Abstract

It is shown that for some classes of functions all epiderivatives and subdifferentials of F.H.Clarke type and P.Michel–J.-P.Penot type and others coincide. Several rules of calculation of subdifferentials for the difference of two convex functions are obtained. Some examples are considered.

Author information

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

Evgeny Sergeevich Polovinkin

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

polovinkin@mail.mipt.ru

Keywords

  • Set-valued map
  • tangent cone
  • convex function
  • semiregular function
  • Lipschitz continuity
  • epiderivatives
  • subdifferentials

Mathematics Subject Classification

49K22, 49K24; 49J22, 49J24

References

  1. J.-P. Aubin: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Adv. Math. Suppl. Stud. 7A (1981) 160–272.Google Scholar
  2. J.-P. Aubin, H. Frankovska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
  3. F. H. Clarke: Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975) 247–262.https://doi.org/10.1090/S0002-9947-1975-0367131-6Google Scholar
  4. F. H. Clarke: Optimization and Nonsmooth Analysis, John Wiley and Sons, New York (1983).Google Scholar
  5. V. F. Demyanov, A. M. Rubinov: Nonsmooth Analysis and Quasidifferential Calculus, Nauka, Moscow (1990) (in Russian).Google Scholar
  6. E. S. Polovinkin: Differential inclusions with measurable-pseudo-Lipschitz right-hand side, Proc. Steklov Inst. Math. 283 (2013) 116–135.https://doi.org/10.1134/S0081543813080099Google Scholar
  7. E. S. Polovinkin: On necessary conditions for optimality of solutions of differential inclusions on the interval, in: Sovr. Matem. v Fiz.-Teh. Zadachah, Izd. MFTI, Moscow (1986) 87–94 (in Russian).Google Scholar
  8. E. S. Polovinkin: On some properties of derivatives of set-valued maps, Trudy MFTI 4(4) (2012) 141–154 (in Russian).Google Scholar
  9. E. S. Polovinkin: Theory of Set-Valued Maps, Izd. MFTI, Moscow (1983) (in Russian).Google Scholar
  10. E. S. Polovinkin, M. V. Balashov: Elements of Convex and Strongly Convex Analysis, 2nd Ed., Fizmatlit, Moscow (2007) (in Russian).Google Scholar
  11. B. N. Pshenichny: Convex Analysis and Extremal Problems, Nauka, Moscow (1980) (in Russian).Google Scholar
  12. B. M. Makarov, A. N. Podkorytov: Lectures on Real Analysis, BHV-Petersburg, St. Petersburg (2011) (in Russian).Google Scholar
  13. P. Michel, J.-P. Penot: Calcul sous-différentiel pour les fonctions lipschitziennes et non-lipschitziennes, C. R. Acad. Sci., Paris, Sér. I 298 (1984) 269–272.Google Scholar

Suggested citation

E. S. Polovinkin. “Calculation of Subdifferentials for the Difference of Two Convex Functions.” Journal of Convex Analysis 24 (2017), No. 1, 287–303. https://doi.org/10.68381/jca24019

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