Advertisement

Mathematical Programming

, Volume 69, Issue 1–3, pp 111–147 | Cite as

New variants of bundle methods

  • Claude Lemaréchal
  • Arkadii Nemirovskii
  • Yurii Nesterov
Article

Abstract

In this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization, convex—concave games and variational inequalities. We outline the ideas underlying these methods and present rate-of-convergence estimates.

Keywords

Nonsmooth convex optimization Bundle methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Auslender,Problèmes de Minimax via l'Analyse Convexe et les Inegalités Variationnelles: Théorie et Algorithmes, Lecture Notes in Economics and Mathematical Systems (Springer, Berlin, 1972).Google Scholar
  2. [2]
    E.W. Cheney and A.A. Goldstein, “Newton's methods for convex programming and Tchebytcheff approximation,”Numerische Mathematik 1 (1959) 253–268.Google Scholar
  3. [3]
    Yu.M. Ermoliev, “Methods for solving nonlinear extremal problems,”Cybernetics 4 (1966) 1–17 (in Russian).Google Scholar
  4. [4]
    M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees: Part II,”Mathematical Programming 1 (1971) 6–25.Google Scholar
  5. [5]
    P. Huard, “Resolution of mathematical programming problems with nonlinear constraints by the method of centers,” in: J. Abadie, ed.,Nonlinear Programming (North-Holland, Amsterdam, 1967) 206–219.Google Scholar
  6. [6]
    J.E. Kelley, “The cutting plane method for solving convex programs,”Journal of the SIAM 8 (1960) 703–712.Google Scholar
  7. [7]
    K.C. Kiwiel, “An aggregate subgradient method for nonsmooth convex minimization,”Mathematical Programming 27 (1983) 320–341.Google Scholar
  8. [8]
    K.C. Kiwiel, “Proximity control in bundle methods for convex nondifferentiable minimization,”Mathematical Programming 46 (1990) 105–122.Google Scholar
  9. [9]
    V.Yu. Lebedev, “On the convergence of the method of loaded functional as applied to a convex programming problem,”Journal of Numerical Mathematics and Mathematical Physics 12 (1977) 765–768 (in Russian).Google Scholar
  10. [10]
    C. Lemaréchal, “An extension of Davidon methods to non-differentiable problems,”Mathematical Programming Study 3 (1975) 95–109.Google Scholar
  11. [11]
    C. Lemaréchal, “Nonsmooth optimization and descent methods,” Research Report 78-4, IIASA, Laxenburg, Austria, 1978.Google Scholar
  12. [12]
    C. Lemaréchal, A. Nemirovskii and Yu. Nesterov, “New variants of bundle methods,” Research Report # 1508, Institut National de Recherche en Informatique et en Automatique, Le Chesnay, 1991.Google Scholar
  13. [13]
    C. Lemaréchal, J.J. Strodiot and A. Bihain, “On a bundle algorithm for nonsmooth optimization,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 4 (Academic Press, New York, 1981) 245–282.Google Scholar
  14. [14]
    P. Marcotte and J.P. Dussault, “A sequential linear programming algorithm for solving monotone variational inequalities,”SIAM Journal on Control and Optimization 27 (1989) 1260–1278.Google Scholar
  15. [15]
    R. Mifflin, “A modification and an extension of Lemaréchal's algorithm for nonsmooth minimization,”Mathematical Programming Study 17 (1982) 77–90.Google Scholar
  16. [16]
    A.S. Nemirovskij and D.B. Yudin,Problem Complexity and Method Efficiency in Optimization (Wiley-Interscience, New York, 1983).Google Scholar
  17. [17]
    M.J.D. Powell, “ZQPCVX, a Fortran subroutine for convex programming,” Report NA17, DAMTP, Cambridge, 1983.Google Scholar
  18. [18]
    B.N. Pshenichny and Yu.M. Danilin,Numerical Methods for Extremal Problems (Mir, Moscow, 1978).Google Scholar
  19. [19]
    S.M. Robinson, “Extension of Newton's method to nonlinear functions with values in a cone,”Numerische Mathematik 9 (1972) 341–347.Google Scholar
  20. [20]
    R.T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,”Transactions of the American Mathematical Society 149 (1970) 75–88.Google Scholar
  21. [21]
    H. Schramm and J. Zowe, “A version of the bundle idea for minimizing a non-smooth function: conceptual idea, convergence analysis, numerical results,”SIAM Journal on Optimization 2 (1992) 121–152.Google Scholar
  22. [22]
    P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions,”Mathematical Programming Study 3 (1975) 145–173.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1995

Authors and Affiliations

  • Claude Lemaréchal
    • 1
  • Arkadii Nemirovskii
    • 2
  • Yurii Nesterov
    • 2
  1. 1.Institut National de Recherche en Informatique et en AutomatiqueLe ChesnayFrance
  2. 2.Central Economical and Mathematical InstituteMoscowRussia

Personalised recommendations