φ-regular functions in Asplund spaces

• Autorzy: Ngai H. V. , Thera M.
• Języki publikacji: EN

Abstrakty

EN

We introduce in the context of Asplund spaces, a new class of (φ-regular functions. This new concept generalizes the one of prox-regularity introduced by Poliquin & Rockafellar (2000) in Rn and extended to Banach spaces by Bernard & Thibault (2004). In particular, the class of φ-regular functions includes all lower semi-continuous convex functions, all lower-C2 functions, and convexly C1,0-composite functions as well. Geometrical and subdifferential characterizations for this new class of functions are investigated.

Słowa kluczowe

EN

φ-regularity; Frechet subdifferential; limiting sub-differential; prox-regularity; lower-C2 function; convexly composite function; Moreau envelope; regularization

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2007
• Tom: Vol. 36, no 3
• Strony: 755--774
• Opis fizyczny: Bibliogr. 52 poz.

Twórcy

autor

Ngai H. V.

autor

Thera M.

• Department of Mathematics, Pedagogical University of Quynhon, 170 An Duong Vuong, Qui Nhon, Vietnam

Bibliografia

• ASPLUND, E. (1968) Fréchet differentiability of convex functions. Acta Math., 121, 31-47.
• ATTOUCH, H. (1984) Variational convergence of functions and operators. Pitman, London.
• AUBIN, J.-P. and FRANKOWSKA, H. (1990) Set-Valued Analysis. Birkhäuser, Boston.
• AUSSEL, D., DANIILIDIS, A. and THIBAULT, L. (2005) Subsmooth sets: functional characterizations and related concepts. Trans. Am. Math. Soc. 357, 1275-1301.
• BENYAMINI, Y. and LINDENSTRAUSS, J. (2000) Geometric Nonlinear Functional Analysis. Amer. Math. Soc. Colloquium Publications 48, Providence.
• BERNARD, F. and THIBAULT, L. (2004) Prox-regularity of functions and sets in Banach spaces. Set-Valued Anal. 12, 25-47.
• BERNARD, F. and THIBAULT, L. (2005) Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303, 1-14.
• BORWEIN, J.M. and FITZPATRlCK, S. (1989) Existence of nearest points in Banach spaces. Can. J. Math. (XLI) 4, 702-720.
• BORWEIN, J.M and GILES, J.R. (1987) The proximal normal formula in Banach space. Trans. Amer. Math. Soc. 302 (1), 371-381.
• CLARKE, F.H. (1983) Optimization and Nonsmooth Analysis. Wiley Interscience, New York.
• CLARKE, F.H., STERN, R.J. and WOLENSKI, P.R. (1995) Proximal smoothness and the lower-C2 property. J. Convex Anal. 2 (1,2), 117-144.
• COLOMBO, G. and GONCHAROV, V. (2001) Variational inequalities and regularity properties of closed sets in Hilbert spaces. J. Convex Anal. 8, 197-221.
• CORREA, R., JOFRE, A. and THIBAULT, L. (1994) Subdifferential monotonicity as a characterization of convex functions. Numer. Fund. Anal. Optim. 15, 531-535.
• COMBARI, C., POLIQUIN, R.A. and THIBAULT, L. (1999) Convergence of subdifferentials of convexly composite functions. Canad. J. Math. 51, 250-265.
• DANIILIDIS, A. and GEORGIEV, P. (2004) Approximate convexity and submonotonicity. J. Math. Anal. Appl. 291, 292-301.
• DANIILIDIS, A., GEORGIEV, P. and PENOT, J.-P. (2003) Integration of multivalued operators and cyclic submonotonicity. Trans. Amer. Math. Soc. 355, 177-195.
• DEGIOVANNI, M., MARINO, A. and TOSQUES, M. (1985) Evolution equations with lack of convexity. Nonlinear Anal. 9, 1401-1443.
• DIESTEL, J. (1975) Geometry of Banach spaces, Selected topics. Lecture Notes in Math., Springer-Verlag.
• EKELAND, I. (1974) On the variational principle. J. Math. Anal, and Appl. 47, 324-353.
• FABIAN, M. (1989) Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carolinae, 30, 51-56.
• GINSBURG, B. and IOFFE, A.D. (1996) The maximum principle in optimal control of systems governed by semilinear equations. In: B.S. Mordukhovich et al., eds., Proceedings of the IMA workshop on Nonsmooth Analysis Non-smooth analysis and geometric methods in deterministic optimal control. Minneapolis, MN, USA, Springer IMA Vol. Math. Appl. 78, New York, 81-110.
• HOLMES, R. B. (1975) Geometric Functional Analysis and Its Applications. Graduate Texts in Maths. 24, Springer-Verlag, New York-Heidelberg-Berlin.
• IOFFE, A.D. (1983) On Subdifferentiability spaces. New York Acad. Sci. 410, 107-119.
• IOFFE, A.D. (1984) Subdifferentiability spaces and nonsmooth analysis. Bull. Am. Math. Soc., New Ser. 10, 87-90.
• IOFFE, A.D. (1990) Proximal analysis and approximate subdifferentials. J. London Math. Soc. 41, 175-192.
• IVANOV, M. and ZLATEVA, N. (2001) On primal lower nice property. C. R. Acad. Bulgare Sci. 54 (11), 5-10.
• JOURANI, A. and THIBAULT, L. (1995) Metric inequality and subdifferential calculus in Banach spaces. Set-valued Anal. 3, 87-100.
• JOURANI, A. (2006) Weak regularity of functions and sets in Asplund spaces. Nonlinear Anal. 65 (3), 660-676.
• KRUGER, A.Y. and MORDUKHOVICH, B. S. (1980) Extremal points and the Euler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24, 684-687.
• LUC, D.T., NGAI, H.V. and THERA, M. (1999) On ε-convexity and ε-monotonicity. In: A. Ioffe, S. Reich and I. Shafrir, eds., Calculus of Variations and Differential Equations, Research Notes in Maths. Chapman & Hall, 82-100.
• MARINO, A. and TOSQUES, M. (1990) Some variational problems with lack of convexity and some partial differential inequalities. Methods of nonconvex analysis, Lect. Notes Math. 1446, Springer, 58-83.
• MOREAU, J.-J. (1965) Proximité et dualité dans un espace hilbertien. Bull. Soc. Mat. France 93, 273-299.
• MORDUKHOVICH, B. S. and SHAO,Y. (1996) Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348 (4), 1235-1280.
• MORDUKHOVICH, B.S (2006) Variational Analysis and Generalized Differentiation. I. Basic theory, II. Applications. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330, 331. Springer-Verlag, Berlin.
• NGAI, H.V., LUC, D.T. and THERA, M. (2000) Approximate convex functions. J. Nonlinear and Convex Anal. 1 (2), 155-176.
• NGAI, H.V. and PENOT, J.-P. (2006) Paraconvex functions and paraconvex sets. To appear in Studio, Math.
• NGAI, H.V. and PENOT, J.-P. (2007) Approximately convex functions and approximately monotone operators. Nonlinear Anal. 66, 547-564.
• NGAI, H. and THERA, M. (2001) Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9, 187-216.
• PENOT, J.-P. (1996) Favorable classes of mappings and multimappings in nonlinear analysis and optimization. J. Convex Analysis 3, 97-116.
• PENOT, J.-P. (2004) Calmness and stability properties of marginal and performance functions. Numer. Functional Anal. Optim. 25 (3-4), 287-308.
• PHELPS, R.R. 1993 Convex Functions, Monotone Operators and Differentiability. Lect. Notes in Math. 1364, Springer-Verlag, Berlin.
• POLIQUIN, R.A. (1991) Integration of subdifferentials of nonconvex functions. Nonlinear Anal. 17, 385-398.
• POLOQUIN, R. A. and ROCKAFELLAR, R. T. (2000) Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 352, 5231-5249.
• POLOQUIN, R.A., ROCKAFELLAR, R.T. and THIBAULT, L. (2000) Local differentiability of distance functions. Trans. Amer. Math. Soc. 307, 5231-5249.
• ROCKAFELLAR, R.T. and WETS, R.J.-B. (1998) Variational Analysis. Springer.
• ROCKAFELLAR, R.T. and WETS, R.J.-B. (2002) Variational Analysis. Grundlehren der Mathematischen Wissenschaften 317, Springer-Verlag, Berlin.
• ROLEWICZ, S. (2000) On α(-)-paraconvex and strongly α(-)-paraconvex functions. Control and Cybernet. 29 (1), 367-377.
• ROLEWICZ, S. (2001) On the coincidence of some subdifferentials in the class of α(-)-paraconvex functions. Optimization 50, 353-360.
• SPINGARN, J.E. (1981) Submonotone subdifferentials of Lipschitz functions. Trans. Amer. Math. Soc. 264, 77-89.
• VIAL, J.-P. (1983) Strong and weak convexity of sets and functions. Math. Oper. Research. 8 (2), 231-259.
• XU, Z.-B. and ROACH, G.F (1991) Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157, 189-210.
• ZĂLINESCU, C. (2002) Convex Analysis in General Vector Spaces. World Scientific, Singapore.