Variational analysis of paraconvex multifunctions. (English) Zbl 1532.90142
In this paper, the authors aim to study the class of so-called \(\rho\)-paraconvex multifunctions from a Banach space \(X\) into the subsets of another Banach space \(Y\). These multifunctions are defined in relation with a modulus function \(\rho: X \to [0,+\infty)\) satisfying some suitable conditions. This class of multifunctions generalizes the class of \(\gamma\)-paraconvex multifunctions with \(\gamma > 1\) introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. The authors establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such a class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study.
Reviewer: Paulo Mbunga (Kiel)
MSC:
| 90C48 | Programming in abstract spaces |
| 90C30 | Nonlinear programming |
| 49J27 | Existence theories for problems in abstract spaces |
| 49J52 | Nonsmooth analysis |
| 49J53 | Set-valued and variational analysis |
Keywords:
weak convexity; lower \(C^2\) functions; paraconvexity; paramonotonicity; approximate convex function; graphical tangent cone regularity; normal cone regularity; Fréchet subdifferential; Fréchet normal cone; coderivatives; fuzzy mean value theoremReferences:
| [1] | Alberti, G.; Ambrosio, L.; Cannarsa, P., On the singularities of convex functions, Manuscripta Math., 76, 3-4, 421-435 (1992) · Zbl 0784.49011 · doi:10.1007/BF02567770 |
| [2] | Aussel, D.; Corvellec, J-N; Lassonde, M., Mean value property and subdifferential criteria for lower semicontinuous functions, Trans. Amer. Math. Soc., 347, 10, 4147-4161 (1995) · Zbl 0849.49016 · doi:10.1090/S0002-9947-1995-1307998-0 |
| [3] | Aussel, D.; Daniilidis, A.; Thibault, L., Subsmooth sets: functional characterizations and related concepts, Trans. Amer. Math. Soc., 357, 4, 1275-1301 (2005) · Zbl 1094.49016 · doi:10.1090/S0002-9947-04-03718-3 |
| [4] | Borwein, JM; Moors, WB; Wang, X., Generalized subdifferentials: a Baire categorical approach, Trans. Amer. Math. Soc., 353, 10, 3875-3893 (2001) · Zbl 0979.49016 · doi:10.1090/S0002-9947-01-02820-3 |
| [5] | Bounkhel, M.; Thibault, L., Directionally pseudo-Lipschitz set-valued mappings, J. Math. Anal. Appl., 266, 2, 269-287 (2002) · Zbl 1012.49016 · doi:10.1006/jmaa.2001.7707 |
| [6] | Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser Boston Inc, Boston, MA (2004) · Zbl 1095.49003 |
| [7] | Clarke, F. H.: Optimization and nonsmooth analysis, second ed., Classics in Applied Mathematics, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1990) · Zbl 0696.49002 |
| [8] | Conway, J.B.: A course in functional analysis, second ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, (1990) · Zbl 0706.46003 |
| [9] | Daniilidis, A., Malick, J.: Filling the gap between lower-\(C^1\) and lower-\(C^2\) functions. J. Convex Anal. 12(2), 315-329 (2005) · Zbl 1107.26013 |
| [10] | De Giorgi, E.; Marino, A.; Tosques, M., \((p,\, q)\)-convex functions Atti Accad, Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., 73, 1-4, 6-14 (1982) · Zbl 0521.49011 |
| [11] | Degiovanni, A.; Marino, M.; Tosques, M., Evolution equations with lack of convexity, Nonlinear Anal., 9, 12, 1401-1443 (1985) · Zbl 0545.46029 · doi:10.1016/0362-546X(85)90098-7 |
| [12] | Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353 (1974) · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0 |
| [13] | Fabián, M.: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, vol. 30, 17th Winter School on Abstract Analysis (Srní, 1989), pp. 51-56, (1989) · Zbl 0714.49022 |
| [14] | Giannessi, F.: Constrained Optimization and Image Space Analysis. Vol. 1, Mathematical Concepts and Methods in Science and Engineering, vol. 49, Springer, New York, 2005, Separation of sets and optimality conditions · Zbl 1082.49001 |
| [15] | Hörmander, L., Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Ark. Mat., 3, 181-186 (1955) · Zbl 0064.10504 · doi:10.1007/BF02589354 |
| [16] | Huang, H., Coderivative conditions for error bounds of \(\gamma \)-paraconvex multifunctions, Set-Valued Var. Anal., 20, 4, 567-579 (2012) · Zbl 1254.49009 · doi:10.1007/s11228-012-0210-7 |
| [17] | Janin, R., Sur une classe de fonctions sous-linéarisables, C, R. Acad. Sci. Paris Sér. A-B, 277, A265-A267 (1973) · Zbl 0286.90050 |
| [18] | Jourani, A., Open mapping theorem and inversion theorem for \(\gamma \)-paraconvex multivalued mappings and applications, Studia Math., 117, 2, 123-136 (1996) · Zbl 0841.46002 · doi:10.4064/sm-117-2-123-136 |
| [19] | Jourani, A., Subdifferentiability and subdifferential monotonicity of \(\gamma \)-paraconvex functions, Control Cybernet., 25, 4, 721-737 (1996) · Zbl 0862.49018 |
| [20] | Ledyaev, YS; Zhu, QJ, Implicit multifunction theorems, Set-Valued Anal., 7, 3, 209-238 (1999) · Zbl 0952.49017 · doi:10.1023/A:1008775413250 |
| [21] | Mazure, M-L; Volle, M., Équations inf-convolutives et conjugaison de Moreau-Fenchel, Ann. Fac. Sci. Toulouse Math., 12, 1, 103-126 (1991) · Zbl 0760.49013 · doi:10.5802/afst.721 |
| [22] | Mifflin, R., Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15, 6, 959-972 (1977) · Zbl 0376.90081 · doi:10.1137/0315061 |
| [23] | Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330, Springer-Verlag, Berlin, Basic theory, (2006) |
| [24] | Mordukhovich, BS; Nam, MN, Subgradient of distance functions with applications to Lipschitzian stability, Math. Program., 104, 2-3, 635-668 (2005) · Zbl 1077.49019 · doi:10.1007/s10107-005-0632-1 |
| [25] | Van Ngai, H.; The Luc, D.; Théra, M., Approximate convex functions, J. Nonlinear Convex Anal., 1, 2, 155-176 (2000) · Zbl 1033.49029 |
| [26] | Van Ngai, H.; Penot, J-P, Approximately convex functions and approximately monotonic operators, Nonlinear Anal., 66, 3, 547-564 (2007) · Zbl 1113.26011 · doi:10.1016/j.na.2005.11.045 |
| [27] | Van Ngai, H., Penot, J.-P.: Rambling Through Local Versions of Generalized Convex Functions and Generalized Monotone Operators, Generalized convexity and related topics, Lecture Notes in Econom. and Math. Systems, vol. 583, Springer, Berlin, 2007, pp. 379-397 · Zbl 1123.26015 |
| [28] | Van Ngai, H.; Penot, J-P, Paraconvex functions and paraconvex sets, Studia Math., 184, 1, 1-29 (2008) · Zbl 1181.49014 · doi:10.4064/sm184-1-1 |
| [29] | Van Ngai, H.; Théra, M., Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal., 12, 1-2, 195-223 (2004) · Zbl 1058.49017 · doi:10.1023/B:SVAN.0000023396.58424.98 |
| [30] | Van Ngai, H.; Tron, NH; Théra, M., Implicit multifunction theorems in complete metric spaces, Math. Program., 139, 1-2, 301-326 (2013) · Zbl 1268.49018 |
| [31] | Penot, JP, Generalized convexity in the light of nonsmooth analysis, Recent developments in optimization (Dijon) Lecture Notes in Econom. and Math. Systems, 269-290 (1994), Berlin: Springer, Berlin · Zbl 0849.49015 |
| [32] | Penot, J-P, Calculus without derivatives, Graduate Texts in Mathematics (2013), New York: Springer, New York · Zbl 1264.49014 · doi:10.1007/978-1-4614-4538-8 |
| [33] | Rockafellar, R. T.: Favorable classes of Lipschitz-continuous functions in subgradient optimization, Progress in nondifferentiable optimization, IIASA Collaborative Proc. Ser. CP-82, vol. 8, Internat. Inst. Appl. Systems Anal., Laxenburg, 1982, pp. 125-143 · Zbl 0511.26009 |
| [34] | Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer-Verlag, Berlin (1998) · Zbl 0888.49001 |
| [35] | Rolewicz, S.: On \(\gamma \)-paraconvex multifunctions, Math. Japon. 24(3), 293-300, (1979/80) · Zbl 0434.54009 |
| [36] | Rolewicz, S.: On conditions warranting \(\Phi_2\)-subdifferentiability, Math. Programming Stud. no. 14, 215-224, (1981) · Zbl 0444.90106 |
| [37] | Rolewicz, S., On \(\alpha (\cdot )\)-monotone multifunctions and differentiability of \(\gamma \)-paraconvex functions, Studia Math., 133, 1, 29-37 (1999) · Zbl 0920.47047 · doi:10.4064/sm-133-1-29-37 |
| [38] | Rolewicz, S., On \(\alpha (\cdot )\)-paraconvex and strongly \(\alpha (\cdot )\)-paraconvex functions, Control Cybernet., 29, 1, 367-377 (2000) · Zbl 1030.90092 |
| [39] | Rolewicz, S.: \( \Phi \)-convex functions defined on metric spaces, vol. 115, 2003, Optimization and related topics, 2, pp. 2631-2652 · Zbl 1057.49017 |
| [40] | Rolewicz, S., Paraconvex analysis, Control Cybernet., 34, 3, 951-965 (2005) · Zbl 1167.49306 |
| [41] | Spingarn, JE, Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc., 264, 1, 77-89 (1981) · Zbl 0465.26008 · doi:10.1090/S0002-9947-1981-0597868-8 |
| [42] | Thibault, L., On subdifferentials of optimal value functions, SIAM J. Control Optim., 29, 5, 1019-1036 (1991) · Zbl 0734.90093 · doi:10.1137/0329056 |
| [43] | Thibault, L., Sweeping process with regular and nonregular sets, J. Differ. Equ., 193, 1, 1-26 (2003) · Zbl 1037.34007 · doi:10.1016/S0022-0396(03)00129-3 |
| [44] | Thibault, L.: Unilateral Variational Analysis in Banach Space, World Scientific, (2022) |
| [45] | Vial, J-P, Strong and weak convexity of sets and functions, Math. Oper. Res., 8, 2, 231-259 (1983) · Zbl 0526.90077 · doi:10.1287/moor.8.2.231 |
| [46] | Zagrodny, D., Approximate mean value theorem for upper subderivatives, Nonlinear Anal., 12, 12, 1413-1428 (1988) · Zbl 0689.49017 · doi:10.1016/0362-546X(88)90088-0 |
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