Skip to main content
SpringerLink
Log in

Parametric Proximal-Point Methods

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The main purpose of the present work is to introduce two parametric proximal-point type algorithms involving the gradient (or subdifferential) of a convex function. We take advantage of some properties of maximal monotone operators to prove monotonicity and convergence rate conditions. One example in Hilbert spaces and two numerical examples with program realizations are presented.

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. Ivanov, R., Raykov, I.: A parametric Newton method for optimization problems in Hilbert spaces. Numer. Funct. Anal. Optim. 24, 243–255 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ivanov, R.P., Raykov, I.L.: Parametric Lyapunov function method for solving nonlinear systems in Hilbert spaces. Numer. Funct. Anal. Optim. 17, 893–901 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Raykov, I.: Parameterized Newton method for optimization problems. In: Guddat, J., Jongen, H. Th., Ruckmann, J.-J., Todorov, M. (eds.) Parametric Optimization and Related Topics VII, pp. 187–205 (2004)

  4. Pavel, N., Raykov, I.: Applications of parametric proximal-point algorithms. In: Momcheva, G., Madjarova, S. Slavov, Z. (eds.) Proceedings of the Conference of the Union of Bulgarian Mathematicians, December 2007, Varna. Mathematics and Computer Sciences—Reality and Perspectives, pp. 36–50 (2007). ISBN: 978-964-715-376-9

  5. De Marchi, S., Raykov, I.: Parametric method for global optimization. J. Optim. Theory Appl. 130(3), 411–430 (2006)

    Article  Google Scholar 

  6. Raykov, I.: Uniform convergence for multivalued algorithms. Indian J. Pure Appl. Math. 28, 1491–1504 (1997)

    MATH  MathSciNet  Google Scholar 

  7. Raykov, I.L., Ivanov, R.P.: Sufficient conditions for the uniform convergence of parametric Lyapunov function method for solving nonlinear systems of equations. In: Proceedings of the Conference on Mathematical Modeling and Scientific Computation, Sozopol, Bulgaria, pp. 149–153 (1993)

  8. Raykov, I.: Sufficient conditions for the uniform convergence for algorithms modeled by point-to-set maps with respect to an independent parameter. In: Proceedings of the 21st Conference of the Union of Bulgarian Mathematicians, Sofia, Bulgaria, pp. 127–135 (1992)

  9. Dontchev, A.L.: Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems. Lecture Notes in Control and Information Sciences, vol. 52. Springer, Berlin (1983)

    MATH  Google Scholar 

  10. Dontchev, A.L., Hager, W.W., Malanowski, K.: Error bounds for the Euler approximation of a state and control constrained optimal control problem. Numer. Funct. Anal. Optim. 21, 653–682 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dontchev, A.L., Hager, W.W., Veliov, V.M.: On quantitative stability in optimization and optimal control. Set-Valued Anal. 8, 31–50 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dontchev, A.L., Hager, W.W., Veliov, V.M.: Uniform convergence and mesh independence of Newton’s method for discretized variational problems. SIAM J. Control Optim. 39, 961–980 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nashed, M.Z.: Inner, outer, and generalized inverses in Banach and Hilbert spaces. Numer. Funct. Anal. Optim. 9, 261–325 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nashed, M.Z., Chen, X.: Convergence of Newton-like methods for singular operator equations using outer inverses. Numer. Math. 66, 235–257 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Rockafellar, R.T.: Nonsmooth analysis and parametric optimization. In: Cellina, A. (ed.) Methods of Nonconvex Analysis. Lecture Notes in Mathematics, vol. 1446, pp. 137–151. Springer, Berlin (1990)

    Chapter  Google Scholar 

  17. Rockafellar, R.T., King, A.J.: Sensitivity analysis for nonsmooth generalized equations. Math. Program. 55, 193–212 (1992)

    Article  MathSciNet  Google Scholar 

  18. Rockafellar, R.T., Poliquin, R.A.: Second-order nonsmooth analysis in nonlinear programming. In: Du, D., Qi, L., Womersley, R. (eds.) Recent Advances in Optimization, pp. 322–350. World Scientific, Singapore (1995)

    Google Scholar 

  19. Chen, X., Nashed, Z., Qi, L.: Convergence of Newton’s method for singular smooth and nonsmooth equations using adaptive outer inverses. SIAM J. Optim. 7, 445–462 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hager, W.W.: The LP dual active set algorithm. In: De Leone, R., Murli, A., Pardalos, P.M., Toraldo, G. (eds.) High Performance Algorithms and Software in Nonlinear Optimization, pp. 243–254. Kluwer, Dordrecht (1998)

    Google Scholar 

  21. Pang, J.S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Qi, L., Chen, X.: A globally convergent successive approximation method for severely nonsmooth equations. SIAM J. Control Optim. 33, 402–418 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ralph, D.: Global convergence of damped Newton’s method for nonsmooth equations, via the path search. Math. Oper. Res. 19, 352–389 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Robinson, S.M.: Newton’s method for a class of nonsmooth functions. Set-Valued Anal. 2, 291–305 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Motreanu, D., Pavel, N.H.: Tangency, Flow Invariance for Differential Equations and Optimization Problems. Monographs and Textbooks in Pure and Appl. Mathematics, vol. 219. Marcel Dekker, New York (1999)

    MATH  Google Scholar 

  26. Pavel, N.H.: Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations. Lecture Notes in Math., vol. 1260. Springer, Heidelberg (1987)

    MATH  Google Scholar 

  27. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  28. Ginchev, I., Guerraggio, A.: Second Order Optimality Conditions in Nonsmooth Unconstrained Optimization. Pliska Studia Mathematica Bulgarica, vol. 12, pp. 39–50 (1998)

  29. Georgiev, P., Zlateva, N.: Second-order subdifferentials of C 1,1 functions and optimality conditions. Set-Valued Anal. 4, 101–117 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hiriart-Urruty, J.B., Strodiot, J.J., Nguyen, V.H.: Generalized Hessian matrix and second-order optimality conditions for problems with C 1,1 data. Appl. Math. Optim. 11, 43–56 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mordukhovich, B.S.: Calculus of second-order subdifferentials in infinite dimensions. Control Cybern. 31, 558–573 (2002)

    MathSciNet  Google Scholar 

  32. Mordukhovich, B.S., Outrata, J.V.: On second-order subdifferentials and their applications. SIAM J. Optim. 12, 139–169 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Filippov, A.F.: Differential Equations with Discontinues Right Hand Side. Nauka, Moscow (1985) (in Russian)

    Google Scholar 

  34. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley-Interscience, New-York (1984)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. Deimling, K.: Multivalued Differential Equations. De Gruyter, Berlin (1992)

    MATH  Google Scholar 

  37. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio University, Athens, OH, USA

    N. Pavel & I. Raykov

Authors
  1. N. Pavel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Raykov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Raykov.

Additional information

Communicated by T.L. Vincent.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pavel, N., Raykov, I. Parametric Proximal-Point Methods. J Optim Theory Appl 139, 85–107 (2008). https://doi.org/10.1007/s10957-008-9408-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-008-9408-0

Keywords

Search

Navigation