American Institute of Mathematical Sciences

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

Stability analysis of parametric variational systems

Shengji Li 1, , Chunmei Liao 1, and  Minghua Li 1,

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.
Keywords: H$\ddot{o}$lder-likeness, Parametric variational systems, metric regularity of a positive order., Robinson's metric regularity of a positive order.
Mathematics Subject Classification: Primary: 49J53, 90C31, 49K40; Secondary: 90C2.
Citation: Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

E. Bednarczuk, Upper Hölder continuity of minimal points, Journal on Convex Analysis, 9 (2002), 327-338.  Google Scholar

[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.  Google Scholar

[6]

J. M. Borwein, Stability and regular points of inequality systems, Journal of Optimization Theory and Applications, 48 (1986), 9-52.  Google Scholar

[7]

J. M. Borwein and Q. J. Zhu, "Techniques of Variational Analysis," Springer, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. L. Dontchev, M. Quincampoix and N. Zlateva, Aubin criterion for metric regularity, Journal of Convex Analysis, 13 (2006), 281-297.  Google Scholar

[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.  Google Scholar

[12]

N. Q. Huy and J. C. Yao, Stability of implicit multifunctions in Asplund spaces, Taiwanese Journal of Mathematics, 13 (2009), 47-65.  Google Scholar

[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.  Google Scholar

[14]

B. T. Kien, On the lower semicontinuity of optimal solution sets, Optimization, 54 (2005), 123-130. doi: doi:10.1080/02331930412331330379.  Google Scholar

[15]

Y. S. Ledyaev and Q. J. Zhu, Implicit multifunctions theorems, Set-Valued Analysis, 7 (1999), 209-238. doi: doi:10.1023/A:1008775413250.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications," Springer, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

S. M. Robinson, Generalized equations and their solutions, Part I, Basic theory, Mathematical Programming Study, 10 (1979), 128-141.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

E. Bednarczuk, Upper Hölder continuity of minimal points, Journal on Convex Analysis, 9 (2002), 327-338.  Google Scholar

[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.  Google Scholar

[6]

J. M. Borwein, Stability and regular points of inequality systems, Journal of Optimization Theory and Applications, 48 (1986), 9-52.  Google Scholar

[7]

J. M. Borwein and Q. J. Zhu, "Techniques of Variational Analysis," Springer, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. L. Dontchev, M. Quincampoix and N. Zlateva, Aubin criterion for metric regularity, Journal of Convex Analysis, 13 (2006), 281-297.  Google Scholar

[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.  Google Scholar

[12]

N. Q. Huy and J. C. Yao, Stability of implicit multifunctions in Asplund spaces, Taiwanese Journal of Mathematics, 13 (2009), 47-65.  Google Scholar

[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.  Google Scholar

[14]

B. T. Kien, On the lower semicontinuity of optimal solution sets, Optimization, 54 (2005), 123-130. doi: doi:10.1080/02331930412331330379.  Google Scholar

[15]

Y. S. Ledyaev and Q. J. Zhu, Implicit multifunctions theorems, Set-Valued Analysis, 7 (1999), 209-238. doi: doi:10.1023/A:1008775413250.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications," Springer, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

S. M. Robinson, Generalized equations and their solutions, Part I, Basic theory, Mathematical Programming Study, 10 (1979), 128-141.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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