Prediction-correction method with BB step sizes

Inertial projection and contraction algorithms with larger step sizes for solving quasimonotone variational inequalities

“…In addition, being analogous to the sequence {θ k } and {τ k } in Algorithm 2 of [16], the role of sequences {θ n } and {μ n } is to relax the parameters γ and μ, respectively. However, we only require that the positive real sequences {θ n } and {μ n } satisfy lim n→∞ μ n = lim n→∞ θ n = 0, other than ∞ n=1 μ n < +∞ and ∞ n=1 θ n < +∞ as in Algorithm 2 of [16]. In contrast to Algorithm 2 of [16], this implies Algorithm 3.1 has larger step sizes.…”

“…The Barzilai-Borwein (BB) step size strategy is first proposed for the steepest-descent method for unconstrained optimization problems in [11]. Though the global convergence of the BB step size methods is still an open problem for general case, thanks to its high efficiency in practical computations, it has been applied to unconstrained optimization problems, box constrained optimization problems and variational inequalities etc, see [12][13][14][15][16][17]. Especially, by the utilizing BB step size strategy, which is,…”

“…in the predictor step, Dong et al [16] modify the line search condition in the PC Method-II and obtain the modified prediction-correction method (MPCM) and the improved predictioncorrection method with BB step sizes (IPCM-BB+) for solving pseudomonotone and Lipschitzian variational inequalities in the finite dimensional space R n , furthermore show that the sequences generated by these methods converge globally to some solution of VI(A, C). The monotonicity of operators plays a key role on the convergence analysis of algorithms for variational inequalities.…”

“…In this paper, to solve VI(A, C), we adopt the inertial technique and the BB step size strategy, and relax the line search condition, that is, r n ≤ ν and the parameters ν and γ in the PC Method-II to further improve its numerical efficiency. We point out that the relaxing form of the line search condition and the parameters ν and γ is the similar to the one of Algorithm 2 in [16]. However, the corresponding conditions used are different, for details, see Remark 3.1.…”

An improvement LQP method for nonlinear complementarity problems

“…x T F(x) = 0, (1.1) where F is a nonlinear mapping from R n into itself. Recently, various iterative methods, which are powerful techniques for solving the NCP effectively, are under the spotlight of researchers; see, e.g., [2,3,8,9,10,11,12,13,14,15,17,20,25,26] and the references therein.…”