Abstract

We consider the gradient projection algorithm for a strongly convex function with the Lipschitz continuous gradient and a proximally smooth (nonconvex in general) set in a real Hilbert space. We prove that the problem of minimization of such function on a proximally smooth set has unique solution if the constant of proximal smoothness of the set is sufficiently large. The considered algorithm converges with the rate of geometric progression.

Author information

Contact details are reproduced from the original publication and may be historical.

Maxim V. Balashov

Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny, Moscow region, Russia 141700

balashov73@mail.ru

Keywords

  • Hilbert space
  • strongly convex set of radius r
  • proximally smooth set with constant R
  • Lipschitz continuous gradient
  • gradient projection algorithm
  • continuous optimization

Mathematics Subject Classification

46C05, 52A07, 49J52; 46N10, 90C26, 26B25

References

  1. M. V. Balashov: Maximization of a function with Lipschitz continuous gradient, J. Math. Sci., New York 209(1) (2015) 12–18.https://doi.org/10.1007/s10958-015-2482-6Google Scholar
  2. M. V. Balashov, G. E. Ivanov: Weakly convex and proximally smooth sets in Banach spaces, Izv. Math. 73(3) (2009) 455–499.https://doi.org/10.1070/IM2009v073n03ABEH002454Google Scholar
  3. M. V. Balashov, G. E. Ivanov: Properties of the metric projection on weakly Vial-convex sets and parametrization of set-valued mappings with weakly convex images, Math. Notes 80(4) (2006) 461–467.https://doi.org/10.1007/s11006-006-0163-yGoogle Scholar
  4. M. V. Balashov, E. S. Polovinkin: M -strongly convex subsets and their generating sets, Sb. Math. 191(1) (2000) 25–60.https://doi.org/10.1070/SM2000v191n01ABEH000447Google Scholar
  5. F. Bernard, L. Thibault, N. Zlateva: Characterizations of prox-regular sets in uniformly convex Banach spaces, J. Convex Analysis 13(3–4) (2006) 525–559.https://doi.org/10.68381/jca13044Google Scholar
  6. A. Canino: On p-convex sets and geodesics, J. Differ. Equations 75 (1988) 118–157.https://doi.org/10.1016/0022-0396(88)90132-5Google Scholar
  7. F. H. Clarke, R. J. Stern, P. R. Wolenski: Proximal smoothness and lower–C 2 property, J. Convex Analysis 2(1–2) (1995) 117–144.https://doi.org/10.68381/jca02008Google Scholar
  8. H. Frankowska, Ch. Olech: R-convexity of the integral of the set-valued functions, in: Contributions to Analysis and Geometry (Baltimore, 1980), D. N. Clark et al. (ed.), John Hopkins Univ. Press, Baltimore(1981) 117–129.Google Scholar
  9. Yu. Nesterov: Inroductory Lectures on Convex Optimization: A Basic Course, Kluwer, Boston (2004).https://doi.org/10.1007/978-1-4419-8853-9Google Scholar
  10. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  11. B. T. Polyak: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Sov. Math., Dokl. 7 (1966) 72–75.Google Scholar
  12. J.-Ph. Vial: Strong and weak convexity of sets and functions, Math. Oper. Res. 8(2) (1983) 231–259.https://doi.org/10.1287/moor.8.2.231Google Scholar

Suggested citation

M. V. Balashov. “About the Gradient Projection Algorithm for a Strongly Convex Function and a Proximally Smooth Set.” Journal of Convex Analysis 24 (2017), No. 2, 493–500. https://doi.org/10.68381/jca24031

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.