Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

References

• Bourbaki N. General topology: chapter 5-10. Berlin: Springer; 1989.
   Google Scholar
• Ekeland I, Temam R. Convex analysis and variational problems. Amsterdam: North-Holland; 1976.
   Google Scholar
• Kutateladze SS, Rubinov AM. Minkowski duality and its applications. Russian Math Surveys. 1972;27(3):137–192. doi: 10.1070/RM1972v027n03ABEH001380
   Google Scholar
• Kutateladze SS, Rubinov AM. Minkowski duality and its applications. Novosibirsk: Nauka; 1976. Russian.
   Google Scholar
• Glover BM, Rubinov AM, Craven BD. Solvability theorems involving inf-convex functions. J Math Anal Appl. 1995;191:305–330. doi: 10.1006/jmaa.1995.1132
   Web of Science ®Google Scholar
• Rubinov AM. Supremal generators of spaces of homogeneous functions. In: Eberhard A, Glover B, Ralph D, editors. Progress in optimization: contribution from Australasia Dordrecht: Kluwer Academic Publishers; 1999. p. 91–100.
   Google Scholar
• Rubinov AM. Abstract convexity and global optimization. Dordrecht: Kluwer Academic Publishers; 2000.
   Google Scholar
• Pallaschke D, Rolewicz S. Foundations of mathematical optimization (Convex analysis without linearity). Dordrecht: Kluwer Academic Publishers; 1997.
   Google Scholar
• Singer I. Abstract convex analysis. New York (NY): Wiley-Interscience Publication; 1997.
   Google Scholar
• Demyanov VF, Rubinov AM. Elements of quasidifferential calculus. In: Demyanov VF, Rubinov AM, editors. Nonsmooth problems of optimization theory and control. Leningrad: Leningrad University Press; 1982. p. 5–127. Russian.
   Google Scholar
• Demyanov VF, Rubinov AM. Osnovy negladkogo analiza i kvazidifferentsial'noe ischislenie [Foundations of nonsmooth analysis and quasidifferential calculus]. Moscow: Nauka Publ.; 1990. Russian.
   Google Scholar
• Demyanov VF, Rubinov AM. Constructive nonsmooth analysis. Frankfurt: Verlag Peter Lang; 1995.
   Google Scholar
• Uderzo A. Convex approximators, convexificators and exhausters: applications to constrained extremum problems. In: Demyanov VF, Rubinov AM, editors. Quasidifferentiability and related topics. Dordrecht: Kluwer Academic Publishers; 2000. p. 297–327.
   Google Scholar
• Gorokhovik VV. On the representation of upper semicontinuous functions defined on infinite-dimensional normed spaces as lower envelopes of families of convex functions. Trudy Inst Mat Mekh UrO RAN. 2017;23(1):88–102. doi: 10.21538/0134-4889-2017-23-1-88-102
   Google Scholar
• Gorokhovik VV, Starovoitava MA. Characteristic properties of primal exhausters for various classes of positively homogeneous functions. Proc Inst Math Natl Acad Sci Belarus. 2011;19(2):12–25. Russian.
   Google Scholar
• Gorokhovik VV, Trafimovich MA. Positively homogeneous functions revisited. J Optimiz Theory Appl. 2016;171(2):481–503. doi: 10.1007/s10957-016-0891-4
   Web of Science ®Google Scholar
• Demyanov VF. Exhausters of a positively homogeneous function. Optimization. 1999;45:13–29. doi: 10.1080/02331939908844424
   Google Scholar
• Demyanov VF. Exhausters and convexificators — new tools in nonsmooth analysis. In: Demyanov VF, Rubinov AM, editors. Quasidifferentiability and related topics. Dordrecht: Kluwer Academic Publishers; 2000. p. 85–137.
   Google Scholar
• Rockafellar RT. Convex analysis. Princeton (NJ): Princeton University Press; 1970.
   Google Scholar
• Penot J-P. Calculus without derivatives. New York (NY): Springer; 2013.
   Google Scholar
• Ioffe AD. Variational analysis of regular mappings: theory and applications. Cham: Springer International Publishing AG; 2017.
   Google Scholar
• Rockafellar RT, Wets RJ-B. Variational analysis. Berlin: Springer; 1998.
   Google Scholar
• Mordukhovich BS. Variational analysis and generalized differentiation I: basic theory. Berlin: Springer; 2005.
   Google Scholar
• Hiriart-Urruty B, Lemarechal C. Convex analysis and minimization algorithms I: fundamentals. Berlin: Springer; 1993.
   Google Scholar
• Hiriart-Urruty JB, Lemarechal C. Convex analysis and minimization algorithms II: advanced theory and bundle methods. Berlin: Springer; 1993.
   Google Scholar
• Penot J-P. Calcul sous-differential et optimization. J Funct Anal. 1978;27:248–276. doi: 10.1016/0022-1236(78)90030-7
   Web of Science ®Google Scholar
• Kruger AY. On Fréchet subdifferentials. J Math Sci. 2003;116(3):3325–3358. doi: 10.1023/A:1023673105317
   Google Scholar
• Bonnans JF, Shapiro A. Perturbation analysis of optimization problems. Berlin: Springer; 2000.
   Google Scholar
• Gorokhovik VV. Demyanov-Rubinov subdifferentials of real-valued functions. In: Polyakova LN, editor. 2017 Constructive Nonsmooth Analysis and Related Topics (Dedicated to the memory of V.F. Demyanov) (CNSA); 2017 May 22–27; Russia, Saint-Petersburg. Proceedings. Institute of Electrical and Electronics Engineers (IEEE); 2017. p. 122-125. DOI:10.1109/cnsa.2017.7973962
   Google Scholar
• Kelley JL. General topology. New York (NY): Springer-Verlag; 1975.
   Google Scholar
• Valentine AF. Convex sets. New York (NY): McGraw-Hill Book Company; 1964.
   Google Scholar
• Smith CR. A characterization of star-shaped sets. Amer Math Monthly. 1968;75:386. doi: 10.2307/2313423
   Web of Science ®Google Scholar
• Gorokhovik VV, Zorko OI. Piecewise affine functions and polyhedral sets. Optimization. 1994;31:209–221. doi: 10.1080/02331939408844018
   Google Scholar
• Castellani M. A dual representation for proper positively homogeneous functions. J Global Optim. 2000;16(4):393–400. doi: 10.1023/A:1008394516838
   Web of Science ®Google Scholar
• Castellani M. Dual representations of positively homogeneous functions. In: Demyanov VF, Rubinov AM, editors. Quasidifferentiability and related topics. Dordrecht: Kluwer Academic Publishers; 2000. p. 73–84.
   Google Scholar
• Bauschke HH, Combetters PL. Convex analysis and monotone operator theory in Hilbert spaces. New York (NY): Springer; 2011.
   Google Scholar
• Hiriart-Urruty JB. Extension of Lipschitz functions. J Math Anal Appl. 1980;77:539–554. doi: 10.1016/0022-247X(80)90246-2
   Web of Science ®Google Scholar
• Gorokhovik VV, Trafimovich MA. On methods for converting exhausters of positively homogeneous functions. Optimization. 2016;65(3):589–608. doi: 10.1080/02331934.2015.1032285
   Web of Science ®Google Scholar
• Borwein JM, Vanderwerff JD. Convex functions: constructions, characterizations and counterexamples. Cambridge: Cambridge University Press; 2010.
   Google Scholar
• Demyanov VF, Roshchina V. Generalized subdifferentials and exhausters in nonsmooth analysis. Dokl Math. 2007;76(2):652–655. doi: 10.1134/S1064562407050043
   Web of Science ®Google Scholar
• Demyanov VF, Roshchina V. Exhausters and subdifferentials in nonsmooth analysis. Optimization. 2008;57(1):41–56. doi: 10.1080/02331930701778825
   Web of Science ®Google Scholar
• Clarke F. Optimization and nonsmooth analysis. New York: Wiley; 1983.
   Google Scholar
• Michel P, Penot J-P. Calcul sous-differential pour les fonctions lipschitzienness et non-lipschitziennes. C R Acad Sci Paris Ser I. 1984;298:269–272.
   Google Scholar
• Baier R, Farkhi E, Roshchina V. Directed subdifferentiable functions and the directed subdifferential without DC structure. J Optim Theory Appl. 2014;160:391–414. doi: 10.1007/s10957-013-0401-x
   Web of Science ®Google Scholar