WEAK CONVERGENCE OF THE OPERATOR EXTRAPOLATION METHOD FOR VARIATIONAL INEQUALITIES IN UNIFORMLY CONVEX BANACH SPACES
Abstract
This work is devoted to the study of new iterative algorithms for solving variational inequalities in uniformly convex Banach spaces. The first algorithm is a modification of the forward-reflectedbackward algorithm, which uses the Alber generalized projection instead of the metric one. The second algorithm is an adaptive version of the first one, where the monotone step size update rule is used, which does not require knowledge of Lipschitz constants and linear search procedure.
References
Nagurney A. Network economics: A variational inequality approach. Dordrecht: Kluwer Academic Publishers, 1999. 325 p.
Kinderlehrer D. Stampacchia G. An introduction to variational inequalities and their applications. New York: Academic Press, 1980. Russian transl., Moscow: Mir, 1983. 256 p.
Facchinei F., Pang J.-S. Finite-Dimensional Variational Inequalities and Complementarily Problem. V. 2. New York: Springer, 2003. 666 p.
Nemirovski A. 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. on Optim. 2004. Vol. 15. P. 229–251.
Gidel G., Berard H., Vincent P., Lacoste-Julien S. A Variational Inequality Perspective on Generative Adversarial Networks. preprint arXiv:1802.10551. 2018.
Vedel Y., Semenov V. Adaptive Extraproximal Algorithm for the Equilibrium Problem in Hadamard Spaces. In: Olenev N., Evtushenko Y., Khachay M., Malkova V. (eds.) Optimization and Applications. OPTIMA 2020. Lecture Notes in Computer Science, vol 12422. Springer, Cham, 2020. P. 287–300.
Semenov V. V., Denisov S. V., Kravets A. V. Adaptive Two-Stage Bregman Method for Variational Inequalities. Cybernetics and Systems Analysis. 2021. Vol. 57. Issue 6. P. 959–967.
Vedel Y., Semenov V., Denisov S. A Novel Algorithm with Self-adaptive Techni-que for Solving Variational Inequalities in Banach Spaces. In: Olenev N. N., Evtushenko Y. G., Jacimovic M., Khachay M., Malkova V. (eds.) Advances in Optimization and Applications. OPTIMA 2021. Communications in Computer and Information Science, vol 1514. Springer, Cham, 2021. P. 50–64.
Malitsky Y., Tam M. K. A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity. SIAM J. on Optim. 2020. Vol. 30. P. 1451–1472.
Alber Y., Ryazantseva I. Nonlinear Ill Posed Problems of Monotone Type. Dordrecht: Springer, 2006. 410 p.
Alber Y. I. Metric and generalized projection operators in Banach spaces: properties and applications. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178. New York: Dekker, 1996. P. 15–50.
Beauzamy B. Introduction to Banach Spaces and Their Geometry. Amsterdam: North-Holland, 1985. 307 p.
Aoyama K., Kohsaka F. Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings. Fixed Point Theory Appl. 2014. 95. https://doi.org/10.1186/1687-1812-2014-95.
Xu H. K. Inequalities in Banach spaces with applications. Nonlinear Anal. 1991.Vol. 16. Iss. 12. P. 1127–1138.
