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New variants of bundle methods

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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.

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References

  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. 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. Yu.M. Ermoliev, “Methods for solving nonlinear extremal problems,”Cybernetics 4 (1966) 1–17 (in Russian).

    Google Scholar 

  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. 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. J.E. Kelley, “The cutting plane method for solving convex programs,”Journal of the SIAM 8 (1960) 703–712.

    Google Scholar 

  7. K.C. Kiwiel, “An aggregate subgradient method for nonsmooth convex minimization,”Mathematical Programming 27 (1983) 320–341.

    Google Scholar 

  8. K.C. Kiwiel, “Proximity control in bundle methods for convex nondifferentiable minimization,”Mathematical Programming 46 (1990) 105–122.

    Google Scholar 

  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. C. Lemaréchal, “An extension of Davidon methods to non-differentiable problems,”Mathematical Programming Study 3 (1975) 95–109.

    Google Scholar 

  11. C. Lemaréchal, “Nonsmooth optimization and descent methods,” Research Report 78-4, IIASA, Laxenburg, Austria, 1978.

    Google Scholar 

  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. 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. 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. R. Mifflin, “A modification and an extension of Lemaréchal's algorithm for nonsmooth minimization,”Mathematical Programming Study 17 (1982) 77–90.

    Google Scholar 

  16. A.S. Nemirovskij and D.B. Yudin,Problem Complexity and Method Efficiency in Optimization (Wiley-Interscience, New York, 1983).

    Google Scholar 

  17. M.J.D. Powell, “ZQPCVX, a Fortran subroutine for convex programming,” Report NA17, DAMTP, Cambridge, 1983.

    Google Scholar 

  18. B.N. Pshenichny and Yu.M. Danilin,Numerical Methods for Extremal Problems (Mir, Moscow, 1978).

    Google Scholar 

  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. 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. 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. P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions,”Mathematical Programming Study 3 (1975) 145–173.

    Google Scholar 

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Lemaréchal, C., Nemirovskii, A. & Nesterov, Y. New variants of bundle methods. Mathematical Programming 69, 111–147 (1995). https://doi.org/10.1007/BF01585555

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  • DOI: https://doi.org/10.1007/BF01585555

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