A new low-cost double projection method for solving variational inequalities
A new low-cost double projection method for solving variational inequalities
复制标题
一种解决变分不等式的新型低成本双投影方法
DOI：
10.1007/s11081-020-09490-2
复制
发表时间：
2020
影响因子：
2.1
通讯作者：
Duong Viet Thong
中科院分区：
文献类型：
--
作者：
A. Gibali;Duong Viet Thong
在这项工作中，我们关注真实希尔伯特空间中的变分不等式，并引入一种新的双投影方法来解决它。该算法受到 Korpelevich 外梯度方法、Gibali 等人的次梯度外梯度方法的启发。和波波夫方法。所提出的方案结合了上述方法的一些优点，首先它只需要在问题的可行集上进行一次正交投影，而下一次计算具有封闭的公式。其次，每次迭代仅需要一次映射评估，并且还使用自适应步长规则，以避免需要知道相关映射的 Lipschitz 常数。我们提出了该方法的两个收敛定理，弱收敛结果需要伪单调性、Lipschitz 和相关映射的顺序弱连续性，以及具有收敛速度的强收敛定理，仅需要 Lipschitz 连续性和强伪单调性。初步数值实验和比较证明了新方案的优势和潜在适用性。
In this work we are concerned with variational inequalities in real Hilbert spaces and introduce a new double projection method for solving it. The algorithm is motivated by the Korpelevich extragradient method, the subgradient extragradient method of Gibali et al. and Popov’s method. The proposed scheme combines some of the advantages of the methods mentioned above, first it requires only one orthogonal projection onto the feasible set of the problem while the next computation has a closed formula. Second, only one mapping evaluation is required per each iteration and there is also a usage of an adaptive step size rule that avoids the need to know the Lipschitz constant of the associated mapping. We present two convergence theorems of the proposed method, weak convergence result which requires pseudomonotonicity, Lipschitz and sequentially weakly continuity of the associated mapping and strong convergence theorem with rate of convergence which requires Lipschitz continuity and strongly pseudomonotone only. Primary numerical experiments and comparisons demonstrate the advantages and potential applicability of the new scheme.