Skip to main content
SpringerLink
Log in

A smooth method for the finite minimax problem

  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider unconstrained minimax problems where the objective function is the maximum of a finite number of smooth functions. We prove that, under usual assumptions, it is possible to construct a continuously differentiable function, whose minimizers yield the minimizers of the max function and the corresponding minimum values. On this basis, we can define implementable algorithms for the solution of the minimax problem, which are globally convergent at a superlinear convergence rate. Preliminary numerical results are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.W. Bandler and C. Charalambous, “Practical leastpth optimization of networks,”IEEE Transactions on Microwave Theory and Techniques (1972 Symposium Issue) MTT-20 (1972) 834–840.

    Google Scholar 

  2. D.P. Bertsekas, “Nondifferentiable optimization via approximation,”Mathematical Programming Study 3 (1975) 1–25.

    Google Scholar 

  3. C. Charalambous, “Acceleration of the leastpth algorithm for minimax optimization with engineering applications,”Mathematical Programming 17 (1979) 270–297.

    Google Scholar 

  4. C. Charalambous and A.R. Conn, “An efficient method to solve the minimax problem directly,”SIAM Journal on Numerical Analysis 15 (1978) 162–187.

    Google Scholar 

  5. F.H. Clarke, V.F. Dem'yanov and F. Giannessi, eds.,Nonsmooth Optimization and Related Topics (Plenum, New York, 1989).

    Google Scholar 

  6. V.F. Dem'yanov and V.N. Malozemov,Introduction to Minimax (Wiley, New York, 1974).

    Google Scholar 

  7. V.F. Dem'yanov and V. Vasil'ev,Nondifferentiable Optimization (Springer, New York, 1986).

    Google Scholar 

  8. G. Di Pillo and L. Grippo, “An exact penalty method with global convergence properties for nonlinear programming problems,”Mathematical Programming 36 (1986) 1–18.

    Google Scholar 

  9. G. Di Pillo, L. Grippo and S. Lucidi, “Superlinearly convergent algorithms for finite minimax problems,” Technical Report, IASI, National Research Council (Rome), to appear.

  10. R.A. El-Attar, M. Vidyasagar and S.R.K. Dutta, “An algorithm forl l-approximation,”SIAM Journal on Numerical Analysis 16 (1979) 70–86.

    Google Scholar 

  11. R. Fletcher, “A model algorithm for composite nondifferentiable optimization problems,”Mathematical Programming Study 17 (1982) 67–76.

    Google Scholar 

  12. R. Fletcher, “Second order correction for nondifferentiable optimization,” in: G.A. Watson, ed.,Numerical Analysis (Springer, Berlin, 1982) pp. 85–114.

    Google Scholar 

  13. M. Gaudioso and M.F. Monaco, “A bundle type approach to the unconstrained minimization of convex nonsmooth functions,”Mathematical Programming 23 (1982) 216–226.

    Google Scholar 

  14. C. Gigola and S. Gomez, “A regularization method for solving the finite convex min—max problem,”SIAM Journal and Numerical Analysis 27 (1990) 1621–1634.

    Google Scholar 

  15. T. Glad and E. Polak, “A multiplier method with automatic limitation of penalty growth,”Mathematical Programming 17 (1979) 140–155.

    Google Scholar 

  16. A. Griewank, “On automatic differentiation,” in: M. Iri and K. Tanabe, eds.,Mathematical programming (Kluwer Academic Publishers, Tokyo) pp. 83–107.

  17. L. Grippo, F. Lampariello and S. Lucidi, “A class of nonmonotone stabilization methods in unconstrained optimization,”Numerische Mathematik 59 (1991) 779–805.

    Google Scholar 

  18. J. Hald and K. Madsen, “Combined LP and quasi-Newton methods for minimax optimization,”Mathematical Programming 20 (1981) 49–62.

    Google Scholar 

  19. S.P. Han, “Variable metric methods for minimizing a class of nondifferentiable functions,”Mathematical Programming 20 (1981) 1–13.

    Google Scholar 

  20. M.R. Hestenes,Optimization Theory. The Finite Dimensional Case (Wiley, New York, 1975).

    Google Scholar 

  21. K.C. Kiwiel,Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics No. 1133 (Springer, Berlin, 1985).

    Google Scholar 

  22. C. Lemaréchal, “An extension of Davidson method to nondifferentiable problems,”Mathematical Programming Study 3 (1975) 95–109.

    Google Scholar 

  23. C. Lemaréchal, “Numerical experiments in nonsmooth optimization,” in: E.A. Nurminskii, ed.,Progress in Nondifferentiable Optimization (IIASA Proceedings, Laxenburg, 1982).

    Google Scholar 

  24. C. Lemaréchal, “Nondifferentiable optimization,” in: G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd, eds.,Handbooks in Operations Research and Management Science, Vol. 1: Optimization (North-Holland, Amsterdam, 1989) pp. 529–572.

    Google Scholar 

  25. C. Lemaréchal and R. Mifflin, eds.,Nonsmooth Optimization (Pergamon, Oxford, 1978).

    Google Scholar 

  26. C. Lemaréchal and R. Mifflin, “A set of nonsmooth optimization test problems,” in: C. Lemaréchal and R. Mifflin, eds.,Nonsmooth Optimization (Pergamon, Oxford, 1978).

    Google Scholar 

  27. S. Lucidi, “New results on a continuously differentiable exact penalty function,” Report R-277, IASI, National Research Council (Rome, 1989).

    Google Scholar 

  28. K. Madsen, “Minimization of non-linear approximation functions,” Ph.D. Thesis, Technical University of Denmark (Lyngby, 1985).

  29. M.M. Mäkelä, “Nonsmooth optimization,” Ph.D. Thesis, University of Jyväskylä (Jyväskylä, 1990).

    Google Scholar 

  30. R. Mifflin, “An algorithm for constrained optimization with semismooth functions,”Mathematics of Operations Research 2 (1977) 191–207.

    Google Scholar 

  31. E. Polak, “Basics of minimax algorithms,” in: F.H. Clarke, V.F. Dem'yanov and F. Giannessi, eds.,Nonsmooth Optimization and Related Topics (Plenum, New York, 1989) pp. 343–369.

    Google Scholar 

  32. E. Polak, D.H. Mayne and J.E. Higgins, “Superlinearly convergent algorithm for min—max problems,”Journal of Optimization Theory and Applications 69 (1991) 407–439.

    Google Scholar 

  33. R.A. Polyak, “Smooth optimization methods for minimax problems,”SIAM Journal on Control and Optimization 26 (1988) 1274–1286.

    Google Scholar 

  34. B.N. Pshenichny and Y.M. Danilin,Numerical Methods in Extremal Problems (Mir, Moscow, 1978).

    Google Scholar 

  35. H. Schramm and J. Zowe, “A version of the bundle ideal for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results,” Report N.206, DFG-Schwerpunktprogramms “Anwendungsbezogene Optimierung und Steuerung,” Universität Bayreuth (Bayreuth, 1990).

    Google Scholar 

  36. N.Z. Shor,Minimization Methods for Nondifferentiable Functions (Springer, Berlin, 1985).

    Google Scholar 

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

    Google Scholar 

  38. Y. Yuan, “On the superlinear convergence of a trust region algorithm for nonsmooth optimization,”Mathematical Programming 31 (1985) 269–285.

    Google Scholar 

  39. I. Zang, “A smoothing-out technique for min—max optimization,”Mathematical Programming 19 (1980) 61–77.

    Google Scholar 

  40. J. Zowe, “Nondifferentiable optimization,” in: K. Schittkowski, eds.,Computational Mathematical Programming (Springer, Berlin, 1985).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Via Buonarrota 12, 00185, Roma, Italy

    G. Di Pillo & L. Grippo

  2. Istituto di Analisi dei Sistemi e Informatica del CNR, Roma, Italy

    S. Lucidi

Authors
  1. G. Di Pillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Grippo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Lucidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was partially supported by the National Research Program on “Metodi di ottimizzazione per le decisioni”, Ministero dell'Università e della Ricerca Scientifica e Tecnologica, Italy.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Di Pillo, G., Grippo, L. & Lucidi, S. A smooth method for the finite minimax problem. Mathematical Programming 60, 187–214 (1993). https://doi.org/10.1007/BF01580609

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01580609

Key words

Search

Navigation