Abstract
In this paper, we deal with the sensitivity analysis in vector optimization. More specifically, formulae for inner and outer evaluating the S-derivative of the efficient point multifunction in parametric vector optimization problems are established. These estimating formulae are presented via the set of efficient/weakly efficient points of the S-derivative of the original multifunction, a composite multifunction of the objective function and the constraint map**. The elaboration of the formulae in vector optimization problems, having multifunction constraints and semiinfinite constraints, is also undertaken. Furthermore, examples are provided for analyzing and illustrating the obtained results.
Similar content being viewed by others
References
Tanino, T.: Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56, 479–499 (1988)
Tanino, T.: Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim. 26, 521–536 (1988)
Aubin, J.P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Mathematical analysis and applications, pp. 159–229. Academic Press, New York (1981)
Shi, D.S.: Contingent derivative of the perturbation map in multiobjective optimization. J. Optim. Theory Appl. 70, 385–396 (1991)
Shi, D.S.: Sensitivity analysis in convex vector optimization. J. Optim. Theory Appl. 77, 145–159 (1993)
Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in parametrized convex vector optimization. J. Math. Anal. Appl. 202, 511–522 (1996)
Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in vector optimization. J. Optim. Theory Appl. 89, 713–730 (1996)
Rockafellar, R.T.: Proto-differentiability of set-valued map**s and its applications in optimization. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, 449–482 (1989)
Lee, G.M., Huy, N.Q.: On sensitivity analysis in vector optimization. Taiwan. J. Math. 11, 945–958 (2007)
Chen, L.: Generalized tangent epiderivative and applications to set-valued map optimization. J. Nonlinear Convex Anal. 3, 303–313 (2002)
Chuong, T.D., Yao, J.-C.: Generalized Clarke epiderivatives of parametric vector optimization problems. J. Optim. Theory Appl. 146, 77–94 (2010)
Huy, N.Q., Mordukhovich, B.S., Yao, J.-C.: Coderivatives of frontier and solution maps in parametric multiobjective optimization. Taiwan. J. Math. 12, 2083–2111 (2008)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Springer, Berlin (2006)
Chuong, T.D.: Clarke coderivatives of efficient point multifunctions in parametric vector optimization. Nonlinear Anal. 74, 273–285 (2011)
Gopfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)
Lee, G.M., Huy, N.Q.: On proto-differentiability of generalized perturbation maps. J. Math. Anal. Appl. 324, 1297–1309 (2006)
Taa, A.: Set-valued derivatives of multifunctions and optimality conditions. Numer. Funct. Anal. Optim. 19, 121–140 (1998)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Basel (1990)
Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)
Reemtsen, R., Rückmann, J.-J. (eds.): Semi-Infinite Programming. Kluwer Academic, Boston (1998)
Chuong, T.D., Huy, N.Q., Yao, J.-C.: Stability of semi-infinite vector optimization problems under functional perturbations. J. Glob. 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., Huy, N.Q., Yao, J.-C.: Subdifferentials of marginal functions in semi-infinite programming. SIAM J. Optim. 20, 1462–1477 (2009)
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.: Coderivatives of efficient point multifunctions in parametric vector optimization. Taiwan. J. Math. 13, 1671–1693 (2009)
Dinh, N., Mordukhovich, B.S., Nghia, T.T.A.: Qualification and optimality conditions for DC programs with infinite constraints. Acta Math. Vietnam. 34, 123–153 (2009)
Li, C., Ng, K.F., Pong, T.K.: Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM J. Optim. 19, 163–187 (2008)
Acknowledgements
This work was supported in part by the project “Joint research and training on Variational Analysis and Optimization Theory, with oriented applications in some technological areas” (Vietnam–USA). The author thanks an anonymous referee for valuable comments and suggestions which improved the original presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Nguyen Dong Yen.
Rights and permissions
About this article
Cite this article
Chuong, T.D. Derivatives of the Efficient Point Multifunction in Parametric Vector Optimization Problems. J Optim Theory Appl 156, 247–265 (2013). https://doi.org/10.1007/s10957-012-0099-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-012-0099-1