Abstract
The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bednarczuk, E.M.: Some stability results for vector optimization problems in partially ordered topological vector. In: Proceedings of First World Congress Nonlinear Analysis (vol. III, pp. 2371–2382). Tampa, FL (1996)
Bednarczuk E.M.: Upper Hölder continuity of minimal points. J. Convex Anal. 9(2), 327–338 (2002)
Bednarczuk E.M.: Hölder-like properties of minimal points in vector optimization. Control Cybernet. 31(3), 423–438 (2002)
Chinchuluun A., Pardalos P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)
Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds): Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)
Chuong T.D., Huy N.Q., Yao J.-C.: Stability of semi-infinite vector optimization problems under functional purtubations. J. Global Optim. 45, 583–595 (2009)
Chuong T.D., Huy N.Q., Yao J.-C.: Pseudo-Lipschitz property of linear semi-infinite vector optimization problems. Eur. J. Oper. Res. 200, 639–644 (2010)
Chuong T.D., Yao J.-C.: Sufficient conditions for pseudo-Lipschitz property in convex semi-infinite vector optimization problems. Nonlinear Anal. 71, 6312–6322 (2009)
Chuong T.D., Yao J.-C., Yen N.D.: Further results on the lower semicontinuity of efficient point multifunctions. Pacific J. Optim. 6, 405–422 (2010)
Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Floudas, C.A., Pardalos, P.M. (eds): Encyclopedia of Optimization. Springer, Berlin (2009)
Gopfert A., Riahi H., Tammer C., Zălinescu C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)
Henrion R., Jourani A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13, 520–534 (2002)
Henrion R., Jourani A., Outrata J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)
Henrion R., Outrata J.: Calmness of constraint systems with applications. Math. Program. 104, 437–464 (2005)
Huy N.Q., Yao J.-C.: Stability of implicit multifunctions in Asplund spaces. Taiwan. J. Math. 13, 47–65 (2009)
Ioffe A.D., Outrata J.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16(2–3), 199–227 (2008)
Klatte D., Kummer B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer Academic Publisher, Dordrecht-Boston-London (2002)
Kruger A.Y.: On Fréchet subdifferentials. J. Math. Sci. (N.Y.) 116(3), 3325–3358 (2003)
Ledyaev Y.S., Zhu Q.J.: Implicit multifunctions theorems. Set-Valued Anal. 7, 209–238 (1999)
Lee G.M., Tam N.N., Yen N.D.: Normal coderivative for multifunctions and implicit function theorems. J. Math. Anal. Appl. 338, 11–22 (2008)
Luc D.T.: Theory of Vector Optimization. Lecture Notes in Economics, Math Syst 319. Springer, Berlin (1989)
Mordukhovich B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Springer, Berlin (2006)
Phelps R.R.: Convex Functions, Monotone Operators and Differentiability, 2nd edn. Lecture Notes in Mathematics, vol. 1364. Springer, Berlin (1993)
Rockafellar R.T., Wets R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Robinson S.M.: Generalized equations and their solutions, Part I: Basic theory. Math. Program. Study 10, 128–141 (1979)
Sawaragi Y., Nakayama H., Tanino T.: Theory of Multiobjective Optimization. Mathematics in Science and Engineering, vol. 176. Academic Press, Orlando, FL (1985)
Xiang S.W., Zhou Y.H.: Continuity properties of solutions of vector optimization. Nonlinear Anal. 64, 2496–2506 (2006)
Xiang S.W., Yin W.S.: Stability results for efficient solutions of vector optimization problems. J. Optim. Theory Appl. 134, 385–398 (2007)
Zheng X.Y., Ng K.F.: Metric subregularity and constraint qualifications for convex generalized equations in Banach spaces. SIAM J. Optim. 18(2), 437–460 (2007)
Zheng X.Y., Ng K.F.: Calmness for L-subsmooth multifunctions in Banach spaces. SIAM J. Optim. 19(4), 1648–1673 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chuong, T.D., Kruger, A.Y. & Yao, JC. Calmness of efficient solution maps in parametric vector optimization. J Glob Optim 51, 677–688 (2011). https://doi.org/10.1007/s10898-011-9651-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9651-z