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International Theoretical Science Journal
Volume 58 >>> № 5 SEPTEMBER — OCTOBER 2022
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UDC 517.988
V.V. Semenov1, S.V. Denisov2, G.V. Sandrakov3, O.S. Kharkov4


1 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

 semenov.volodya@gmail.com

2 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

 sireukr@gmail.com

3 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

 gsandrako@gmail.com

4 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

 olehharek@gmail.com

CONVERGENCE OF THE OPERATOR EXTRAPOLATION METHOD
FOR VARIATIONAL INEQUALITIES IN BANACH SPACES

Abstract. New iterative algorithms for solving variational inequalities in uniformly convex Banach spaces are analyzed. The first algorithm is a modification of the forward-reflected-backward 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 the Lipschitz constants and linear search procedure. For variational inequalities with monotone, Lipschitz continuous operators acting in a 2-uniformly convex and uniformly smooth Banach space, theorems on the weak convergence of the methods are proved. Also, for the first algorithm, an efficiency estimate in terms of the gap function is proved.

Keywords: variational inequality, monotone operator, Alber generalized projection, 2-uniformly convex Banach space, uniformly smooth Banach space, operator extrapolation method, weak convergence, gap function.


FULL TEXT

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