Efficient implementation of a modified and relaxed hybrid steepest-descent method for a type of variational inequality

Abstract

To reduce the difficulty and complexity in computing the projection from a real Hilbert space onto a nonempty closed convex subset, researchers have provided a hybrid steepest-descent method for solving VI(F, K) and a subsequent three-step relaxed version of this method. In a previous study, the latter was used to develop a modified and relaxed hybrid steepest-descent (MRHSD) method. However, choosing an efficient and implementable nonexpansive mapping is still a difficult problem. We first establish the strong convergence of the MRHSD method for variational inequalities under different conditions that simplify the proof, which differs from previous studies. Second, we design an efficient implementation of the MRHSD method for a type of variational inequality problem based on the approximate projection contraction method. Finally, we design a set of practical numerical experiments. The results demonstrate that this is an efficient implementation of the MRHSD method.

The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

“…From Tables 1 to 3, we found that the iteration numbers and CPU time of PRH under Condition 12 are more efficient than that under Condition 10. In Table 4 of our method, the tests' results give out that the PRH method under some descent directions is more slightly efficient than those of the MRHSD method [14,16], and it is easy to obtain these descent directions. Furthermore, it is important to find , , and by Tables 2 and 4.…”

“…Recently, Ding et al [7] proposed a threestep relaxed hybrid steepest-descent method for variational inequalities, and the simple proof of three-step relaxed hybrid steepest-descent methods under different conditions was introduced by Yao et al [24]. The literature [14,16] described a modified and relaxed hybrid steepest-descent (MRHSD) method and the different convergence of the MRHSD method under the different conditions. A set of practical numerical experiments in the literature [16] demonstrated that the MRHSD method has different efficiency under different conditions.…”

“…The literature [14,16] described a modified and relaxed hybrid steepest-descent (MRHSD) method and the different convergence of the MRHSD method under the different conditions. A set of practical numerical experiments in the literature [16] demonstrated that the MRHSD method has different efficiency under different conditions. Subsequently, the prediction-correction and relaxed hybrid steepest-descent method (PRH method) [15] makes more use of the history information and less decreases the loss of information than the methods [7,14].…”

“…Subsequently, the prediction-correction and relaxed hybrid steepest-descent method (PRH method) [15] makes more use of the history information and less decreases the loss of information than the methods [7,14]. The PRH method introduced more descent directions than the MRHSD method [14,16], and computing these descent directions only needs the history information.…”

“…Moreover, the proof of strong convergence is different from the previous proof in [15], which is not similar to that in [7] in Step 2. And more importantly, numerical experiments verify that the PRH method under Condition 12 is more efficient than that under Condition 10, and the PRH method under some descent directions is more slightly efficient than that of the MRHSD method [14,16]. Furthermore, it is easy to obtain these descent directions.…”