Local Lipschitz Continuity of the Metric Projection Operator

Abstract

In this survey, we consider the metric projection operator from the real Hilbert space onto a closed subset. We discuss the following question: When is this operator Lipschitz continuous? First, we consider the class of strongly convex sets of radius R, i.e., each set from this class is a nonempty intersection of closed balls of radius R. We prove that the restriction of the metric projection operator on the complement of the neighborhood of radius r of a strongly convex set of radius R is Lipschitz continuous with Lipschitz constant C = R/(r + R) ∈ (0, 1). Vice versa, if for a closed convex set from the real Hilbert space the metric projection operator is Lipschitz continuous with Lipschitz constant C ∈ (0, 1) on the complement of the neighborhood of radius r of the set, then the set is strongly convex of radius R = Cr/(1 − C).

It is known that if a closed subset of a real Hilbert space has Lipschitz continuous metric projection in some neighborhood, then this set is proximally smooth. We show that if a closed subset of the real Hilbert space has Lipschitz continuous metric projection on the neighborhood of radius r with Lipschitz constant C > 1, then this set is proximally smooth with constant of proximal smoothness R = Cr/(C − 1), and, if the constant C is the smallest possible, then the constant R is the largest possible.

We apply the obtained results to the question concerning the rate of convergence for the gradient projection algorithm.

References

1. T. J. Abatzoglou, "The minimum norm projection on C ²-manifolds in ℝ ⁿ ," Trans. Am. Math. Soc., 243, 115–122 (1978).

   MathSciNet  MATH  Google Scholar

2. T. J. Abatzoglou, "The Lipschitz continuity of the metric projection," J. Approx. Theory, 26, 212–218 (1979).

   Article  MathSciNet  MATH  Google Scholar

3. M. V. Balashov, "Proximal smoothness of a set with the Lipschitz metric projection," J. Math. Anal. Appl., 406, No. 1, 360–363 (2013).

   Article  MathSciNet  MATH  Google Scholar

4. M. V. Balashov, "Maximization of a function with Lipschitz continuous gradient," J. Math. Sci., 209, No. 1, 12–18 (2015).

   Article  MathSciNet  MATH  Google Scholar

5. M. V. Balashov, "About the gradient projection algorithm for a strongly convex function and a proximally smooth set," J. Convex Anal., 24, No. 2, 493–500 (2017).

   MathSciNet  MATH  Google Scholar

6. 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).

   Article  MathSciNet  MATH  Google Scholar

7. M. V. Balashov and 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, No. 3, 461–467 (2006).

   Article  MathSciNet  MATH  Google Scholar

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

   Article  MathSciNet  MATH  Google Scholar

9. M. V. Balashov and E. S. Polovinkin, "M-strongly convex subsets and their generating sets," Sb. Math., 191, No. 1, 27–64 (2000).

   Article  MathSciNet  MATH  Google Scholar

10. M. V. Balashov and D. Repovˇs, "Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order," J. Math. Anal. Appl., 377, No. 2, 754–761 (2011).

    Article  MathSciNet  MATH  Google Scholar

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

    MathSciNet  MATH  Google Scholar

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

    Article  MathSciNet  MATH  Google Scholar

13. 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  MATH  Google Scholar

14. J. W. Daniel, "The continuity of metric projection as function of data," J. Approx. Theory, 12, No. 3, 234–240 (1974).

    Article  MathSciNet  MATH  Google Scholar

15. N. Dunford and J. Schwartz, Linear Operators. Part I: General Theory, Interscience Publishers, New York (1958).

16. H. Frankowska and Ch. Olech, "R-convexity of the integral of the set-valued functions," in: Contributions to Analysis and Geometry, John Hopkins Univ. Press, Baltimore (1981), pp. 117–129.

17. J. Lindenstrauss, "On nonlinear projection in Banach spaces," Michigan Math. J., 11, No. 3, 263–287 (1964).

    Article  MathSciNet  MATH  Google Scholar

18. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I. Sequence Spaces, Springer, New York (1977).

19. Yu. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer, Dordrecht (2004).

    Book  MATH  Google Scholar

20. R. R. Phelps, "Convex sets and nearest points," Proc. Am. Math. Soc., 8, 790–797 (1957).

    Article  MathSciNet  MATH  Google Scholar

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

    Article  MathSciNet  MATH  Google Scholar

22. E. S. Polovinkin, "Strongly convex analysis," Sb. Math., 187, No. 2, 259–286 (1996).

    Article  MathSciNet  MATH  Google Scholar

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

24. S. B. Stechkin, "Approximative properties of sets in linear normed spaces," Rev. Math. Pures Appl., 8, No. 1, 5–18 (1963).

    MathSciNet  Google Scholar

25. J.-P. Vial, 'Strong and weak convexity of sets and functions," Math. Oper. Res., 8, No. 2, 231–259 (1983).

26. L. P. Vlasov, "Chebyshev sets and approximately convex sets," Math. Notes, 2, No. 2, 600–605 (1967).

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Authors and Affiliations

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

   M. V. Balashov

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Correspondence to M. V. Balashov.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 13–29, 2018.

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Balashov, M.V. The Lipschitz Property of the Metric Projection in the Hilbert Space. J Math Sci 250, 391–403 (2020). https://doi.org/10.1007/s10958-020-05022-6

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• Published: 17 September 2020

• Issue Date: October 2020

• DOI : https://doi.org/10.1007/s10958-020-05022-6

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