Abstract
In this paper we propose a class of differentiable gap functions in order to formulate a generalized variational inequality (GVI) problem, involving a set-valued map with closed and convex graph, as an optimization problem. We also show that under appropriate assumptions on the set-valued map, any stationary point of the equivalent optimization problem is a global optimal solution and solves the GVI. Finally, we describe descent methods for solving the optimization problem equivalent to the GVI and we prove its global convergence.
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References
Auslender, A.: Optimisation: Méthodes Numériques. Masson, Paris (1976)
Bureau of Public Roads: Traffic Assignment Manual. U.S. Department of Commerce, Urban Planning Division, Washington (1964)
Dafermos, S.: Traffic equilibria and variational inequalities. Transp. Sci. 14, 42–54 (1980)
Dietrich, H.: A smooth dual gap function solution to a class of quasivariational inequalities. J. Math. Anal. Appl. 235, 380–393 (1999)
Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)
Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, New York (2001)
Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)
Lu, S.: Sensitivity of static traffic user equilibria with perturbations in arc cost function and travel demand. Transp. Sci. 42, 105–123 (2008)
Panicucci, B., Pappalardo, M., Passacantando, M.: On finite-dimensional generalized variational inequalities. J. Ind. Manag. Optim. 2, 43–53 (2006)
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)
Zangwill, W.I.: Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood Cliffs (1969)
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Paper presented at the Erice 2007 Workshop on Nonlinear Optimization. This work has been supported by the National Research Program PRIN/2005017083 “Innovative Problems and Methods in Nonlinear Optimization”.
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Panicucci, B., Pappalardo, M. & Passacantando, M. Descent methods for a class of generalized variational inequalities. Comput Optim Appl 45, 415–425 (2010). https://doi.org/10.1007/s10589-008-9230-5
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DOI: https://doi.org/10.1007/s10589-008-9230-5