×
Uderzo, Amos

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

J. Appl. Anal. 28, No. 1, 11-33 (2022).
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.

MSC:

49J53 Set-valued and variational analysis
90C30 Nonlinear programming

Keywords:

tangential approximation; decrease principle; prederivative; fan; contingent cone; optimality condition
PDF BibTeX XML Cite
\textit{A. Uderzo}, J. Appl. Anal. 28, No. 1, 11--33 (2022; Zbl 1490.49014)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[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.
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.