On tangential approximations of the solution set of set-valued inclusions. (English) Zbl 1490.49014

Summary: In the present paper, the problem of estimating the contingent cone to the solution set associated with certain set-valued inclusions is addressed by variational analysis methods and tools. As a main result, inner (resp. outer) approximations, which are expressed in terms of outer (resp. inner) prederivatives of the set-valued term appearing in the inclusion problem, are provided. For the analysis of inner approximations, the evidence arises that the metric increase property for set-valued mappings turns out to play a crucial role. Some of the results obtained in this context are then exploited for formulating necessary optimality conditions for constrained problems, whose feasible region is defined by a set-valued inclusion.


49J53 Set-valued and variational analysis
90C30 Nonlinear programming
Full Text: DOI


[1] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd ed., Springer, Berlin, 2006. · Zbl 1156.46001
[2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Mod. Birkhäuser Class., Birkhäuser, Boston, 2009.
[3] D. Azé and J.-N. Corvellec, Nonlinear local error bounds via a change of metric, J. Fixed Point Theory Appl. 16 (2014), no. 1-2, 351-372. · Zbl 1317.49017
[4] D. Azé, J.-N. Corvellec and R. E. Lucchetti, Variational pairs and applications to stability in nonsmooth analysis, Nonlinear Anal. 49 (2002), no. 5, 643-670. · Zbl 1035.49014
[5] H. T. Banks and M. Q. Jacobs, A differential calculus for multifunctions, J. Math. Anal. Appl. 29 (1970), 246-272. · Zbl 0191.43302
[6] J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, CMS Books Math./Ouvrages Math. SMC 3, Springer, New York, 2000. · Zbl 0953.90001
[7] J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, CMS Books Math./Ouvrages Math. SMC 20, Springer, New York, 2005. · Zbl 1076.49001
[8] G. Bouligand, Sur les surfaces dépourvues de points hyperlimites, Ann. Soc. Polon. Math. 9 (1930), 32-41. · JFM 57.0097.01
[9] M. Castellani, Error bounds for set-valued maps, Generalized Convexity and Optimization for Economic and Financial Decisions (Verona 1998), Pitagora, Bologna (1999), 121-135. · Zbl 1055.49509
[10] E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 68 (1980), no. 3, 180-187. · Zbl 0465.47041
[11] V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Approx. Optim. 7, Peter Lang, Frankfurt am Main, 1995. · Zbl 0887.49014
[12] A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, 2nd ed., Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 2014. · Zbl 1337.26003
[13] M. Gaydu, M. H. Geoffroy and Y. Marcelin, Prederivatives of convex set-valued maps and applications to set optimization problems, J. Global Optim. 64 (2016), no. 1, 141-158. · Zbl 1331.49022
[14] A. D. Ioffe, Nonsmooth analysis: Differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), no. 1, 1-56. · Zbl 0651.58007
[15] A. G. Kusraev and S. S. Kutateladze, Calculus of tangents and beyond, Vladikavkaz. Mat. Zh. 19 (2017), no. 4, 27-34. · Zbl 1448.49023
[16] L. A. Lyusternik, On the conditional extrema of functionals (in Russian), Mat. Sb. 41 (1934), no. 3, 390-401. · Zbl 0011.07401
[17] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I. Basic Theory, Grundlehren Math. Wiss. 330, Springer, Berlin, 2006.
[18] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. II. Applications, Grundlehren Math. Wiss. 331, Springer, Berlin, 2006.
[19] D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets. Fractional Arithmetic with Convex Sets, Math. Appl. 548, Kluwer Academic, Dordrecht, 2002. · Zbl 1027.46001
[20] C. H. J. Pang, Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity, Math. Oper. Res. 36 (2011), no. 3, 377-397. · Zbl 1242.90250
[21] J.-P. Penot, Calculus Without Derivatives, Grad. Texts in Math. 266, Springer, New York, 2013. · Zbl 1264.49014
[22] S. M. Robinson, An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res. 16 (1991), no. 2, 292-309. · Zbl 0746.46039
[23] W. Schirotzek, Nonsmooth Analysis, Universitext, Springer, Berlin, 2007. · Zbl 1120.49001
[24] F. Severi, Su alcune questioni di topologia infinitesimale, Ann. Soc. Polon. Math. 9 (1930), 97-108. · JFM 57.0754.01
[25] A. Uderzo, On some generalized equations with metrically C-increasing mappings: Solvability and error bounds with applications to optimization, Optimization 68 (2019), no. 1, 227-253. · Zbl 1430.90526
[26] A. Uderzo, Solution analysis for a class of set-inclusive generalized equations: A convex analysis approach, Pure Appl. Funct. Anal. 5 (2020), no. 3, 769-790. · Zbl 1472.46077
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