Skip to main content
SpringerLink
Log in

Computing proximal points of nonconvex functions

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal envelopes (Moreau envelopes), will have a natural extension from convex optimization to nonconvex optimization. For example, the inexact proximal point methods for convex optimization might be redesigned to work for nonconvex functions. In order to begin the practical implementation of proximal points in a nonconvex setting, a first crucial step would be to design efficient methods of approximating nonconvex proximal points. This would provide a solid foundation on which future design and analysis for nonconvex proximal point methods could flourish. In this paper we present a methodology based on the computation of proximal points of piecewise affine models of the nonconvex function. These models can be built with only the knowledge obtained from a black box providing, for each point, the function value and one subgradient. Convergence of the method is proved for the class of nonconvex functions that are prox-bounded and lower-\({\mathcal{C}}^2\) and encouraging preliminary numerical testing is 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. Auslender A. (1987). Numerical methods for nondifferentiable convex optimization. Math. Program. Stud. 30: 102–126

    MathSciNet  MATH  Google Scholar 

  2. Auslender A., Crouzeix J.P. and Fedit P. (1987). Penalty-proximal methods in convex programming. J. Optim. Theory Appl. 55(1): 1–21

    Article  MathSciNet  MATH  Google Scholar 

  3. Auslender A. and Haddou M. (1995). An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities. Math. Program. 71(1, Ser. A): 77–100

    Article  MathSciNet  Google Scholar 

  4. Bellman R., Kalaba R. and Lockett J. (1966). Numerical Inversion of the Laplace Transform. Elsevier, Amsterdam

    MATH  Google Scholar 

  5. Bernard F. and Thibault L. (2004). Prox-regularity of functions and sets in Banach spaces. Set Valued Anal. 12(1-2): 25–47

    Article  MathSciNet  MATH  Google Scholar 

  6. Bernard F. and Thibault L. (2005). Prox-regular functions in Hilbert spaces. J. Math Anal. Appl. 303(1): 1–14

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonnans J.F., Gilbert J., Lemaréchal C. and Sagastizábal C. (1995). A family of variable metric proximal methods. Math. Program. Ser. A 68: 15–47

    Google Scholar 

  8. Cominetti R. and Courdurier M. (2003). Coupling general penalty schemes for convex programming with the steepest descent and the proximal point algorithm. SIAM J. Optim. 13(3): 745–765 (electronic)

    Article  MathSciNet  Google Scholar 

  9. Correa R. and Lemaréchal C. (1993). Convergence of some algorithms for convex minimization. Math. Program. 62(2): 261–275

    Article  Google Scholar 

  10. Dolan E. and Moré J. (2002). Benchmarking optimization software with performance profiles. Math. Program. 91(2, Ser. A): 201–213

    Article  MathSciNet  MATH  Google Scholar 

  11. Eckstein J. (1993). Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Math. Oper. Res. 18: 202–226

    MathSciNet  MATH  Google Scholar 

  12. Frangioni A. (1996). Solving semidefinite quadratic problems within nonsmooth optimization algorithms. Comput. Oper. Res. 23(11): 1099–1118

    Article  MathSciNet  MATH  Google Scholar 

  13. Fuduli A., Gaudioso M. and Giallombardo G. (2003). Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J. Optim. 14(3): 743–756 (electronic)

    Article  MathSciNet  Google Scholar 

  14. Güler O. (1992). New proximal point algorithms for convex minimization. SIAM J. Optim. 2: 649–664

    Article  MathSciNet  MATH  Google Scholar 

  15. Hare, W., Lewis, A.: Identifying active contraints via partial smoothness and prox-regularity. J. Convex Anal. 11, 251–266

  16. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. No. 305–306 in Grund. der math. Wiss. Springer, Heidelberg (two volumes, 1993)

  17. Kiwiel K. (1986). A method for solving certain quadratic programming problems arising in nonsmooth optimization. IMA J. Numer. Anal. 6: 137–152

    Article  MathSciNet  MATH  Google Scholar 

  18. Kiwiel K. (1991). Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization. Math. Program. 52(2, Ser. B): 285–302

    Article  MathSciNet  MATH  Google Scholar 

  19. Lemaréchal C. and Sagastizábal C. (1997). Variable metric bundle methods: from conceptual to implementable forms. Math. Program. Ser. A 76: 393–410

    Article  Google Scholar 

  20. Lemaréchal C., Strodiot J.J. and Bihain A. (1981). On a bundle method for nonsmooth optimization. In: Mangasarian, O., Meyer, R., and Robinson, S. (eds) Nonlinear Programming, vol. 4, pp 245–282. Academic, New York

    Google Scholar 

  21. Lukšan L. and Vlček J. (1998). A bundle-Newton method for nonsmooth unconstrained minimization. Math. Program. 83(3, Ser. A): 373–391

    Article  MATH  Google Scholar 

  22. Lukšan L. and Vlček J. (2001). Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization. J. Optim. Theory Appl. 2: 407–430

    Google Scholar 

  23. Makela, M., Neittaanmaki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific, Singapore (1992)

  24. Martinet B. (1970). Régularisation d’inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle 4(Ser. R-3): 154–158

    MathSciNet  Google Scholar 

  25. Mifflin R. (1977). Semi-smooth and semi-convex functions in constrained optimization. SIAM J. Control Optim. 15: 959–972

    Article  MathSciNet  MATH  Google Scholar 

  26. Mifflin, R.: Convergence of a modification of lemaréchal’s algorithm for nonsmooth optimization. In: Progress in Nondifferentiable Optimization, IIASA Collaborative Proc., vol. 8, pp. 85–95. (Ser. CP-82) (1982)

  27. Mifflin R. (1982). A modification and extension of Lemaréchal’s algorithm for nonsmooth minimization. Math. Program. Stud. 17: 77–90

    MathSciNet  MATH  Google Scholar 

  28. Mifflin R. and Sagastizábal C. (2003). Primal-dual gradient structured functions: second-order results; links to epi-derivatives and partly smooth functions. SIAM J. Optim. 13(4): 1174–1194

    Article  MathSciNet  MATH  Google Scholar 

  29. Mifflin R., Sagastizábal C. (2004) \({\mathcal{VU}}\) -smoothness and proximal point results for some nonconvex functions. Optim. Meth. Softw. 19(5): 463–478

    Article  MATH  Google Scholar 

  30. Mordukhovich B.S. (1976). Maximum principle in the problem of time optimal response with nonsmooth constraints. Prikl. Mat. Meh. 40(6): 1014–1023

    MathSciNet  Google Scholar 

  31. Moreau J. (1965). Proximité et dualité dans un espace Hilbertien. Bulletin de la Société Mathématique de France 93: 273–299

    MathSciNet  MATH  Google Scholar 

  32. Poliquin R.A. and Rockafellar R.T. (1996). Generalized Hessian properties of regularized nonsmooth functions. SIAM J. Optim. 6(4): 1121–1137

    Article  MathSciNet  MATH  Google Scholar 

  33. Poliquin R.A. and Rockafellar R.T. (1996). Prox-regular functions in Variational Analysis. Trans. Am. Math. Soc. 348(5): 1805–1838

    Article  MathSciNet  MATH  Google Scholar 

  34. Popova, N., Tarasov, V.: A modification of the cutting-plane method with accelerated convergence. In: Nondifferentiaible Optimization: Motivations and application, Lecture Notes in Economics and Mathematical Systems, pp. 284–290 (1984)

  35. Rockafellar R. (1976). Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1: 97–116

    Article  MathSciNet  MATH  Google Scholar 

  36. Rockafellar R. (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14: 877–898

    Article  MathSciNet  MATH  Google Scholar 

  37. Rockafellar, R., Wets, R.B.: Variational Analysis. No. 317 in Grund. der math. Wiss. Springer, Heidelberg (1998)

  38. Yosida K. (1964) Functional Analysis. Springer, Heidelberg

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing and Software, McMaster University, 1200 Main St. W, Hamilton, ON, L8N 3Z5, Canada

    Warren Hare

  2. IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, 22460-320, Brazil

    Claudia Sagastizábal

Authors
  1. Warren Hare
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claudia Sagastizábal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudia Sagastizábal.

Additional information

Dedicated to Alfred Auslender on the occasion of his 65th birthday, with special thanks for his important contributions in proximal point methods.

Research of Warren Hare was supported by CNPq Grant no. 150234/2004-0.

Claudia Sagastizábal was on leave from INRIA, France. Her research was supported by CNPq Grant no. 383066/2004-2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hare, W., Sagastizábal, C. Computing proximal points of nonconvex functions. Math. Program. 116, 221–258 (2009). https://doi.org/10.1007/s10107-007-0124-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-007-0124-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation