Properties of gap functions for mixed variational inequalities

I. V. Konnov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 3 (2000) no. 3, pp. 259-270.

Résumé

Various gap functions for a class of mixed variational inequalities containing a $P$-mapping function and a convex separable function which are not necessarily differentiable are considered. Such problems have a number of applications in mathematical physics, economics, and operations research. The initial problem is shown to be equivalent to a constrained optimization problem for a conventional gap function which may be non-differentiable. At the same time, the $D$-gap function allows one to reduce the initial problem to the problem of finding stationary points of a continuously differentiable function. This latter problem can be solved by standard unconstrained differentiable optimization methods.

Bibliographie

[1] Browder F. E., “On the unification of the calculus of variations and the theory of monotone nonlinear operators in Banach spaces”, Proc. of the National Academy of Sciences of the USA, 56:2 (1966), 419–425 | DOI | MR | Zbl

[2] Duvaut G., Lions J.-L., Les Inéquations en Mechanique et Physique, Dunod, Paris, 1972 | MR | Zbl

[3] Baiocchi C., Capelo A., Variational and Quasi-variational Inequalities. Applications to Free Boundary Problems, John Wiley and Sons, New York, 1984 | MR | Zbl

[4] Patriksstm M., Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Kluwer Academic Publishers, Dordrecht, 1999 | MR

[5] Ortega J. M., Rheinboldt W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970 | MR

[6] Opoitsev V. I., Nonlinear System Statics, Nauka, Moscow, 1986 | MR

[7] Polterovich V. M., Equilibrium and Economic Mechanism, Nauka, Moscow, 1990

[8] Lapin A., Iterative solution for two classes of mesh variational inequalities, Preprint, University of Oulu, Oulu, 1999

[9] More J. J., “Classes of functions and feasibility conditions in nonlinear complementarity problems”, Math. Progr., 6:3 (1974), 327–338 | DOI | MR | Zbl

[10] Kanzow C., Fukushima M., “Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities”, Math. Progr., 83:1 (1998), 55–87 | DOI | MR | Zbl

[11] Facchinei F., Kanzow C., “Beyond monotonicity in regularization methods for nonlinear complementarity problems”, SIAM J. Control and Optimiz., 37:4 (1999), 1150–1161 | DOI | MR | Zbl

[12] Peng J.-M., “Equivalence of variational inequality problems to unconstrained minimization”, Math. Progr., 78:3 (1997), 347–355 | DOI | MR | Zbl

[13] Yamashita N., Taji K., Fukushima M., “Unconstrained optimization formulations of variational inequality problems”, J. of Optimiz. Theory and Appl., 92:3 (1997), 439–456 | DOI | MR | Zbl

[14] Rockafellar R. T., Convex Analysis, Princeton University Press, Princeton, 1970 | MR

[15] Dem'yanov V. F., Vasil'yev L. V., Non-differentiable Optimization, Nauka, Moscow, 1981

[16] Konnov I. V., Methods for Solving Finite-Dimensional Variational Inequalities, DAS, Kazan, 1998 | MR