# American Institute of Mathematical Sciences

December  2011, 31(4): 1383-1396. doi: 10.3934/dcds.2011.31.1383

## Solving strongly monotone variational and quasi-variational inequalities

 1 Center for Operations Research and Econometrics (CORE), Catholic University of Louvain, 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium 2 Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania

Received  November 2009 Revised  September 2010 Published  September 2011

In this paper we develop a new and efficient method for variational inequality with Lipschitz continuous strongly monotone operator. Our analysis is based on a new strongly convex merit function. We apply a variant of the developed scheme for solving quasivariational inequalities. As a result, we significantly improve the standard sufficient condition for existence and uniqueness of their solutions. Moreover, we get a new numerical scheme, whose rate of convergence is much higher than that of the straightforward gradient method.
Citation: Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383
##### References:
 [1] C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalities: Applications to Free Boundary Problems," A Wiley Interscience Publication, John Wiley & Sons, Inc., New York, 1984. [2] A. Bensoussan, M. Goursat and J.-L- Lions, Contrôle impulsionnel et inéquations quasi-variationnelle stationnaires, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1279-A1284. [3] A. Bensoussan, Points de Nash dans le cas de fontionnelles quadratiques et jeux différentiels linéaires à N personnes, SIAM J. Control, 12 (1974), 460-499. doi: 10.1137/0312037. [4] M. Bliemer and P. Bovy, Quasi-variational inequality formulation of the multiclass dynamic traffic assignment problem, Transportation Res. Part B, 37 (2003), 501-519. doi: 10.1016/S0191-2615(02)00025-5. [5] A. Causa and F. Raciti, Lipschitz continuity results for a class of variational inequalities and applications: A geometric approach, J. Optim. Theory Appl., 145 (2010), 235-248. doi: 10.1007/s10957-009-9622-4. [6] D. Chan and J. S. Pang, The generalized quasivariational inequality problem, Math. Oper. Res., 7 (1982), 211-222. doi: 10.1287/moor.7.2.211. [7] F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems," Springer Series in Operations Research, Springer-Verlag, New York, 2003. [8] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Program., 53 (1992), 99-110. doi: 10.1007/BF01585696. [9] F. Giannessi, A. Maugeri and P. M. Pardalos, eds., "Variational Analysis and Applications," Kluwer Academic Publishers, 2005. [10] P. T. Harker, Generalized Nash games and quasi-variational inequalities, Eur. J. Oper. Res., 54 (1991), 81-94. doi: 10.1016/0377-2217(91)90325-P. [11] J. Haslinger, Aproximation of the Signorini problem with friction, obeying Coulomb law, Math. Methods Appl. Sci., 5 (1983), 422-437. doi: 10.1002/mma.1670050127. [12] J. Haslinger and P. D. Panagiotopoulos, The reciprocal variational approach to the Signorini problem with friction. Approximation results, Proc. Royal Soc.of Edinburgh, Sect.A, 98 (1984), 365-383. [13] M. Kocvara and J. V. Outrata, On a class of quasi-variational inequalities, Optim. Methods Softw., 5 (1995), 275-295. doi: 10.1080/10556789508805617. [14] A. Maugeri and L. Scrimali, Global Lipschitz continuity of solutions to parameterized variational inequalities, Boll. Unione Mat. Ital. (9), 2 (2009), 45-69. [15] G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 314-321. doi: 10.1090/S0002-9904-1967-11732-4. [16] U. Mosco, Implicit variational problems and quasi variational inequalities, in "Nonlinear Operators and Calculus of Variations" (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Lectures Notes Math, 543, Springer, Berlin, (1976), 83-156. [17] A. Nagurney, "Network Economics: A Variational Inequality Approach," Advances in Computational Economics, 1, Kluwer Academic Publishers Group, Dordrecht, 1993. [18] A. Nemirovski, Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM J. Optim., 15 (2004), 229-251. doi: 10.1137/S1052623403425629. [19] A. Nemirovsky and D. Yudin, "Problem Complexity and Method Efficiency in Optimization," A Wiley Interscience Publication, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1983. [20] Yu. Nesterov, "Introductory Lectures on Convex Optimization. A Basic Course," Applied Optimization, 87, Kluwer Academic Publishers, Boston, MA, 2004. [21] Yu. Nesterov, Smooth minimization of nonsmooth function, Math. Program., 103 (2005), 127-152. doi: 10.1007/s10107-004-0552-5. [22] Yu. Nesterov, Dual extrapolation and its applications to solving variational inequalities and related problems, Math. Program., 109 (2007), 319-344. doi: 10.1007/s10107-006-0034-z. [23] Yu. Nesterov, Minimizing functions with bounded variation of the gradient, CORE DP 2005/79. [24] M. A. Noor and Z. A. Memon, Algorithms for general mixed quasi variational inequalities, J. Inequal. Pure Appl. Math., 3 (2002), 7 pp. [25] M. A. Noor and W. Oettli, On general nonlinear complementarity problems and quasi-equilibria, Le Matematiche (Catania), 49, (1994) 313-331. [26] J. Outrata and J. Zowe, A numerical approach to optimization problems with variational inequality constraints, Math. Program., 68 (1995), 105-130. doi: 10.1007/BF01585759. [27] J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria and multi-leader-follower games, Comput. Manag. Sci., 1 (2005), 21-56. doi: 10.1007/s10287-004-0010-0. [28] Salahuddin, Projection methods for quasi-variational inequalities, Mathematical and Computational Applications, 9 (2004), 125-131. [29] L. Scrimali, The financial equilibrium problem with implicit budget constraints, Cent. Eur. J. Oper. Res., 16 (2008), 191-203. doi: 10.1007/s10100-007-0046-7. [30] L. Scrimali, Mixed behavior network equilibria and quasi-variational inequalities, J. Ind. Manag. Optim., 5 (2009), 363-379. doi: 10.3934/jimo.2009.5.363. [31] J. C. Yao, The generalized quasi-variational inequality problem with applications, J. Math. Anal. Appl., 158 (1991), 139-160. doi: 10.1016/0022-247X(91)90273-3. [32] J. Y. Wei and Y. Smeers, Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices, Oper. Res., 47 (1999), 102-112. doi: 10.1287/opre.47.1.102.

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##### References:
 [1] C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalities: Applications to Free Boundary Problems," A Wiley Interscience Publication, John Wiley & Sons, Inc., New York, 1984. [2] A. Bensoussan, M. Goursat and J.-L- Lions, Contrôle impulsionnel et inéquations quasi-variationnelle stationnaires, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1279-A1284. [3] A. Bensoussan, Points de Nash dans le cas de fontionnelles quadratiques et jeux différentiels linéaires à N personnes, SIAM J. Control, 12 (1974), 460-499. doi: 10.1137/0312037. [4] M. Bliemer and P. Bovy, Quasi-variational inequality formulation of the multiclass dynamic traffic assignment problem, Transportation Res. Part B, 37 (2003), 501-519. doi: 10.1016/S0191-2615(02)00025-5. [5] A. Causa and F. Raciti, Lipschitz continuity results for a class of variational inequalities and applications: A geometric approach, J. Optim. Theory Appl., 145 (2010), 235-248. doi: 10.1007/s10957-009-9622-4. [6] D. Chan and J. S. Pang, The generalized quasivariational inequality problem, Math. Oper. Res., 7 (1982), 211-222. doi: 10.1287/moor.7.2.211. [7] F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems," Springer Series in Operations Research, Springer-Verlag, New York, 2003. [8] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Program., 53 (1992), 99-110. doi: 10.1007/BF01585696. [9] F. Giannessi, A. Maugeri and P. M. Pardalos, eds., "Variational Analysis and Applications," Kluwer Academic Publishers, 2005. [10] P. T. Harker, Generalized Nash games and quasi-variational inequalities, Eur. J. Oper. Res., 54 (1991), 81-94. doi: 10.1016/0377-2217(91)90325-P. [11] J. Haslinger, Aproximation of the Signorini problem with friction, obeying Coulomb law, Math. Methods Appl. Sci., 5 (1983), 422-437. doi: 10.1002/mma.1670050127. [12] J. Haslinger and P. D. Panagiotopoulos, The reciprocal variational approach to the Signorini problem with friction. Approximation results, Proc. Royal Soc.of Edinburgh, Sect.A, 98 (1984), 365-383. [13] M. Kocvara and J. V. Outrata, On a class of quasi-variational inequalities, Optim. Methods Softw., 5 (1995), 275-295. doi: 10.1080/10556789508805617. [14] A. Maugeri and L. Scrimali, Global Lipschitz continuity of solutions to parameterized variational inequalities, Boll. Unione Mat. Ital. (9), 2 (2009), 45-69. [15] G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 314-321. doi: 10.1090/S0002-9904-1967-11732-4. [16] U. Mosco, Implicit variational problems and quasi variational inequalities, in "Nonlinear Operators and Calculus of Variations" (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Lectures Notes Math, 543, Springer, Berlin, (1976), 83-156. [17] A. Nagurney, "Network Economics: A Variational Inequality Approach," Advances in Computational Economics, 1, Kluwer Academic Publishers Group, Dordrecht, 1993. [18] A. Nemirovski, Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM J. Optim., 15 (2004), 229-251. doi: 10.1137/S1052623403425629. [19] A. Nemirovsky and D. Yudin, "Problem Complexity and Method Efficiency in Optimization," A Wiley Interscience Publication, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1983. [20] Yu. Nesterov, "Introductory Lectures on Convex Optimization. A Basic Course," Applied Optimization, 87, Kluwer Academic Publishers, Boston, MA, 2004. [21] Yu. Nesterov, Smooth minimization of nonsmooth function, Math. Program., 103 (2005), 127-152. doi: 10.1007/s10107-004-0552-5. [22] Yu. Nesterov, Dual extrapolation and its applications to solving variational inequalities and related problems, Math. Program., 109 (2007), 319-344. doi: 10.1007/s10107-006-0034-z. [23] Yu. Nesterov, Minimizing functions with bounded variation of the gradient, CORE DP 2005/79. [24] M. A. Noor and Z. A. Memon, Algorithms for general mixed quasi variational inequalities, J. Inequal. Pure Appl. Math., 3 (2002), 7 pp. [25] M. A. Noor and W. Oettli, On general nonlinear complementarity problems and quasi-equilibria, Le Matematiche (Catania), 49, (1994) 313-331. [26] J. Outrata and J. Zowe, A numerical approach to optimization problems with variational inequality constraints, Math. Program., 68 (1995), 105-130. doi: 10.1007/BF01585759. [27] J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria and multi-leader-follower games, Comput. Manag. Sci., 1 (2005), 21-56. doi: 10.1007/s10287-004-0010-0. [28] Salahuddin, Projection methods for quasi-variational inequalities, Mathematical and Computational Applications, 9 (2004), 125-131. [29] L. Scrimali, The financial equilibrium problem with implicit budget constraints, Cent. Eur. J. Oper. Res., 16 (2008), 191-203. doi: 10.1007/s10100-007-0046-7. [30] L. Scrimali, Mixed behavior network equilibria and quasi-variational inequalities, J. Ind. Manag. Optim., 5 (2009), 363-379. doi: 10.3934/jimo.2009.5.363. [31] J. C. Yao, The generalized quasi-variational inequality problem with applications, J. Math. Anal. Appl., 158 (1991), 139-160. doi: 10.1016/0022-247X(91)90273-3. [32] J. Y. Wei and Y. Smeers, Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices, Oper. Res., 47 (1999), 102-112. doi: 10.1287/opre.47.1.102.
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