Abstract Convexity of Functions with Respect to the Set of Lipschitz (Concave) Functions

V. V. Gorokhovik ; A. S. Tykoun

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 3, pp. 73-85

Résumé

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. The notion of regular ${\mathcal H}$-convex functions is introduced. These are functions representable as the upper envelopes of the set of their maximal (with respect to the pointwise order) ${\mathcal H}$-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 both a necessary condition and a sufficient one 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 lower bounded by a Lipschitz function, the set of 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 Brøndsted–Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the wider class of lower semicontinuous functions and go back to the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set, which is one of the most important results of classical convex analysis.

Détail

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

Bibliographie

[1] Singer I., Abstract Convex Analysis, Wiley-Interscience Publ., N Y, 1997, 491 pp. | MR | Zbl

[2] Soltan V.P., Vvedenie v aksiomaticheskuyu teoriyu vypuklosti, Shtiintsa, Kishinev, 1984, 223 pp.

[3] Kutateladze C.S., Rubinov A.M., “Dvoistvennost Minkovskogo i ee prilozheniya”, Uspekhi mat. nauk, 27:3(165) (1972), 127–176 | MR | Zbl

[4] Kutateladze C.S., Rubinov A.M., Dvoistvennost Minkovskogo i ee prilozheniya, Nauka. Sib. otd-nie, Novosibirsk, 1976, 254 pp. | MR

[5] Pallaschke D., Rolewicz S., Foundations of mathematical optimization (Convex analysis without linearity), Kluwer Acad. Publ., Dordrecht, 1997, 596 pp. | DOI | MR | Zbl

[6] Rubinov A.M., Abstract convexity and global optimization, Kluwer Acad. Publ., Dordrecht, 2000, 490 pp. | MR | Zbl

[7] Ekland I., Temam R., Vypuklyi analiz i variatsionnye problemy, Mir, M., 1979, 399 pp.

[8] Brøndsted A., Rockafellar R.T., “On the subdifferentiability of convex functions”, Proc. Amer. Math. Soc., 16:4 (1965), 605–611 | DOI | MR

[9] Polovinkin E.S., Balashov M.V., Elementy vypuklogo i silno vypuklogo analiza, Fizmatlit, M., 2004, 416 pp.

[10] Bishop E., Phelps R.R., “The support functionals of convex sets”, Convexity, Proc. of Symposia in Pure Mathematics, VII, ed. V. Klee, American Math. Soc., Providence, Rhode Island, 1963, 27–35 | DOI | MR

[11] Gorokhovik V.V., “O predstavlenii polunepreryvnykh sverkhu funktsii, opredelennykh na beskonechnomernykh normirovannykh prostranstvakh, v vide nizhnikh ogibayuschikh semeistv vypuklykh funktsii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 23:1 (2017), 88–102 | DOI | MR

[12] Gorokhovik V.V., “Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials”, Published online: 09 Sep, Optimization. J. Math. Programming and Operations Research, 2018 | DOI | MR

[13] Gorokhovik V.V., “Demyanov–Rubinov subdifferentials of real-valued functions”, Constructive Nonsmooth Analysis and Related Topics, Dedicated to the memory of V.F. Demyanov(CNSA), ed. Polyakova, Institute of Electrical and Electronics Engineers (IEEE), Piscataway, New Jersey, 2017, 122–125, Proc. Sonf. | DOI | MR

[14] Ekeland I., “Nonconvex minimization problems”, Bull. Amer. Math. Soc., 1:3 (1979), 432–467 | DOI | MR

[15] Penot J.P., Calculus without derivatives, Springer, N Y, 2013, 524 pp. | DOI | MR | Zbl

[16] Magaril-Ilyaev G.G., Tikhomirov V.M., Vypuklyi analiz i ego prilozheniya, Editorial URSS, M., 2003, 176 pp. | MR