AI 中文总结
研究希尔伯特空间上变分包含问题，提出三种含双惯性技术的投影收缩算法，\(\algoO\)弱收敛，\(\algoT\)在强单调性下\(R\)-线性收敛，\(\algoDIM\)强收敛到最小范数解，通过数值模拟验证\(\algoT\)收敛性，并应用于相关问题与现有方法比较。
AI 中文摘要
本文提出三种投影收缩算法来解决希尔伯特空间中的变分包含问题，每种算法都将双惯性技术纳入投影收缩框架。第一种算法\(\algoO\)在单值算子的单调性和利普希茨连续性下实现弱收敛，具有无需利普希茨常数先验知识的自适应步长规则。改进变体\(\algoT\)在强单调性假设下达到\(R\)-线性收敛。第三种算法\(\algoDIM\)无需强单调性即可强收敛到最小范数解。通过抽象变分包含问题说明所提方法，并应用于分裂可行性问题和弹性网正则化问题，与文献中现有方法比较。还通过数值模拟验证了\(\algoT\)的\(R\)-线性收敛。
英文摘要
In this paper, we propose three projection--contraction algorithms for solving variational inclusion problems in the setting of Hilbert spaces, each incorporating a double inertial technique into the projection--contraction framework. The first algorithm \algoO achieves weak convergence under monotonicity and Lipschitz continuity of the single-valued operator, with an adaptive stepsize rule that does not require prior knowledge of the Lipschitz constant. A modified variant \algoT of \algoO attains $R$-linear convergence under the strong monotonicity assumption. The third algorithm \algoDIM obtains strong convergence to the minimum-norm solution without requiring strong monotonicity. We illustrate the proposed methods on an abstract variational inclusion problem and apply them to the split feasibility problem and the elastic net regularization problem, comparing them with existing methods in the literature. The $R$-linear convergence of \algoT is also verified through numerical simulations.
Comments41 pages, 4 figures, accepted by JCAM
