A hybrid method without extrapolation step for solving variational inequality problems

Abstract

In this paper, we introduce a new method for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. The iterative process is based on two well-known projection method and the hybrid (or outer approximation) method. However we do not use an extrapolation step in the projection method. The absence of one projection in our method is explained by slightly different choice of sets in hybrid method. We prove a strong convergence of the sequences generated by our method.2010 Mathematics Subject Classification: 47J20

Modified hybrid projection methods for finding common solutions to variational inequality problems

“…However, in infinite dimensional Hilbert spaces, the extragradient method only converges weakly. In recent years, the extragradient method has received a lot of attention, see, for example, [10,14,15,24,30,31] and the references therein. Nadezhkina and Takahashi [32] introduced the following hybrid extragradient method ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y n = P K (x n − λA(x n )), z n = P K (x n − λA(y n )), C n = {z ∈ C : ||z − z n || ≤ ||z − x n ||} , Q n = {z ∈ C : x 0 − x n , z − x n ≤ 0} , x n+1 = P C n ∩Q n (x 0 ), (4) where λ ∈ (0, 1 L ).…”

A modified hybrid method for solving variational inequality problems in Banach spaces

“…Inspired by Malitsky and Semenov' results [17], we propose the Algorithm 3.1 to extend the Algorithm 1.2 from Hilbert spaces to Banach spaces and prove a strong convergence theorem, which is different from the scheme proposed by Nakajo [2].…”

“…This might seriously affect the efficiency of the Algorithm 1.1. In this paper, we will construct a new iterative algorithm based on the idea in [17] as follows.…”

“…As remarked in [17], the sets C n and Q n in Algorithm 1.2 are halfspaces and hence it is much more simpler to calculate P C n Q n x 0 which can be found by …”

Convergence of the Modified Extragradient Method for Variational Inequalities with Non-Lipschitz Operators

“…The convergence without problem modification is provided in iterative extragradient methods first proposed by Korpelevich in [21]. These methods were analyzed in many studies [22][23][24][25][26][27][28][29][30][31][32][33][34]. For variational inequalities and equilibrium programming problems, modifications of the Korpelevich algorithm with one metric projection onto feasible set were proposed [27,28].…”