditions in semismooth optimization,”Mathematical Programming 40 (1988) 95–109.Google Scholar
  4. [4]
    F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).Google Scholar
  5. [5]
    R. Correa and A. Jofre, “Some properties of semismooth and regular functions in nonsmooth analysis,” in: L. Contesse, R. Correa and A. Weintraub, eds.,Proceedings of the IFIP Working Conference (Springer, Berlin, 1987).Google Scholar
  6. [6]
    R. Correa and A. Jofre, “Tangent continuous directional derivatives in nonsmooth analysis,”Journal of Optimization Theory and Applications 6 (1989) 1–21.Google Scholar
  7. [7]
    P.T. Harker and B. Xiao, “Newton's method for the nonlinear complementarity problem: A B-differentiable equation approach,”Mathematical Programming 48 (1990) 339–357.Google Scholar
  8. [8]
    M. Kojima and S. Shindo, “Extensions of Newton and quasi-Newton methods to systems ofPC 1 Equations,”Journal of Operations Research Society of Japan 29 (1986) 352–374.Google Scholar
  9. [9]
    B. Kummer, “Newton's method for non-differentiable functions,” in: J. Guddat, B. Bank, H. Hollatz, P. Kall, D. Karte, B. Kummer, K. Lommatzsch, L. Tammer, M. Vlach and K. Zimmermann, eds.,Advances in Mathematical Optimization (Akademi-Verlag, Berlin, 1988) pp. 114–125.Google Scholar
  10. [10]
    R. Mifflin, “Semismooth and semiconvex functions in constrained optimization,”SIAM J. Control and Optimization 15 (1977) 957–972.Google Scholar
  11. [11]
    R. Mifflin, “An algorithm for constrained optimization with semismooth functions,”Mathematics of Operations Research 2 (1977) 191–207.Google Scholar
  12. [12]
    J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).Google Scholar
  13. [13]
    J.S. Pang, “Newton's method for B-differentiable equations,”Mathematics of Operations Research 15 (1990) 311–341.Google Scholar
  14. [14]
    J.S. Pang, “A B-differentiable equation-based, globally, and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems,”Mathematical Programming 51 (1991) 101–131.Google Scholar
  15. [15]
    L. Qi, “Semismoothness and decomposition of maximal normal operators,”Journal of Mathematical Analysis and Applications 146 (1990) 271–279.Google Scholar
  16. [16]
    L. Qi and J. Sun, “A nonsmooth version of Newton's method and an interior point algorithm for convex programming,” Applied Mathematics Preprint 89/33, School of Mathematics, The University of New South Wales (Kensington, NSW, 1989).Google Scholar
  17. [17]
    S.M. Robinson, “Local structure of feasible sets in nonlinear programming, part III: stability and sensitivity,”Mathematical Programming Study 30 (1987) 45–66.Google Scholar
  18. [18]
    S.M. Robinson, “An implicit function theorem for B-differentiable functions,”Industrial Engineering Working Paper, University of Wisconsin (Madison, WI, 1988).Google Scholar
  19. [19]
    S.M. Robinson, “Newton's method for a class of nonsmooth functions,”Industrial Engineering Working Paper, University of Wisconsin (Madison, WI, 1988).Google Scholar
  20. [20]
    R.T. Rockafellar, “Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization,”Mathematics of Operations Research 6 (1981) 427–437.Google Scholar
  21. [21]
    R.T. Rockafellar, “Favorable classes of Lipschitz-continuous functions in subgradient optimization,” in: E. Nurminski, ed.,Nondifferentiable Optimization (Pergamon Press, New York, 1982) pp. 125–143.Google Scholar
  22. [22]
    R.T. Rockafellar, “Linear-quadratic programming and optimal control,”SIAM Journal on Control and Optimization 25 (1987) 781–814.Google Scholar
  23. [23]
    R.T. Rockafellar, “Computational schemes for solving large-scale problems in extended linearquadratic programming,”Mathematical Programming 48 (1990) 447–474.Google Scholar
  24. [24]
    R.T. Rockafellar and R.J-B. Wets, “Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time,”SIAM Journal on Control and Optimization 28 (1990) 810–822.Google Scholar
  25. [25]
    A. Shapiro, “On concepts of directional differentiability,”Journal of Optimization Theory and Applications 66 (1990) 477–487.Google Scholar
  26. [26]
    J.E. Spingarn, “Submonotone subdifferentials of Lipschitz functions,”Transactions of the American Mathematical Society 264 (1981) 77–89.Google Scholar
  27. [27]
    J. Sun, “An affine-scaling method for linearly constrained convex programs,” Preprint, Department of IEMS, Northwestern University (Evanston, IL, 1990).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • Liqun Qi
    • 1
  • Jie Sun
    • 2
  1. 1.School of MathematicsThe University of New South WalesKensingtonAustralia
  2. 2.Department of Industrial Engineering and Management SciencesNorthwestern UniversityEvanstonUSA

Personalised recommendations