无直线搜索的单调绝对值方程坐标邻近预测-校正方法
A Line-Search-Free Coordinate Proximal Predictor-Corrector Method for Monotone Absolute Value Equations
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中文总结 AI 辅助
该研究针对残差单调性下的绝对值方程，提出无需直线搜索的坐标邻近预测-校正方法，证明其收敛性与线性收敛条件，数值实验显示该方法可缩短求解时间。
中文摘要 AI 辅助
我们在残差单调性条件下研究绝对值方程（AVE）$Ax-|x|=b$，通过系数矩阵的对称部分刻画该性质，从而允许非对称性，并基于AVE特有的前向邻近分解提出坐标邻近预测-校正（CPPC）方法。对最大邻近残差坐标进行精确标量邻近更新生成预测点，而正对齐恒等式则保证后续全残差分离超平面校正无需回溯。在缓存当前矩阵乘积的情况下，每次迭代仅需一次新的全矩阵-向量乘积。对于非空解集，我们证明其满足Fejér单调性、全序列收敛性，以及$O(K^{-1/2})$最优迭代残差界。正单调裕度进一步确保对任意右端项均存在唯一解，且全局线性收敛。数值结果表明，减少的每次迭代工作量可缩短求解时间。
英文摘要
We consider the absolute value equation (AVE) $Ax-|x|=b$ under residual monotonicity. We characterize this property through the symmetric part of the coefficient matrix, thereby allowing nonsymmetry, and propose a coordinate proximal predictor-corrector (CPPC) method based on an AVE-specific forward-proximal decomposition. An exact scalar proximal update on the largest proximal-residual coordinate generates a predictor point, while a positive-alignment identity certifies the ensuing full-residual separating-hyperplane correction without backtracking. With the current matrix product cached, each iteration requires one new full matrix-vector product. For a nonempty solution set, we prove Fejér monotonicity, whole-sequence convergence, and an $O(K^{-1/2})$ best-iterate residual bound. A positive monotonicity margin further ensures unique solvability for every right-hand side and global linear convergence. Numerical results identify regimes in which the reduced per-iteration work yields shorter solution times.
