V.V. Gorokhovik, A.S. Tykоun. Abstract convexity of functions with respect to the set of Lipschitz (concave) functions ... P. 73-85

The paper is devoted to the abstract ${\mathcal H}$-convexity of functions (where ${\mathcal H}$ is a given set of elementary functions) and its realization in the cases when ${\mathcal H}$ is the space of Lipschitz functions or the set of Lipschitz concave functions. We introduce the notion of regular ${\mathcal H}$-convex functions. These are functions representable as the upper envelopes of the set of their maximal (with respect to the pointwise ordering) ${\mathcal H}$-minorants. As a generalization of the global subdifferential of a convex function, we introduce the set of maximal support ${\mathcal H}$-minorants at a point and the set of lower ${\mathcal H}$-support points. Using these tools, we formulate necessary as well as sufficient conditions for global minima of nonsmooth functions. In the second part of the paper, the abstract notions of ${\mathcal H}$-convexity are realized in the specific cases when functions are defined on a metric or normed space $X$ and the set of elementary functions is the space ${\mathcal L}(X,{\mathbb{R}})$ of Lipschitz functions or the set ${\mathcal L}\widehat{C}(X,{\mathbb{R}})$ of Lipschitz concave functions, respectively. An important result of this part of the paper is the proof of the fact that, for a lower semicontinuous function bounded from below by a Lipschitz function, the set of its lower ${\mathcal L}$-support points and the set of lower ${\mathcal L}\widehat{C}$-support points coincide and are dense in the effective domain of the function. These results extend the known Brondsted-Rockafellar theorem on the existence of a subdifferential of convex lower semicontinuous functions to the wider class of lower semicontinuous functions and go back to the Bishop-Felps theorem on the density of support points in the boundary of a closed convex set, which is one of most important results of classical convex analysis.

Keywords: abstract convexity, support minorants, support points, global minimum, semicontinuous functions, Lipschitz functions, concave Lipschitz functions, density of support points

Received April 20, 2019

Revised May 15, 2019

Accepted May 20, 2019

Funding Agency: This work was supported by the National Program for Scientific Research of the Republic of Belarus for 2016–2020 “Convergence 2020” (project no. 1.4.01).

Valentin Vikent’evich Gorokhovik, Dr. Phys.-Math. Sci., Corresponding Member of NAS of Belarus, Prof., Institute of Mathematics, The National Academy of Sciences of Belarus, Minsk, 220072 Belarus, e-mail: gorokh@im.bas-net.by

Alexander Stanislavovich Tykoun, Cand. Sci. (Phys.-Math.), Belarusian State University, Minsk, 220030 Belarus, e-mail: tykoun@bsu.by

REFERENCES

1.   Singer I. Abstract convex analysis. N Y: Wiley-Interscience Publ., 1997, 491 p. ISBN: 978-0471160151 .

2.   Soltan V.P. Vvedenie v aksiomaticheskuyu teoriyu vypuklosti [Introduction to axiomatic convexity theory]. Kishinev: Shtiintsa Publ., 1984, 223 p.

3.   Kutateladze S.S., Rubinov A.M. Minkowski duality and its applications. Russian Math Surveys, 1972, vol. 27, no. 3, pp. 137–191. doi: 10.1070/RM1972v027n03ABEH001380 

4.   Kutateladze S.S., Rubinov A.M. Dvoistvennost’ Minkovskogo i ee prilozheniya [Minkowski Duality and Its Applications]. Novosibirsk: Nauka Publ., 1976, 254 p.

5.   Pallaschke D., Rolewicz S. Foundations of mathematical optimization [Convex analysis without linearity]. Dordrecht: Kluwer Acad. Publ., 1997, 596 p. DOI: 10.1007/978-94-017-1588-1 

6.   Rubinov A.M. Abstract convexity and global optimization. Dordrecht: Kluwer Acad. Publ., 2000, 490 p. ISBN: 978-1-4757-3200-9 .

7.   Ekeland I, Temam R. Convex analysis and variational problems. Amsterdam: North-Holland, 1976, 402 p. Translated to Russian under the title Vypuklyi analiz i variatsionnye problemy. Moscow: Mir Publ., 1979, 399 p.

8.   Brondsted A., Rockafellar R.T. On the subdifferentiability of convex functions. Proc. Amer. Math. Soc., 1965, vol. 16, no. 4, pp. 605–611. doi: 10.2307/2033889 

9.   Polovinkin E.S., Balashov M.V. Elementy vypuklogo i sil’no vypuklogo analiza [Elements of convex and strongly convex analysis]. Moscow: Fizmatlit Publ., 2004, 416 p. ISBN: 5-9221-0499-3 .

10.   Bishop E., Phelps .R. The support functionals of convex sets. In: V. Klee (ed.), Convexity: Proc. of Symposia in Pure Mathematics, vol. VII. Providence, RI: American Math. Soc., 1963, pp. 27–35. doi: 10.1090/pspum/007/0154092 

11.   Gorokhovik V.V. On the representation of upper semicontinuous functions defined on infinite-dimensional normed spaces as lower envelopes of families of convex functions. Trudy Inst. Mat. i Mekh. UrO RAN, 2017, vol. 23, no. 1, pp. 88–102 (in Russian). doi: 10.21538/0134-4889-2017-23-1-88-102 

12.   Gorokhovik V.V. Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials. Optimization. J. Math. Programming and Operations Research. Published online: 09 Sep 2018. doi: 10.1080/02331934.2018.1518446 

13.   Gorokhovik V.V. Demyanov–Rubinov subdifferentials of real-valued functions. In: Polyakova L.N. (ed). Constructive Nonsmooth Analysis and Related Topics (Dedicated to the memory of V.F. Demyanov) (CNSA), Proc. Conf., New Jersey: Institute of Electrical and Electronics Engineers (IEEE), 2017, pp. 122–125. doi: 10.1109/cnsa.2017.7973962 

14.   Ekeland I. Nonconvex minimization problems. Bull. Amer. Math. Soc., 1979, vol. 1, no. 3, pp. 443–474.

15.   Penot J.P. Calculus without Derivatives. N Y: Springer, 2013, 524 p. doi: 10.1007/978-1-4614-4538-8 

16.   Magaril-Ilyaev G.G., Tikhomirov V.M. Convex analysis: Theory and applications. N Y: American Math. Soc., 2003, 183 p. ISBN: 978-0821835258 . Original Russian text published in Magaril-Il’yaev G.G., Tikhomirov V.M. Vypuklyi analiz i ego prilozheniya. Moscow: Editorial URSS, 2003, 176 p.

Cite this article as: V.V. Gorokhovik, A.S. Tykoun. Abstract convexity of functions with respect to the set of Lipschitz (concave) functions, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 3, pp. 73–85.