Abstract
In this paper we give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second-order objects are envisaged and the assumptions we impose in order to get the calculus are in terms of metric subregularity of the assembly of the initial data. This approach is different from those used in alternative recent papers in literature and allows us to avoid compactness conditions. An application to a special type of vector optimization problems, where the objective is given as the sum of two multifunctions, is presented. Furthermore, also as application, a special attention is paid for the case of perturbation set-valued maps which naturally appear in optimization problems.
Similar content being viewed by others
References
Aragón Artacho F.J., Mordukhovich B.S.: Enhanced metric regularity and Lipschitzian properties of variational systems. J. Glob. Optim. 50, 145–167 (2011)
Aubin J.P., Frankowska H.: Set-valued Analysis. Birkäuser, Basel (1990)
Dontchev A.L., Rockafellar R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)
Durea M.: Calculus of the Bouligand derivative of set-valued maps in Banach spaces. Nonlinear Funct. Anal. Appl. 13, 573–585 (2008)
Durea M., Strugariu R.: Quantitative results on openness of set-valued mappings and implicit multifunction theorems. Pac. J. Optim. 6, 533–549 (2010)
Durea M., Strugariu R.: Necessary optimality conditions for weak sharp minima in set-valued optimization. Nonlinear Anal. Theory Methods Appl. 73, 2148–2157 (2010)
Gadhi N., Laghdir M., Metrane A.: Optimality conditions for d.c. vector optimization problems under reverse convex constraints. J. Glob. Optim. 33, 527–540 (2005)
Gadhi, N.: Optimality Conditions for a d.c. Set-Valued Problem Via the Extremal Principle. Optimization with Multivalued Mappings, Springer Optimization and Its Applications, vol. 2, Part III, pp. 251–264 (2006)
Henrion R., Outrata J.V.: Calmness of constraint systems with applications. Math. Programm. Serie B 104, 437–464 (2005)
Klatte D., Kummer B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Nonconvex Optimization and its Applications, vol. 60. Kluwer, Dordrecht (2002)
Luc D.T.: Theory of Vector Optimization. Springer, Berlin (1989)
Li, S.J., Liao, C.M.: Second-order differentiability of generalized perturbation maps. J. Glob. Optim. doi:10.1007/s10898-011-9661-x (published online)
Li S.J., Meng K.W., Penot J.P.: Calculus rules for derivatives of multimaps. Set-Valued Anal. 17, 21–39 (2009)
Lee G.M., Huy N.Q.: On proto-differentiability of generalized perturbation maps. J. Math. Anal. Appl. 324, 1297–1309 (2006)
Mordukhovich B.S.: Metric approximations and necessary optimality conditions for general classes of extremal problems. Sov. Math. Doklady 22, 526–530 (1980)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Volume I: Basic Theory, Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol. 330. Springer, Berlin (2006)
Mordukhovich B.S., Nam N.M., Yen N.D.: Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55, 685–708 (2006)
Mordukhovich B.S., Shao Y.: Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces. Nonlinear Anal. 25, 1401–1424 (1995)
Pardalos, P.M., Rassias, T.M., Khan, A.A. (eds): Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol. 35. Springer, Berlin (2010)
Penot J.P.: Differentiability of relations and differential stability perturbed optimization problems. SIAM J. Control Optim. 22, 529–551 (1984)
Rockafellar R.T.: Proto-differentiability of set-valued mappings and its applications in optimization. Ann. Inst. H. Poincaré 6, 449–482 (1989)
Rockafellar R.T., Wets R.: Variational Analysis. Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol. 317. Springer, Berlin (1998)
Yuan D., Chinchuluun A., Liu X., Pardalos P.M.: Generalized convexities and generalized gradients based on algebraic operations. J. Math. Anal. Appl. 321, 675–690 (2006)
Zheng X.Y., Ng K.F.: Metric subregularity and calmness for nonconvex generalized equations in Banach spaces. SIAM J. Optim. 20, 2119–2136 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Durea, M., Strugariu, R. Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions. J Glob Optim 56, 587–603 (2013). https://doi.org/10.1007/s10898-011-9800-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9800-4
Keywords
- Bouligand (contingent) tangent sets
- Ursescu (adjacent) tangent sets
- Metric regularity
- Set-valued derivatives
- Perturbation maps