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In a very interesting paper (SIAM J. Control Optim. 37(3): 765–776, 1999), Solodov and Svaiter introduced an effective projection algorithm with linesearch for finding a solution of a variational inequality problem (VIP) in Euclidean space. They showed that the iterative sequence generated by their algorithm converges to a solution of (VIP) under the main assumption that the cost mapping is pseudomonotone and continuous. In this paper, we propose to modify this algorithm for solving variational inequality problems in which the cost mapping is not required to be satisfied any pseudomonotonicity. Moreover, we do not use the embedded projection methods as in methods used in literature and the linesearch procedure is not necessary when the cost mapping is Lipschitz. Several numerical examples are also provided to illustrate the efficient of the proposed algorithms. [ABSTRACT FROM AUTHOR]

It is well known that the monotonicity of the underlying mapping of variational inequalities plays a central role in the convergence analysis. In this paper, we propose an infeasible projection algorithm (IPA for short) for nonmonotone variational inequalities. The next iteration point of IPA is generated by projecting a vector onto a half-space. Hence, the computational cost of computing the next iteration point of IPA is much less than the algorithm of Ye and He (Comput. Optim. Appl. 60, 141–150, 2015) (YH for short). Moreover, if the underlying mapping is Lipschitz continuous with its modulus is known, by taking suitable parameters, IPA requires only one projection onto the feasible set per iteration. The global convergence of IPA is obtained when the solution set of its dual variational inequalities is nonempty. Moreover, if in addition error bound holds, the convergence rate of IPA is Q-linear. IPA can be used for a class of quasimonotone variational inequality problems and a class of quasiconvex minimization problems. Comparing with YH and Algorithm 2 in Deng, Hu and Fang (Numer. Algor. 86, 191–221, 2021) (DHF for short) by solving high-dimensional nonmonotone variational inequalities, numerical experiments show that IPA is much more efficient than YH and DHF from CPU time point of view. Moreover, IPA is less dependent on the initial value than YH and DHF. [ABSTRACT FROM AUTHOR]