Global Lipschitz Continuity of Solutions to Parameterized Variational Inequalities
Maugeri, Antonino ; Scrimali, Laura
Bollettino dell'Unione Matematica Italiana, Tome 2 (2009), p. 45-69 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

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.

Publié le : 2009-02-01
@article{BUMI_2009_9_2_1_45_0,
     author = {Antonino Maugeri and Laura Scrimali},
     title = {Global Lipschitz Continuity of Solutions to Parameterized Variational Inequalities},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {2},
     year = {2009},
     pages = {45-69},
     zbl = {1170.49011},
     mrnumber = {2493644},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_1_45_0}
}

Maugeri, Antonino; Scrimali, Laura. Global Lipschitz Continuity of Solutions to Parameterized Variational Inequalities. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 45-69. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_1_45_0/

[1] Arós, A. - Sofonea, M. - Viaño, J. M., A class of evolutionary variational inequalities with Volterra-type term, Math. Models Methods Appl. Sci., 14, No. 4 (2004), 557-577. | MR 2046578 | Zbl 1077.74035

[2] Barbagallo, A., Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems, J. Global Optim., 40, No. 1-3 (2008). | MR 2373537 | Zbl 1149.49032

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

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

[5] Boltzmann, L., Zur Theorie der elastischen Nachwirkung, Ann. Phys. u. Chem., 5 (1878), 430-432. | Zbl 10.0670.01

[6] Bonnans, J. F. - Shapiro, A., Perturbation analysis of optimization problems, Springer, New York (2000). | MR 1756264 | Zbl 0966.49001

[7] Coleman, B. D. - Noll, W., Foundations of linear viscoelasticity, Rev. of Modern Physics, 33, No. 2 (1961), 239-249. | MR 158605 | Zbl 0103.40804

[8] Daniele, P. - Maugeri, A. - Oettli, W., Time-dependent traffic equilibria, J. Optim. Theory Appl., 103, No. 3 (1999), 543-554. | MR 1727254 | Zbl 0937.90005

[9] Daniele, P. - Maugeri, A., Variational inequalities and discrete and continuum models of network equilibrium problems, Math. Comput. Model., 35 (2002), 689-708. | MR 1884027 | Zbl 0994.90033

[10] P. Daniele - F. Giannessi - A. Maugeri (Eds.), Equilibrium problems and variational models, Kluwer Academic Publishers (2003). | MR 2026132

[11] Daniele, P., Dynamic networks and evolutionary variational inequalities, Edward Elgar Publishing (2006). | MR 2273417 | Zbl 1117.49002

[12] Daniele, P. - Giuffrè, S., General infinite dimensional duality and applications to evolutionary network equilibrium problems, Optim. Lett., 1, No. 3 (2007), 227-243. | MR 2340691 | Zbl 1151.90577

[13] Daniele, P. - Giuffrè, S. - Idone, G. - Maugeri, A., Infinite dimensional duality and applications, Math. Ann., 339, No. 1 (2007), 221-239. | MR 2317768 | Zbl 1356.49064

[14] Facchinei, F. - Pang, J. S., Finite-dimensional variational inequalities and complementarity problems, Springer, New York (2003). | MR 1955649 | Zbl 1062.90001

[15] Fiacco, A. V. - Mccormick, G. P., Nonlinear programming: sequential unconstrained minimization techniques, Wiley, New York (1968). | MR 243831 | Zbl 0193.18805

[16] Fiacco, A. V., Sensitivity analysis for nonlinear programming using penality methods, Math. Program., 10 (1976), 287-311. | MR 434444 | Zbl 0357.90064

[17] Fiacco, A. V., An introduction to sensitivity and stability analysis in Nonlinear programming, Academic Press, New York (1983). | MR 721641 | Zbl 0543.90075

[18] Fichera, G., Integro-differential problems of hereditary elasticity, Atti Sem. Mat. Fis. Univ. Modena, XXVI (1977), 363-370. | MR 532388 | Zbl 0407.73005

[19] F. Giannessi - A. Maugeri (Eds.), Variational inequalities and network equilibrium problems, Plenum Publishing, New York (1995). | MR 1331396

[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 2030621

[21] Giannessi, F. - Maugeri, A., Preface [Special issue on the Proeceedings of the First AMS-UMI Joint Meeting], J. Global Optim., 28, No. 3-4 (2004). | MR 2072682

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

[23] Gwinner, J., 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. | MR 2043474 | Zbl 1069.49005

[24] Heinonen, J., Lectures on Lipschitz analysis. In Lectures at the 14th Jyväskylä Summer School (August, 2004). | MR 2177410 | Zbl 1086.30003

[25] Jahn, J., Introduction to the theory of nonlinear optimization, Springer-Verlag, Berlin, Germany (1996). | MR 1429391 | Zbl 0855.49001

[26] Kinderleher, D. - Stampacchia, G., An introduction to variational inequalities and their applications, Academic Press, New York (1980). | MR 567696

[27] Kyparisis, J., Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems, Math. Program., 36 (1986), 105-113. | MR 862072 | Zbl 0613.90096

[28] Kyparisis, J., Solution differentiability for variational inequalities, Math. Program., 48 (1990), 285-301. | MR 1073712 | Zbl 0727.90082

[29] Kyparisis, J., Sensitivity analysis for variational inequalities and nonlinear complementarity problems, Ann. Oper. Res., 27 (1990), 143-174. | MR 1088991 | Zbl 0723.90075

[30] Lions, J. L. - Stampacchia, G., Variational inequalities, Comm. Pure Appl. Math., 22 (1967) 493-519. | MR 216344

[31] Maugeri, A., Variational and quasi-variational inequalities and applications to optmization problems in networks, Boll. Un. Mat. Ital. B, 7, No. 4 (1990), 327-343. | MR 1061221 | Zbl 0705.90023

[32] Maugeri, A., Dynamic Models and generalized equilibrium problems. In Giannessi F. (Ed.), New Trends in Mathematical Programming, Kluwer Academic Publishers (1998), 191-202. | MR 1641319 | Zbl 0908.90120

[33] Maugeri, A. - Vitanza, C., Time-dependent equilibrium problems. In Migdalos A., Pardalos P. and Pitsoulis L. (Eds.), Pareto Optimality Game Theory and Equilibria, Springer (2008), 505-524. | MR 2441391 | Zbl 1191.49009

[34] Mordukhovich, B., Variational analysis and generalized differentiation (Springer-Verlag , 2006). | MR 2191745

[35] Qui, Y. - Magnanti, T. L., Sensitivity analysis for variational inequalities, Math. Oper. Res., 17 (1992), 61-76. | MR 1148778 | Zbl 0756.49009

[36] Robinson, S. M., Some continuity properties of polyhedral multifunctions, Math. Programming Stud., 14 (1981), 206-214. | MR 600130 | Zbl 0449.90090

[37] Robinson, S. M., Generalized equations and their solutions, part II: applications to nonlinear programming, Math. Programming Stud., 19 (1982), 200-221. | MR 669732 | Zbl 0495.90077

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

[39] Robinson, S. M. - Lu, S., Solution continuity in variational conditions, J. Global Optim., 40, No. 1-3 (2008). | MR 2373568 | Zbl 1145.90096

[40] Scrimali, L., Quasi-variational inequalities in transportation networks, Math. Models Methods Appl. Sci., 14, No. 10 (2004), 1541-1560. | MR 2095302 | Zbl 1069.90026

[41] Scrimali, L., The financial equilibrium problem with implicit budget constraints, Cent. Eur. J. Oper. Res., 16, No. 2 (2008), 191-203. | MR 2407047 | Zbl 1145.91028

[42] Scrimali, L., A solution differentiability result for evolutionary quasi-variational inequalities, J. Global Optim., 40, No. 1-3 (2008). | MR 2373569 | Zbl 1282.49008

[43] Shapiro, A., Sensitivity analysis of nonlinear programs and differentiability properties of metric projections, SIAM J. Control Optim., 26 (1988), 628-645. | MR 937676 | Zbl 0647.90089

[44] Shapiro, A., Sensitivity analysis of parameterized variational inequalities, Math. Oper. Res., 30, No. 1 (2005), 109-126. | MR 2125140 | Zbl 1082.49015

[45] Smith, M. J., 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] Steinbach, J., On a variational inequality containing a memory term with an application in electro-chemical machining, J. Convex Anal., 5, No. 1 (1998), 63-80. | MR 1649441 | Zbl 0908.49011

[47] Steinbach, J., A variational inequality approach to free boundary problems with applications in mould filling (Birkhäuser-Verlag, 2002). | MR 1891393 | Zbl 1011.35001

[48] Volterra, V., Sulle equazioni integro-differenziali della teoria della elasticità, Rend. Acc. Naz. Lincei, XVIII, No. 2 (1909), 295-301. | Zbl 40.0870.01

[49] Volterra, V., Sulle equazioni integro-differenziali della elasticità nel caso della entropia, Rend. Acc. Naz. Lincei, XVIII, No. 2 (1909), 577-586. | Zbl 40.0871.01

[50] Volterra, V., Sur les equation intégro-differentielles et leurs applications, Acta Math., XXXV (1912), 295-356. | MR 1555080 | Zbl 43.0431.03

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

[52] Yen, N. D., Hölder continuity of solutions to a parametric variational inequality, Appl. Math. Optim., 31 (1995), 245-255. | MR 1316259

[53] Yen, N. D., Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint, Math. Oper. Res., 20 (1995), 695-708. | MR 1354777 | Zbl 0845.90116