Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Dudov S. I., Osipcev M. A. Distance between strongly and weakly convex sets. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 4, pp. 434-441. DOI: 10.18500/1816-9791-2021-21-4-434-441, EDN: FHSXJE

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Distance between strongly and weakly convex sets

Dudov Sergey Ivanovitch, Saratov State University
Osipcev Mikhail Anatolievich, Saratov State University

The problem of finding the distance between non-intersecting strongly convex and weakly convex (as defined by J.-F. Viall) sets of finite-dimensional space is considered. Three alternative formalizations in the form of extremal problems are used in presenting the results. We obtained the necessary conditions for the solution of the problem taking into account the constants of strong and weak convexity of the sets and their other characteristics. Besides the condition of stationarity, they contain estimates of the growth of the objective functions in alternative formalizations of the problem as the argument moves away from the solution point. These growth estimates are further used to obtain both global and local solution conditions. In this case, the conditions of the local solution are accompanied by the indication of the radius of its neighborhood. The examples that show the importance of the conditions in the theorems being proved are given, as well as the accuracy of the formulas for the radii of the neighborhood of the local solution.

  1. Vasiliev F. P. Metody optimizatsii [Optimization Methods]. Moscow, MCCME, 2011. 624 p. (in Russian).
  2. Dem’yanov V. F., Vasil’ev L. V. Nondifferentiable Optimization. New York, Springer-Verlag, 1985. 452 p. (Russ. ed.: Moscow, Nauka, 1981. 384 p.).
  3. Vial J.-P. Strong and weak convexity of set and funtions. Mathematics of Operations Research, 1983, vol. 8, no. 2, pp. 231–259.
  4. Polovinkin E. S., Balashov M. V. Elementy vypuklogo i sil’no vypuklogo analiza [Elements of Convex and Strongly Convex Analysis]. Moscow, Fizmatlit, 2007. 440 p. (in Russian).
  5. Ivanov G. E. Slabo vypuklye mnozhestva i funktsii [Weakly Convex Sets and Functions]. Moscow, Fizmatlit, 2006. 352 p. (in Russian).
  6. Dudov S. I., Osiptsev M. A. Characterization of solutions of strong-weak convex programming problems. Sbornik: Mathematics, 2021, vol. 212, iss. 6, pp. 782–809. https://doi.org/10.1070/SM9431