Variational Sets of Perturbation Maps and Applications to Sensitivity Analysis for Constrained Vector Optimization
- 288 Downloads
We consider sensitivity analysis in terms of variational sets for nonsmooth vector optimization. First, relations between variational sets, or their minima/weak minima, of a set-valued map and that of its profile map are obtained. Second, given an objective map, relationships between the above sets of this objective map and that of the perturbation map and weak perturbation map are established. Finally, applications to constrained vector optimization are given. Many examples are provided to illustrate the essentialness of the imposed assumptions and some advantages of our results.
KeywordsNonsmooth vector optimization Sensitivity analysis Perturbation maps Weak perturbation maps Variational sets Singular variational sets
This research was supported by the Vietnam National University Hochiminh City. A part of the work of the second author was completed during his stay as a visiting professor at the Vietnam Institute for Advanced Study in Mathematics, whose hospitality is gratefully acknowledged. The authors are indebted to an anonymous referee for many valuable detailed remarks which have helped improve the paper significantly.
- 1.Fiacco, A.V.: Introduction to sensitivity and stability analysis. In: Nonlinear Programming. Academic Press, New York (1983) Google Scholar
- 9.Wang, Q.L., Li, S.J.: Second-order contingent derivative of the perturbation map in multiobjective optimization. Fixed Point Theory Appl. 2011, 857520 (2011) Google Scholar
- 13.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Springer, Berlin (2006) Google Scholar
- 14.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications. Springer, Berlin (2006) Google Scholar
- 31.Ginchev, I.: Higher-order optimality conditions in nonsmooth vector optimization. J. Stat. Manag. Syst. 5, 321–339 (2002). In: Cambini, A., Dass, B.K., Martein, L., Generalized Convexity, Generalized Monotonicity, Optimality Conditions and Duality in Scalar and Vector Optimization MathSciNetzbMATHCrossRefGoogle Scholar
- 32.Khanh, P.Q., Tuan, N.D.: Variational sets of multivalued mappings and a unified study of optimality conditions. J. Optim. Theory Appl. 139, 45–67 (2008) Google Scholar
- 35.Diem, H.T.H., Khanh, P.Q., Tung, L.T.: On higher-order sensitivity analysis in nonsmooth vector optimization. J. Optim. Theory Appl., to appear Google Scholar
- 36.Anh, N.L.H., Khanh, P.Q.: Higher-order radial epiderivatives and optimality conditions in nonsmooth vector optimization. Submitted for publication Google Scholar