Abstract
In this paper, we consider a generalized variational inequality problem which involves the integrable cost mapping and a nonsmooth mapping with convex components. We propose a new gradient-type method which determines a stepsize by using the smooth part of the cost function. Thus, the method does not utilize analogs of derivatives of nonsmooth functions. We show that its convergence does not require additional assumptions.
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References
G. Isac, Complementarity Problems (Springer-Verlag, Berlin, 1992).
J.-C. Yao, “Existence of Generalized Variational Inequalities,” Oper. Res. Lett. 15(1), 35–40 (1995).
I. V. Konnov, “Combined Relaxation Methods for Generalized Variational Inequalities,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 12, 46–64 (2001) [RussianMathematics (Iz. VUZ) 45 (12), 43–51 (2001)].
E. A. Nurminskii and N. B. Shamrai, “Method of Local Convex Majorants for Variational-Like Inequalities,” Comp. Maths. and Math. Phys. 47(3), 355–363 (2007).
B. Martos, Nonlinear Programming. Theory and Methods (Akad émiai Kiado, Budapest, 1975).
A. G. Sukharev, A. V. Timokhov, and V. V. Fedorov, A Course in Optimization Methods (Nauka, Moscow, 1986) [in Russian].
J. V. Burke, “Descent Methods for Composite Nondifferentiable Optimization Problems,” Mathem. Progr. 33(3), 260–279 (1985).
V. Jeyakumar and X. Q. Yang, “Convex Composite Minimization with C 1,1 Functions,” J. Optimiz. Theory Appl. 86(3), 631–648 (1995).
H.W. Kuhn, “On a Theorem of Wald,” in Linear Inequalities and Related Topics, Ed. by H.W. Kuhn and A.W. Tucker (Princeton Univ. Press, Princeton, 1956), pp. 265–273.
I. V. Konnov, “Dual Approach to One Class of Mixed Variational Inequalities,” Comput. Maths. Math. Phys. 42(9), 1276–1288 (2002).
I. V. Konnov, Equilibrium Models and Variational Inequalities (Elsevier, Amsterdam, 2007).
V. F. Dem’yanov and L. V. Vasil’ev, Nondifferentiable Optimization (Nauka, Moscow, 1981) [in Russian].
I. V. Konnov, Methods of Nondifferentiable Optimization (Kazan University Press, Kazan, 1993) [in Russian].
P. D. Panagiotopoulos, Inequality Problems in Mechanics and Their Applications (Birkhauser, Boston, 1985).
W. W. Hogan, “Point-to-Set Maps in Mathematical Programming,” SIAM Review 15(3), 591–603 (1973).
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Original Russian Text © I.V. Konnov, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 1, pp. 66–75.
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Konnov, I.V. A nonlinear descent method for a variational inequality on a nonconvex set. Russ Math. 53, 56–63 (2009). https://doi.org/10.3103/S1066369X09010034
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DOI: https://doi.org/10.3103/S1066369X09010034