Skip to main content

Maximization of a Function with Lipschitz Continuous Gradient

Abstract

In the present paper, we consider (nonconvex in the general case) functions that have Lipschitz continuous gradient. We prove that the level sets of such functions are proximally smooth and obtain an estimate for the constant of proximal smoothness. We prove that the problem of maximization of such function on a strongly convex set has a unique solution if the radius of strong convexity of the set is sufficiently small. The projection algorithm (similar to the gradient projection algorithm for minimization of a convex function on a convex set) for solving the problem of maximization of such a function is proposed. The algorithm converges with the rate of geometric progression.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    M. V. Balashov and M. O. Golubev, “About the Lipschitz property of the metric projection in the Hilbert space,” J. Math. Anal. Appl., 394, 545–551 (2012).

    MathSciNet  Article  Google Scholar 

  2. 2.

    M. V. Balashov and G. E. Ivanov, “Weakly convex and proximally smooth sets in Banach spaces,” Izv. Math., 73, No. 3, 455–499 (2009).

    MathSciNet  Article  Google Scholar 

  3. 3.

    F. Bernard, L. Thibault, and N. Zlatev, “Characterization of proximal regular sets in super reflexive Banach spaces,” J. Convex Anal., 13, No. 3-4, 525–559 (2006).

    MathSciNet  Google Scholar 

  4. 4.

    A. Canino, “On p-convex sets and geodesics,” J. Differ. Equ., 75, 118–157 (1988).

    MathSciNet  Article  Google Scholar 

  5. 5.

    F. H. Clarke, R. J. Stern, and P. R.Wolenski, “Proximal smoothness and lower-C 2 property,” J. Convex Anal., 2, No. 1-2, 117–144 (1995).

    MathSciNet  Google Scholar 

  6. 6.

    G. E. Ivanov, Weakly Convex Sets and Functions [in Russian], Fizmatlit, Moscow (2006).

    Google Scholar 

  7. 7.

    J. B. Hiriart-Urruty and Yu. S. Ledyaev, “A note on the characterization of the global maxima of a (tangentially) convex function over a convex set,” J. Convex Anal., 3, No. 1, 55–61 (1996).

    MathSciNet  Google Scholar 

  8. 8.

    R. A. Poliquin, R. T. Rockafellar, and L. Thibault, “Local differentiability of distance functions,” Trans. Am. Math. Soc., 352, 5231–5249 (2000).

    MathSciNet  Article  Google Scholar 

  9. 9.

    E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis [in Russian], Fizmatlit, Moscow (2007).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. V. Balashov.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 5, pp. 17–25, 2013.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Balashov, M.V. Maximization of a Function with Lipschitz Continuous Gradient. J Math Sci 209, 12–18 (2015). https://doi.org/10.1007/s10958-015-2482-6

Download citation

Keywords

  • Convex Function
  • Iteration Process
  • Real Hilbert Space
  • Projection Algorithm
  • Geometric Progression