# American Institute of Mathematical Sciences

2011, 1(2): 317-331. doi: 10.3934/naco.2011.1.317

## Stability analysis of parametric variational systems

 1 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China, China

Received  October 2010 Revised  May 2011 Published  June 2011

In this paper, Robinson's metric regularity of a positive order around/at some point of parametric variational systems is discussed. Under some suitable conditions, the relationships among H$\ddot{o}$lder-likeness, H$\ddot{o}$lder calmness, metric regularity of a positive order and Robinson's metric regularity of a positive order are discussed for the parametric variational systems. Then, some applications to the stabilities of the optimal value map and the solution map are studied for a parametric vector optimization problem, respectively.
Citation: Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317
##### References:
 [1] F. J. Aragón Artacho and B. S. Mordukhovich, Metric regularity and Lipschitzian stability of parametric variational systems, Nonlinear Analysis, 72 (2010), 1149-1170. doi: doi:10.1016/j.na.2009.07.051. [2] F. J. Aragón Artacho and B. S. Mordukhovich, Enhanced metric regularity and Lipschitzian properties of variational systems, Journal of Global Optimization, 50 (2011), 145-167. doi: doi:10.1007/s10898-011-9698-x. [3] J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," John Wiley and Sons, New York, 1984. [4] E. Bednarczuk, Upper Hölder continuity of minimal points, Journal on Convex Analysis, 9 (2002), 327-338. [5] E. Bednarczuk, Weak sharp efficiency and growth condition for vector-valued functions with applications, Optimization, 53 (2004), 455-474. doi: doi:10.1080/02331930412331330478. [6] J. M. Borwein, Stability and regular points of inequality systems, Journal of Optimization Theory and Applications, 48 (1986), 9-52. [7] J. M. Borwein and Q. J. Zhu, "Techniques of Variational Analysis," Springer, New York, 2005. [8] N. H. Chieu, J. C. Yao and N. D. Yen, Relationships between Robinson metric regularity and Lipschitz-like behavior of implicit multifunctions, Nonlinear Analysis, 72 (2010), 3594-3601. doi: doi:10.1016/j.na.2009.12.039. [9] P. H. Dien and N. D. Yen, On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints, Applied Mathematics and Optimization, 24 (1991), 35-54. doi: doi:10.1007/BF01447734. [10] A. L. Dontchev, M. Quincampoix and N. Zlateva, Aubin criterion for metric regularity, Journal of Convex Analysis, 13 (2006), 281-297. [11] I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications, 47 (1974), 324-353. doi: doi:10.1016/0022-247X(74)90025-0. [12] N. Q. Huy and J. C. Yao, Stability of implicit multifunctions in Asplund spaces, Taiwanese Journal of Mathematics, 13 (2009), 47-65. [13] V. Jeyakumar and N. D. Yen, Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization, SIAM Journal on Optimization, 14 (2004), 1106-1127. doi: doi:10.1137/S1052623402419236. [14] B. T. Kien, On the lower semicontinuity of optimal solution sets, Optimization, 54 (2005), 123-130. doi: doi:10.1080/02331930412331330379. [15] Y. S. Ledyaev and Q. J. Zhu, Implicit multifunctions theorems, Set-Valued Analysis, 7 (1999), 209-238. doi: doi:10.1023/A:1008775413250. [16] G. M. Lee, N. N. Tam and N. D. Yen, Normal coderivative for multifunctions and implicit function theorems, Journal of Mathematical Analysis and Applications, 338 (2008), 11-22. doi: doi:10.1016/j.jmaa.2007.05.001. [17] M. H. Li and S. J. Li, Robinson metric regularity of parametric variational systems, Nonlinear Analysis, 74 (2011), 2262-2271. doi: doi:10.1016/j.na.2010.11.031. [18] B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications," Springer, Berlin, 2006. [19] H. V. Ngai and M. Théra, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Analysis, 12 (2004), 195-223. doi: doi:10.1023/B:SVAN.0000023396.58424.98. [20] S. M. Robinson, Stability theory for systems of inequalities, I. Linear systems, SIAM Journal on Numerical Analysis, 12 (1975), 754-769. doi: doi:10.1137/0712056. [21] S. M. Robinson, Stability theory for systems of inequalities, II. Differentiable nonlinear systems, SIAM Journal on Numerical Analysis, 13 (1976), 497-513. doi: doi:10.1137/0713043. [22] S. M. Robinson, Generalized equations and their solutions, Part I, Basic theory, Mathematical Programming Study, 10 (1979), 128-141. [23] S. M. Robinson, Regularity and stability for convex multivalued functions, Mathematics of Operations Research, 1 (1976), 130-143. doi: doi:10.1287/moor.1.2.130. [24] A. Uderzo, On some regularity properties in variational analysis, Set-Valued and Variational Analysis, 17 (2009), 409-430. doi: doi:10.1007/s11228-009-0121-4. [25] N. D. Yen and J. C. Yao, Point-based sufficient conditions for metric regularity of implicit multifunctions, Nonlinear Analysis, 70 (2009), 2806-2815. doi: doi:10.1016/j.na.2008.04.005. [26] N. D. Yen, J. C. Yao and B. T. Kien, Covering properties at positive-order rates of multifunctions and some related topics, Journal of Mathematical Analysis and Applications, 338 (2008), 467-478. doi: doi:10.1016/j.jmaa.2007.05.041.

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##### References:
 [1] F. J. Aragón Artacho and B. S. Mordukhovich, Metric regularity and Lipschitzian stability of parametric variational systems, Nonlinear Analysis, 72 (2010), 1149-1170. doi: doi:10.1016/j.na.2009.07.051. [2] F. J. Aragón Artacho and B. S. Mordukhovich, Enhanced metric regularity and Lipschitzian properties of variational systems, Journal of Global Optimization, 50 (2011), 145-167. doi: doi:10.1007/s10898-011-9698-x. [3] J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," John Wiley and Sons, New York, 1984. [4] E. Bednarczuk, Upper Hölder continuity of minimal points, Journal on Convex Analysis, 9 (2002), 327-338. [5] E. Bednarczuk, Weak sharp efficiency and growth condition for vector-valued functions with applications, Optimization, 53 (2004), 455-474. doi: doi:10.1080/02331930412331330478. [6] J. M. Borwein, Stability and regular points of inequality systems, Journal of Optimization Theory and Applications, 48 (1986), 9-52. [7] J. M. Borwein and Q. J. Zhu, "Techniques of Variational Analysis," Springer, New York, 2005. [8] N. H. Chieu, J. C. Yao and N. D. Yen, Relationships between Robinson metric regularity and Lipschitz-like behavior of implicit multifunctions, Nonlinear Analysis, 72 (2010), 3594-3601. doi: doi:10.1016/j.na.2009.12.039. [9] P. H. Dien and N. D. Yen, On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints, Applied Mathematics and Optimization, 24 (1991), 35-54. doi: doi:10.1007/BF01447734. [10] A. L. Dontchev, M. Quincampoix and N. Zlateva, Aubin criterion for metric regularity, Journal of Convex Analysis, 13 (2006), 281-297. [11] I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications, 47 (1974), 324-353. doi: doi:10.1016/0022-247X(74)90025-0. [12] N. Q. Huy and J. C. Yao, Stability of implicit multifunctions in Asplund spaces, Taiwanese Journal of Mathematics, 13 (2009), 47-65. [13] V. Jeyakumar and N. D. Yen, Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization, SIAM Journal on Optimization, 14 (2004), 1106-1127. doi: doi:10.1137/S1052623402419236. [14] B. T. Kien, On the lower semicontinuity of optimal solution sets, Optimization, 54 (2005), 123-130. doi: doi:10.1080/02331930412331330379. [15] Y. S. Ledyaev and Q. J. Zhu, Implicit multifunctions theorems, Set-Valued Analysis, 7 (1999), 209-238. doi: doi:10.1023/A:1008775413250. [16] G. M. Lee, N. N. Tam and N. D. Yen, Normal coderivative for multifunctions and implicit function theorems, Journal of Mathematical Analysis and Applications, 338 (2008), 11-22. doi: doi:10.1016/j.jmaa.2007.05.001. [17] M. H. Li and S. J. Li, Robinson metric regularity of parametric variational systems, Nonlinear Analysis, 74 (2011), 2262-2271. doi: doi:10.1016/j.na.2010.11.031. [18] B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications," Springer, Berlin, 2006. [19] H. V. Ngai and M. Théra, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Analysis, 12 (2004), 195-223. doi: doi:10.1023/B:SVAN.0000023396.58424.98. [20] S. M. Robinson, Stability theory for systems of inequalities, I. Linear systems, SIAM Journal on Numerical Analysis, 12 (1975), 754-769. doi: doi:10.1137/0712056. [21] S. M. Robinson, Stability theory for systems of inequalities, II. Differentiable nonlinear systems, SIAM Journal on Numerical Analysis, 13 (1976), 497-513. doi: doi:10.1137/0713043. [22] S. M. Robinson, Generalized equations and their solutions, Part I, Basic theory, Mathematical Programming Study, 10 (1979), 128-141. [23] S. M. Robinson, Regularity and stability for convex multivalued functions, Mathematics of Operations Research, 1 (1976), 130-143. doi: doi:10.1287/moor.1.2.130. [24] A. Uderzo, On some regularity properties in variational analysis, Set-Valued and Variational Analysis, 17 (2009), 409-430. doi: doi:10.1007/s11228-009-0121-4. [25] N. D. Yen and J. C. Yao, Point-based sufficient conditions for metric regularity of implicit multifunctions, Nonlinear Analysis, 70 (2009), 2806-2815. doi: doi:10.1016/j.na.2008.04.005. [26] N. D. Yen, J. C. Yao and B. T. Kien, Covering properties at positive-order rates of multifunctions and some related topics, Journal of Mathematical Analysis and Applications, 338 (2008), 467-478. doi: doi:10.1016/j.jmaa.2007.05.041.
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