SENSITIVITY ANALYSIS FOR A CLASS OF IMPLICIT MULTIFUNCTIONS WITH APPLICATIONS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 49, Issue 2, 2012, pp.249-262
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2012.49.2.249

Title & Authors

SENSITIVITY ANALYSIS FOR A CLASS OF IMPLICIT MULTIFUNCTIONS WITH APPLICATIONS

Li, Shengjie; Li, Minghua;

Li, Shengjie; Li, Minghua;

Abstract

In this paper, under some suitable conditions and in virtue of a selection which depends on a vector-valued function and a feasible set map, the sensitivity analysis of a class of implicit multifunctions is investigated. Moreover, by using the results established, the solution sets of parametric vector optimization problems are studied.

Keywords

selection;sensitivity analysis;implicit multifunction;parametric vector optimization problem;

Language

English

Cited by

References

1.

T. Amaroq and L. Thibault, On proto-differentiability and strict proto-differentiability of multifunctions of feasible points in perturbed optimization problems, Numer. Funct. Anal. Optim. 16 (1995), 1293-1307.

2.

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.

3.

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.

4.

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.

5.

J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), no. 1, 9-52.

6.

A. L. Dontchev, Implicit function theorems for generalized equations, Math. Programming 70 (1995), no. 1, Ser. A, 91-106.

7.

A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Program- ming, Academic Press, New York, New York, 1983.

8.

A. J. King and R. T. Rockafellar, Sensitivity analysis for nonsmooth generalized equa- tions, Math. Programming 55 (1992), no. 2, Ser. A, 193-212.

9.

H. Kuk, T. Tanino, and M. Tanaka, Sensitivity analysis in vector optimization, J. Optim. Theory Appl. 89 (1996), no. 3, 713-730.

10.

A. B. Levy, Implicit multifunction theorems for the sensitivity analysis of variational conditions, Math. Programming 74 (1996), no. 3, Ser. A, 333-350.

11.

A. B. Levy and R. T. Rockafellar, Variational conditions and the proto-differentiation of partial subgradient mappings, Nonlinear Anal. 26 (1996), no. 12, 1951-1964.

12.

M. H. Li and S. J. Li, Second-order differential and sensitivity properties of weak vector variational inequalities, J. Optim. Theory Appl. 144 (2010), no. 1, 76-87.

13.

S. J. Li and K. W. Meng, Contingent derivatives of set-valued maps with applications to vector optimization, submitted.

14.

S. J. Li, K. W. Meng, and J.-P. Penot, Calculus rules for derivatives of multimaps, Set-Valued Var. Anal. 17 (2009), no. 1, 21-39.

15.

S. J. Li, H. Yan, and G. Y. Chen, Differential and sensitivity properties of gap functions for vector variational inequalities, Math. Methods Oper. Res. 57 (2003), no. 3, 377-391.

16.

K. W. Meng and S. J. Li, Differential and sensitivity properties of gap functions for Minty vector variational inequalities, J. Math. Anal. Appl. 337 (2008), no. 1, 386-398.

17.

J.-P. Penot, Differentiability of relations and differential stability of perturbed optimiza- tion problems, SIAM J. Control Optim. 22 (1984), no. 4 529-551.

18.

Y. P. Qiu and T. L. Magnanti, Sensitivity analysis for variational inequalities defined on polyhedral sets, Math. Oper. Res. 14 (1989), no. 3, 410-432.

19.

Y. P. Qiu and T. L. Magnanti, Sensitivity analysis for variational inequalities, Math. Oper. Res. 17 (1992), no. 1, 61-76.

20.

S. M. Robinson, Generalized equations and their solutions. I. Basic theory, Math. Programming Stud. 10 (1979), 128-141.

21.

22.

S. M. Robinson, An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res. 16 (1991), no. 2, 292-309.

23.

R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincare Anal. Non Lineaire 6 (1989), suppl., 449-482.

24.

R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Programming Stud. 17 (1982), 28-66.

25.

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998.

26.

Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of Multiobjective Optimization, Academic Press, New York, 1985.

27.

A. Shapiro, On concepts of directional differentiability, J. Optim. Theory Appl. 66 (1990), no. 3, 477-487.

28.

A. Shapiro, Sensitivity analysis of parameterized variational inequalities, Math. Oper. Res. 30 (2005), no. 1, 109-126.

29.

D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl. 70 (1991), no. 2, 385-396.

30.

T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl. 56 (1988), no. 3, 479-499.

31.

R. L. Tobin, Sensitivity analysis for variational inequalities, J. Optim. Theory Appl. 48 (1986), no. 1 191-204.