New kinds of self-adaptive subgradient extragradient projection methods for solving pseudomonotone variational inequalities

Abstract

Gibali [J. Nonlinear Anal. Optim., 2015, 6(1): 41‒51] presented a self-adaptive subgradient extragradient projection method for solving variational inequalities without Lipschitz continuity, where its next iterative point was obtained by projecting a vector onto a specific half-space. In this paper, we present new kinds of self-adaptive subgradient extragradient projection methods by using a new descent direction. With the help of the techniques in the method of He and Liao [J. Optim. Theory Appl, 2002, 112(1): 111‒128], we get a longer step-size for these kinds of algorithms, which proves the global convergence of the generated sequence. Numerical results show that these kinds of extragradient subgradient projection methods are less dependent on the choice of the initial point, the dimension of the variational inequalities, and the tolerance of accuracy than the known methods. Moreover, the new methods proposed in this paper outperform (with respect to the number of iterations and cpu-time) the method presented by Gibali.

Keywords

Variational inequalities / subgradient extragradient projection method / half-space / pseudomonotone / non-Lipschitz continuous

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