Geometrical and analytical characterizations of positively homogeneous functions

V. V. Gorokhovik ; M. A. Trafimovich

Trudy Instituta matematiki, Tome 23 (2015) no. 1, pp. 27-54.

Résumé

The article provides an overview of results related to the positively homogeneous functions and the spaces formed by various classes of these functions. In the paper we review characteristic properties of positively homogeneous functions of a number of mostly used subspaces of the space of such functions. As such subspaces we consider the subspace of continuous functions, the subspace of Lipschitz functions, the subspace of difference sublinear functions and the subspace of piecewise functions. We give a description of algebraic structures with which endowed these subspaces and establish relationships between them. Along with known results we present a number of new ones with their proofs.

Bibliographie

[1] Abbasov M. E., Demyanov V. F., “Usloviya ekstremuma negladkoi funktsii v terminakh ekzosterov i koekzosterov”, Trudy Instituta matematiki i mekhaniki UrO RAN, 15, no. 3, 2009, 10–19 | MR | Zbl

[2] Aleksandrov A. D., “O poverkhnostyakh, predstavimykh raznostyu vypuklykh funktsii”, Izv. Kazakhskoi SSR. Ser. matematiki i mekhaniki, 1949, no. 3, 3–20

[3] Aleksandrov A. D., “Poverkhnosti, predstavimye raznostyu dvukh vypuklykh funktsii”, Dokl. AN SSSR, 62:4 (1950), 613–616

[4] Burbaki N., Obschaya topologiya. Ispolzovanie veschestvennykh chisel v obschei topologii. Funktsionalnye prostranstva. Svodka rezultatov. Slovar, M., 1975

[5] Gorokhovik V. V., Vypuklye i negladkie zadachi vektornoi optimizatsii, Minsk, 1990; 2-ое изд., М., 2012

[6] Gorokhovik V. V., “Geometricheskie i analiticheskie kharakteristiki kusochno-affinnykh otobrazhenii”, Trudy Instituta matematiki NAN Belarusi, 15:1 (2007), 22–32 | Zbl

[7] Gorokhovik V. V., Gorokhovik S. Ya., “Kriterii globalnoi epilipshitsevosti mnozhestv”, Izv. AN Belarusi. Ser. fiz.-mat. nauk, 1995, no. 1, 118–120 | MR

[8] Gorokhovik V. V., Zorko O. I., “Poliedralnaya kvazidifferentsiruemost veschestvennoznachnykh funktsii”, Dokl. AN Belarusi, 36:5 (1992), 393–397 | MR | Zbl

[9] Gorokhovik V. V., Zorko O. I., “Nevypuklye poliedralnye mnozhestva i funktsii i ikh analiticheskie predstavleniya”, Dokl. AN Belarusi, 39:1 (1995), 5–9

[10] Gorokhovik V. V., Zorko O. I., “Kusochno-affinnye otobrazheniya”, Operatory i operatornye uravneniya, Novocherkasskii gos. tekhn. un-t, Novocherkask, 1995, 3–18 | MR

[11] Gorokhovik V. V., Malashevich D. S., “Ob analiticheskom predstavlenii nevypuklykh mnogogrannykh mnozhestv i kusochno-affinnykh funktsii”, Analiticheskie metody analiza i differentsialnykh uravnenii, Trudy Instituta matematiki NAN Belarusi, 9, 2001, 45–48 | MR

[12] Gorokhovik V. V., Starovoitova M. A., “Kharakteristicheskie svoistva pryamykh ekzosterov razlichnykh klassov polozhitelno odnorodnykh funktsii”, Trudy Instituta matematiki NAN Belarusi, 19:2 (2011), 12–25

[13] Gorokhovik V. V., Trofimovich M. A., “Metod konvertirovaniya pryamykh ekzosterov nepreryvnykh polozhitelno odnorodnykh funktsii”, Dokl. NAN Belarusi, 57:5 (2013), 28–36

[14] Gorokhovik V. V., Trofimovich M. A., “Banakhovy prostranstva polozhitelno odnorodnykh funktsii v negladkom analize i optimizatsii”, Mezhdunar. konf. “Dinamika sistem i protsessy upravleniya”, posvyasch. 90 letiyu akadem. N. N. Krasovskogo, Tez. dokl., Institut matematiki i mekhaniki UrO RAN, Ekaterinburg, 2014, 55–57

[15] Demyanov V. F., Roschina V. A., “Obobschennye subdifferentsialy i ekzostery v negladkom analize”, Dokl. RAN, 416:1 (2007), 18–21 | MR | Zbl

[16] Demyanov V. F., Rubinov A. M. (otv. red.), “Elementy kvazidifferentsialnogo ischisleniya”, Negladkie zadachi teorii optimizatsii i upravleniya, Izd-vo Leningr. un-ta, L., 1982, 5–127 | MR

[17] Demyanov V. F., Rubinov A. M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1990

[18] Zalgaller V. A., “O predstavlenii funktsii dvukh peremennykh raznostyu vypuklykh funktsii”, Vestnik Leningr. un-ta. Ser. Matematika, mekhanika, 18:1 (1963), 44–45 | MR

[19] Zalgaller V. A., “O predstavlenii funktsii neskolkikh peremennykh raznostyu vypuklykh funktsii”, Zapiski nauchnogo seminara POMI, 246, 1997, 36–65 | Zbl

[20] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988

[21] Kutateladze S. S., Rubinov A. M., Dvoistvennost Minkovskogo i ee prilozheniya, Nauka, Novosibirsk, 1976

[22] Makarov V. L., Rubinov A. M., Matematicheskaya teoriya ekonomicheskoi dinamiki i ravnovesiya, Nauka, M., 1973

[23] Polovinkin E. S., Mnogoznachnyi analiz i differentsialnye vklyucheniya, Fizmatlit, M., 2014

[24] Prudnikov I. A., “Neobkhodimye i dostatochnye usloviya predstavimosti polozhitelno odnorodnoi funktsii trekh peremennykh v vide raznosti dvukh vypuklykh funktsii”, Izv. RAN. Ser. matematicheskaya, 56:5 (1992), 1114–1126

[25] Pshenichnyi B. N., Vypuklyi analiz i ekstremalnye zadachi, Nauka, M., 1980

[26] Reshetnyak Yu. G., Kurs matematicheskogo analiza, v. 1, kn. 2, Nauka. Izd-vo In-ta matematiki SO RAN, Novosibirsk, 1999

[27] Rokafellar R. T., Vypuklyi analiz, Mir, M., 1973

[28] Strekalovskii A. S., “O minimizatsii raznosti vypuklykh funktsii na dopustimom mnozhestve”, Zhurn. vychislitelnoi matematiki i matematicheskoi fiziki, 43:3 (2003), 399–409 | MR | Zbl

[29] Subbotin A. I., Minimaksnye neravenstva i uravneniya Gamiltona–Yakobi, Nauka, M., 1991

[30] Subbotin A. I., Shagalova L. G., “Kusochno-lineinoe reshenie zadachi Koshi dlya uravneniya Gamiltona–Yakobi”, Dokl. RAN, 325:5 (1992), 932–936 | Zbl

[31] Ushakov V. N., Lakhtin A. S., Lebedev P. D., “Optimizatsiya khausdorfova rasstoyaniya mezhdu mnozhestvami v evklidovom prostranstve”, Trudy Instituta matematiki i mekhaniki UrO RAN, 20, no. 3, 2014, 291–308

[32] Abbasov M. E., Demyanov V. F., “Proper and adjoint exhausters in nonsmooth analysis: optimality conditions”, J. of Global Optim., 56:2 (2013), 569–585 | DOI | MR | Zbl

[33] Abbasov M. E., Demyanov V. F., “Adjoint Coexhausters in Nonsmooth Analysis and Extremality Conditions”, J. Optimization Theory and Applications, 156:3 (2013), 535–553 | DOI | MR | Zbl

[34] Aliprantis Ch. D., Tourky R., Cones and duality, American Mathematical Society, Providence, Rhode Island, 2007 | MR | Zbl

[35] Bank B., Guddat J., Klatte D., Kummer B., Tummer K., Non-linear parametric optimization, Akademie-Verlag, Berlin, 1982

[36] Bartels S. G., Pallaschke D., “Some remarks on the space of differences of sublinear functions”, Applicationes mathematicae, 22:3 (1994), 419–426 | MR | Zbl

[37] Bauschke H. H., Combetters P. L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011 | MR | Zbl

[38] Castellani M., “A dual representation for proper positively homogeneous functions”, J. of Global Optim., 16:4 (2000), 393–400 | DOI | MR | Zbl

[39] Castellani M., “Dual representation of classes of positively homogeneous functions”, Quasidifferentiability and Related Topics, Nonconvex Optim. Appl., 43, eds. V. Demyanov, A. Rubinov, Kluwer Acad. Publ., Dordrecht, 2000, 73–84 | DOI | MR | Zbl

[40] Clarke F. H., Optimization and Nonsmooth Analysis, Wiley, New York, 1983 | MR | Zbl

[41] Kruger A. Y., “On Frechet subdifferentials”, Journal of Mathematical Sciences, 116:3 (2003), 3325–3358 | DOI | MR | Zbl

[42] Demyanov V. F., “Exhausters of a positively homogeneous function”, Optimization, 45:1 (1999), 13–29 | DOI | MR | Zbl

[43] Demyanov V. F., “Exhausters and convexificators — new tools in nonsmooth analysis”, Quasidifferentiability and Related Topics, Nonconvex Optim. Appl., 43, eds. V. Demyanov, A. Rubinov, Kluwer Acad. Publ., Dordrecht, 2000, 85–137 | DOI | MR | Zbl

[44] Demyanov V. F., Roshchina V. A., “Optimality conditions in terms of upper and lower exhausters”, Optimization, 55:5–6 (2006), 525–540 | DOI | MR | Zbl

[45] Demyanov V. F., Roshchina V. A., “Exhausters and subdifferentials in nonsmooth analysis”, Optimization, 57:1 (2008), 41–56 | DOI | MR | Zbl

[46] Demyanov V. F., Roshchina V. A., “Exhausters, optimality conditions and related problems”, J. of Global Optimization, 40:1 (2008), 71–85 | DOI | MR | Zbl

[47] Demyanov V. F., Rubinov A. M., Quasidifferential calculus, Optimization Software Inc., Publications Division, New York, 1986 | MR | Zbl

[48] Demyanov V. F., Rubinov A. M., Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt am Main, 1995 | MR | Zbl

[49] Demyanov V. F., Rubinov A. M. (eds.), Quasidifferentiability and Related Topics, 43, Kluwer Academic Publishers, Dordrecht, 2000, 297–327 | DOI | MR

[50] Demyanov V. F., Rubinov A. M., “Exhaustive families of approximations revisited”, From convexity to nonconvexity, Nonconvex Optim. Appl., 55, Kluwer Acad. Publ., 2001, 43–50 | DOI | MR | Zbl

[51] Demyanov V. F., Ryabova J. A., “Exhausters, coexhausters and convertors in nonsmooth analysis”, Discrete and Continuous Dynamical Systems, 31:4 (2011), 1273–1292 | DOI | MR | Zbl

[52] Diamond P., Kloeden P., Rubinov A., Vladimirov A., Three metrics in the space of compact convex sets, Research Report 4/97, School of Information Technology and Mathematical Sciences. University of Ballarat, 1997

[53] Diamond P., Kloeden P., Rubinov A., Vladimirov A., “Comparative properties of three metrics in the space of compact convex sets”, Set-Valued Analysis, 5 (1997), 267–289 | DOI | MR | Zbl

[54] Duda J., Vesely L., Zajicek L., “On d.c. functions and mappings”, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 111–138 | MR | Zbl

[55] Gorokhovik V. V., “Piecewise affine mappings: Geometrical properties and analytical representations”, Analiticheskie metody analiza i differentsialnykh uravnenii, Izd. tsentr BGU, Minsk, 2012, 59–68

[56] Gorokhovik V. V., Zorko O. I., “Piecewise affine functions and polyhedral sets”, Optimization, 31:2 (1994), 209–221 | DOI | MR | Zbl

[57] Grzybowski J., Pallaschke D., Urbanski R., “Reduction of finite exhausters”, J. of Global Optimization, 46:4 (2010), 589–601 | DOI | MR | Zbl

[58] Grzybowski J., Pallaschke D., Urbanski R., “Demyanov difference in infinite dimensional spaces”, Constructive Nonsmooth Analysis and Related Topics, Springer Optimization and Its Applications, 87, Springer, New York, 2014, 13–24 | DOI | MR | Zbl

[59] Ha T. X. D., “Banach spaces of d.c. functions and quasidifferentiable functions”, Acta Math. Vietnam, 13:2 (1988), 55–70 | MR | Zbl

[60] Hiriart-Urruty J.-B., “Generalized differentiability, duality and optimization for problems dealing with differences of convex functions”, Convexity and Duality in Optimization, Lecture Notes in Economics and Mathematical Systems, 256, 1985, 37–70 | DOI | MR | Zbl

[61] Hiriart-Urruty J.-B., Lemarechal C., Convex Analysis and Minimization Algorithms, v. I, Fundamentals, Springer-Verlag, Berlin–Heidelberg, 1993 | MR

[62] Kripfhanz A., Schulze R., “Piecewise-affine functions as the difference of two convex functions”, Optimization, 29:1 (1987), 23–29 | DOI | MR

[63] Melzer D., “On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions”, Math. Program. Study, 29 (1986), 118–134 | DOI | MR | Zbl

[64] Michel P., Penot J.-P., “Calcul sous-differentiel pour des fonctions lipschitziennes et non-lipschitziennes”, Comptes Rendus de l'Academie des Sciences de Paris, 298 (1984), 684–687 | MR

[65] Michel P., Penot J.-P., “A generalized derivative for calm and stable functions”, Differential and Integral Equations, 5:2 (1992), 189–196 | MR

[66] Michel P., Penot J.-P., “Second-order moderate derivatives”, Nonlinear Analysis: Theory, Methods and Applications, 22:7 (1994), 809–822 | DOI | MR

[67] Minkowski H., Geometrie der Zhalen, Teubner, Leipzig, 1910

[68] Mordukhovich B. S., Variational Analysis and Generalized Differentiation, v. I, Basic Theory, Springer-Verlag, Berlin, 2006

[69] Ovchinnikov S., “Max-Min representation of piecewise linear functions”, Contributions to Algebra and Geometry, 43 (2002), 297–302 | MR | Zbl

[70] Pallaschke D., Rolewicz S., Foundations of Mathematical Optimization, Kluwer Acad. Publ., Dordrecht, 1997 | MR | Zbl

[71] Pallaschke D., Urbanski R., “Minimal pairs of convex sets, with applications to quasidifferental calculus”, Quasidifferentiability and Related Topics, Nonconvex Optim. Appl., eds. V. Demyanov, A. Rubinov, Kluwer Acad. Publ., Dordrecht, 2000, 297–327 | MR

[72] Pallaschke D., Urbanski R., Pairs of Compact Convex Sets. Fractional Arithmetic with Convex Sets, Kluwer Acad. Publ., Dordrecht, 2002 | MR | Zbl

[73] Polyakova L. N., “On global properties of d.c. functions”, From Convexity to Nonconvexity, Nonconvex Optimization and Its Applications, 55, 2001, 223–231 | DOI | MR | Zbl

[74] Roberts A. W., Varberg D. E., Convex functions, Academic Press, New York–London, 1973 | MR | Zbl

[75] Robinson S., “Some continuity properties of polyhedral multifunctions”, Mathematical Programming Study, 14 (1981), 206–214 | DOI | MR | Zbl

[76] Rockafellar R. T., Wets J.-B., Variational analysis, Springer, Berlin, 1998 | MR | Zbl

[77] Rubinov A. M., Akhundov I. S., “Difference of compact sets in the sense of Demyanov and its applications to non-smooth analysis”, Optimization, 23:3 (1992), 179–188 | DOI | MR | Zbl

[78] Rubinov A. M., Vladimirov A. A., “Difference of convex compacta and metric spaces of convex compacta with applications: a survey”, Quasidifferentiability and Related Topics, Nonconvex Optim. Appl., eds. V. Demyanov, A. Rubinov, Kluwer Acad. Publ., Dordrecht, 2000, 263–296 | DOI | MR | Zbl

[79] Shapiro A., “Quasidifferential calculus and first-order optimality conditions in nonsmooth optimization”, Math. Program. Study, 29 (1986), 59–68 | MR

[80] Stein J. D., “Functions satisfying lipschitz conditions”, Michigan Math. J., 16:4 (1968), 385–396 | MR

[81] Tuy H., Convex Analysis and Global Optimization, Kluwer Acad. Publ., Dordrecht, 1998 | MR | Zbl

[82] Uderzo A., “Convex approximators, convexificators and exhausters: applications to constrained extremum problems”, Quasidifferentiability and Related Topics, eds. Demyanov V. F., Rubinov A. M., Kluwer Academic Publishers, Dordrecht, 2000, 297–327 | DOI | MR | Zbl

[83] Vesely L., Zajicek L., “Delta-convex mappings between Banach spaces and applications”, Dissertationes Mathematicae, 289, 1989, 52 pp. | MR | Zbl