Geodesic
Global Lipschitz Continuity of Solutions to Parameterized Variational Inequalities
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 45-69.

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The question of Lipschitz continuity of solutions to parameterized variational inequalities with perturbed constraint sets is considered. Under the sole Lipschitz continuity assumption on data, a Lipschitz continuity result is proved which, in particular, holds for variational inequalities modeling evolutionary network equilibrium problems. Moreover, in view of real-life applications, a long-term memory is introduced and the corresponding variational inequality model is discussed. 
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Maugeri, Antonino; Scrimali, Laura. Global Lipschitz Continuity of Solutions to Parameterized Variational Inequalities. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 45-69. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_1_a2/

[1] A. Arós - M. Sofonea - J. M. Viaño, A class of evolutionary variational inequalities with Volterra-type term, Math. Models Methods Appl. Sci., 14, No. 4 (2004), 557-577. | DOI | MR | Zbl

[2] A. Barbagallo, Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems, J. Global Optim., 40, No. 1-3 (2008). | DOI | MR | Zbl

[3] L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Sitzber. Kaiserl. Akad. Wiss. Wien, Math.-Naturw. Kl., 70, Sect. II (1874), 275-300.

[4] M. J. Beckman - J. P. Wallace, Continuous lags and the stability of market equilibrium, Economica, New Series, 36, No. 141 (1969), 58-68.

[5] L. Boltzmann, Zur Theorie der elastischen Nachwirkung, Ann. Phys. u. Chem., 5 (1878), 430-432. | Zbl

[6] J. F. Bonnans - A. Shapiro, Perturbation analysis of optimization problems, Springer, New York (2000). | DOI | MR | Zbl

[7] B. D. Coleman - W. Noll, Foundations of linear viscoelasticity, Rev. of Modern Physics, 33, No. 2 (1961), 239-249. | DOI | MR | Zbl

[8] P. Daniele - A. Maugeri - W. Oettli, Time-dependent traffic equilibria, J. Optim. Theory Appl., 103, No. 3 (1999), 543-554. | DOI | MR | Zbl

[9] P. Daniele - A. Maugeri, Variational inequalities and discrete and continuum models of network equilibrium problems, Math. Comput. Model., 35 (2002), 689-708. | DOI | MR | Zbl

[10] P. DANIELE - F. GIANNESSI - A. MAUGERI (Eds.), Equilibrium problems and variational models, Kluwer Academic Publishers (2003). | DOI | MR

[11] P. Daniele, Dynamic networks and evolutionary variational inequalities, Edward Elgar Publishing (2006). | MR | Zbl

[12] P. Daniele - S. Giuffrè, General infinite dimensional duality and applications to evolutionary network equilibrium problems, Optim. Lett., 1, No. 3 (2007), 227-243. | DOI | MR | Zbl

[13] P. Daniele - S. Giuffrè - G. Idone - A. Maugeri, Infinite dimensional duality and applications, Math. Ann., 339, No. 1 (2007), 221-239. | DOI | MR | Zbl

[14] F. Facchinei - J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer, New York (2003). | MR | Zbl

[15] A. V. Fiacco - G. P. Mccormick, Nonlinear programming: sequential unconstrained minimization techniques, Wiley, New York (1968). | MR | Zbl

[16] A. V. Fiacco, Sensitivity analysis for nonlinear programming using penality methods, Math. Program., 10 (1976), 287-311. | DOI | MR | Zbl

[17] A. V. Fiacco, An introduction to sensitivity and stability analysis in Nonlinear programming, Academic Press, New York (1983). | MR | Zbl

[18] G. Fichera, Integro-differential problems of hereditary elasticity, Atti Sem. Mat. Fis. Univ. Modena, XXVI (1977), 363-370. | MR | Zbl

[19] F. GIANNESSI - A. MAUGERI (Eds.), Variational inequalities and network equilibrium problems, Plenum Publishing, New York (1995). | DOI | MR

[20] F. GIANNESSI - A. MAUGERI - P. PARDALOS (Eds.), Equilibrium problems: non-smooth optimization and variational inequality models, Kluwer Academic Publishers, Dordrecht, The Netherlands (2001). | MR

[21] F. Giannessi - A. Maugeri, Preface [Special issue on the Proeceedings of the First AMS-UMI Joint Meeting], J. Global Optim., 28, No. 3-4 (2004). | DOI | MR

[22] F. GIANNESSI - A. MAUGERI (Eds.), Variational inequalities and applications, Springer, New York (2005).

[23] J. Gwinner, Time-dependent variational inequalities. Some recent trends. In Daniele P., Giannessi F. and Maugeri A. (Eds.), Equilibrium problems and variational models, Academic Publishers, Dordrecht, The Netherlands (2003), 225-264. | DOI | MR | Zbl

[24] J. Heinonen, Lectures on Lipschitz analysis. In Lectures at the 14th Jyväskylä Summer School (August, 2004). | MR | Zbl

[25] J. Jahn, Introduction to the theory of nonlinear optimization, Springer-Verlag, Berlin, Germany (1996). | DOI | MR | Zbl

[26] D. Kinderleher - G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York (1980). | MR

[27] J. Kyparisis, Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems, Math. Program., 36 (1986), 105-113. | DOI | MR | Zbl

[28] J. Kyparisis, Solution differentiability for variational inequalities, Math. Program., 48 (1990), 285-301. | DOI | MR | Zbl

[29] J. Kyparisis, Sensitivity analysis for variational inequalities and nonlinear complementarity problems, Ann. Oper. Res., 27 (1990), 143-174. | DOI | MR | Zbl

[30] J. L. Lions - G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 22 (1967) 493-519. | DOI | MR

[31] A. Maugeri, Variational and quasi-variational inequalities and applications to optmization problems in networks, Boll. Un. Mat. Ital. B, 7, No. 4 (1990), 327-343. | MR | Zbl

[32] A. Maugeri, Dynamic Models and generalized equilibrium problems. In Giannessi F. (Ed.), New Trends in Mathematical Programming, Kluwer Academic Publishers (1998), 191-202. | DOI | MR | Zbl

[33] A. Maugeri - C. Vitanza, Time-dependent equilibrium problems. In Migdalos A., Pardalos P. and Pitsoulis L. (Eds.), Pareto Optimality Game Theory and Equilibria, Springer (2008), 505-524. | DOI | MR | Zbl

[34] B. Mordukhovich, Variational analysis and generalized differentiation (Springer-Verlag , 2006). | MR

[35] Y. Qui - T. L. Magnanti, Sensitivity analysis for variational inequalities, Math. Oper. Res., 17 (1992), 61-76. | DOI | MR | Zbl

[36] S. M. Robinson, Some continuity properties of polyhedral multifunctions, Math. Programming Stud., 14 (1981), 206-214. | DOI | MR | Zbl

[37] S. M. Robinson, Generalized equations and their solutions, part II: applications to nonlinear programming, Math. Programming Stud., 19 (1982), 200-221. | DOI | MR | Zbl

[38] S. M. Robinson, Implicit B-Differentiability in generalized equations, Technical Summary Report 2854, Mathematics Research center, University of Wisconsin, Madison, WI (1985).

[39] S. M. Robinson - S. Lu, Solution continuity in variational conditions, J. Global Optim., 40, No. 1-3 (2008). | DOI | MR | Zbl

[40] L. Scrimali, Quasi-variational inequalities in transportation networks, Math. Models Methods Appl. Sci., 14, No. 10 (2004), 1541-1560. | DOI | MR | Zbl

[41] L. Scrimali, The financial equilibrium problem with implicit budget constraints, Cent. Eur. J. Oper. Res., 16, No. 2 (2008), 191-203. | DOI | MR | Zbl

[42] L. Scrimali, A solution differentiability result for evolutionary quasi-variational inequalities, J. Global Optim., 40, No. 1-3 (2008). | DOI | MR | Zbl

[43] A. Shapiro, Sensitivity analysis of nonlinear programs and differentiability properties of metric projections, SIAM J. Control Optim., 26 (1988), 628-645. | DOI | MR | Zbl

[44] A. Shapiro, Sensitivity analysis of parameterized variational inequalities, Math. Oper. Res., 30, No. 1 (2005), 109-126. | DOI | MR | Zbl

[45] M. J. Smith, A new dynamic traffic model and the existence and calculation of dynamic user equilbria on congested capacity-constrained road networks, Transportation Res. B, 27B, No. 1 (1993), 49-63.

[46] J. Steinbach, On a variational inequality containing a memory term with an application in electro-chemical machining, J. Convex Anal., 5, No. 1 (1998), 63-80. | fulltext EuDML | MR | Zbl

[47] J. Steinbach, A variational inequality approach to free boundary problems with applications in mould filling (Birkhäuser-Verlag, 2002). | DOI | MR | Zbl

[48] V. Volterra, Sulle equazioni integro-differenziali della teoria della elasticità, Rend. Acc. Naz. Lincei, XVIII, No. 2 (1909), 295-301. | Zbl

[49] V. Volterra, Sulle equazioni integro-differenziali della elasticità nel caso della entropia, Rend. Acc. Naz. Lincei, XVIII, No. 2 (1909), 577-586. | Zbl

[50] V. Volterra, Sur les equation intégro-differentielles et leurs applications, Acta Math., XXXV (1912), 295-356. | DOI | MR | Zbl

[51] J. G. Wardrop, Some theoretical aspects of road traffic research. In Proceedings of the Institute of Civil Engineers, Part II (1952), 325-378).

[52] N. D. Yen, Hölder continuity of solutions to a parametric variational inequality, Appl. Math. Optim., 31 (1995), 245-255. | DOI | MR

[53] N. D. Yen, Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint, Math. Oper. Res., 20 (1995), 695-708. | DOI | MR | Zbl