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, log in via an institution to check access.
Similar content being viewed by others
References
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).
M. V. Balashov and G. E. Ivanov, “Weakly convex and proximally smooth sets in Banach spaces,” Izv. Math., 73, No. 3, 455–499 (2009).
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).
A. Canino, “On p-convex sets and geodesics,” J. Differ. Equ., 75, 118–157 (1988).
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).
G. E. Ivanov, Weakly Convex Sets and Functions [in Russian], Fizmatlit, Moscow (2006).
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).
R. A. Poliquin, R. T. Rockafellar, and L. Thibault, “Local differentiability of distance functions,” Trans. Am. Math. Soc., 352, 5231–5249 (2000).
E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis [in Russian], Fizmatlit, Moscow (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 5, pp. 17–25, 2013.
Rights and permissions
About this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-015-2482-6