Abstract
This paper establishes an upper estimate for the Fréchet normal cone to the graph of the nonlinearly perturbed polyhedral normal cone mappings in finite dimensional spaces. Under a positive linear independence assumption on the normal vectors of the active constraints at the point in question, the result leads to an upper estimate for values of the Mordukhovich coderivative of such mappings. On the basis, new results on solution stability of parametric affine variational inequalities under nonlinear perturbations are derived.
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References
Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)
Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)
Lu, S., Robinson, S.M.: Variational inequalities over perturbed polyhedral convex sets. Math. Oper. Res. 33, 689–711 (2008)
Nam, N.M.: Coderivatives of normal mappings and the Lipschitzian stability of parametric variational inequalities. Nonlinear Anal. 73, 2271–2282 (2010)
Robinson, S.M.: Solution continuity in monotone affine variational inequalities. SIAM J. Optim. 18, 1046–1060 (2007)
Robinson, S.M., Lu, S.: Solution continuity in variational conditions. J. Glob. Optim. 40, 405–415 (2008)
Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality, Part 1: Basic calculations. Acta Math. Vietnam. 34, 157–172 (2009)
Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality, Part 2: Applications. Pacific. J. Optim. 5, 493–506 (2009)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)
Qui, N.T.: Upper and lower estimates for a Fréchet normal cone. Acta Math. Vietnam. (2011, in press)
Qui, N.T.: Linearly pertubed polyhedral normal cone mappings and applications. Nonlinear Anal. 74, 1676–1689 (2011)
Qui, N.T.: New results on linearly pertubed polyhedral normal cone mappings. J. Math. Anal. Appl. 381, 352–364 (2011)
Robinson, S.M.: Generalized equations and their solutions. I. Basic theory. Point-to-set maps and mathematical programming. Math. Program. Stud. 10, 128–141 (1979)
Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)
Phung, H.T.: On the locally uniform openness of polyhedral sets. Acta Math. Vietnam. 25, 273–284 (2000)
Bartl, D.: A short algebraic proof of the Farkas lemma. SIAM J. Optim. 19, 234–239 (2008)
Tam, N.N., Yen, N.D.: Continuity properties of the Karush–Kuhn–Tucker point set in quadratic programming problems. Math. Program. 85, 193–206 (1999)
Lee, G.M., Yen, N.D.: Fréchet and normal coderivatives of implicit multifunctions. Appl. Anal. 90(6), 1011–1027 (2011)
Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Program. 99, 311–327 (2004)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, a View from Variational Analysis. Springer, Dordrecht (2009)
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Communicated by Nguyen Dong Yen.
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Qui, N.T. Nonlinear Perturbations of Polyhedral Normal Cone Mappings and Affine Variational Inequalities. J Optim Theory Appl 153, 98–122 (2012). https://doi.org/10.1007/s10957-011-9937-9
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DOI: https://doi.org/10.1007/s10957-011-9937-9