AI 中文总结
本文提出离散势能优化框架，在符号向量集上求解绝对值方程，实现多项式复杂度的符号翻转算法，并推广至线性互补问题，显著改进现有指数级最坏情况。
AI 中文摘要
求解绝对值方程 $Ax - |x| = b$ 通常是NP难的。现有方法大多在连续变量空间中操作，但Rohn的符号一致算法是一个显著的例外，其最坏情况迭代次数为指数级的 $2^n$。为了克服这一瓶颈，我们在符号向量集 $\{-1, 1\}^n$ 上开发了一种离散势能优化（DPO）框架。在 $1$-范数条件 $\\|A^{-1}\\|_1 < 1/2$ 下，我们证明了绝对值方程的解恰好对应于该离散势能函数的全局最大化器。当 $\\|A^{-1}\\|_1$ 被 $1/2$ 一致上界约束时，对于最大幅度为 $L$ 的有理数输入，我们开发了一个统一的符号翻转算法多项式时间框架，该框架排除了Rohn的符号一致算法。在该框架内，特定的单翻转机制（包括我们新的最速下降和Gauss-Southwell规则）在 $\mathcal{O}(n^2\log(nL))$ 次迭代内终止，而全翻转更新（等价于经典GNM）仅需 $\mathcal{O}(n\log(nL))$ 次迭代。我们进一步放宽假设，通过有理对角缩放要求谱半径 $\rho(|A^{-1}|)$ 被 $1/2$ 一致上界约束。此外，一种均匀随机的 $m$-翻转方法被证明在无需显式对角预处理的情况下达到 $\mathcal{O}\bigl(n^3 \log(nL)/m\bigr)$ 的期望迭代界限。作为推论，GNM在 $\mathcal{O}(n^2 \log(nL))$ 次迭代内求解绝对值方程，改进了在更严格条件 $\rho(|A^{-1}|)$ 小于 $1/3$ 下的有限终止先前结果。关键的是，通过将线性互补问题（LCPs）等价地重构为绝对值方程，我们将此DPO框架扩展，为LCPs提供了具有多项式迭代复杂度的GNM和枢轴类方法。数值实验验证了GNM的实际效率和所提出的符号翻转方法的结构鲁棒性。
英文摘要
Solving the AVE $Ax - |x| = b$ is generally NP-hard. Existing approaches mostly operate in continuous variable spaces, with the notable exception of Rohn's sign-accord algorithm, which incurs an exponential worst-case bound of $2^n$ iterations. To overcome this bottleneck, we develop a discrete potential optimization (DPO) framework over the sign-vector set $\{-1, 1\}^n$. Under the $1$-norm condition $\|A^{-1}\|_1 < 1/2$, we establish that the AVE solution corresponds exactly to the global maximizer of this discrete potential function. When $\|A^{-1}\|_1$ is uniformly upper bounded by $1/2$, for rational inputs with maximum magnitude $L$, we develop a unified polynomial-time framework for sign-flip algorithms, which excludes Rohn's sign-accord algorithm. Within this framework, specific single-flip mechanisms (including our new steepest and Gauss-Southwell rules) terminate in $\mathcal{O}(n^2\log(nL))$ iterations, while full-flip updates (equivalent to the classical GNM) require only $\mathcal{O}(n\log(nL))$ iterations. We further relax the assumption by requiring that the spectral radius $ρ(|A^{-1}|)$ be uniformly upper bounded by $1/2$ via rational diagonal scaling. Moreover, a uniformly randomized $m$-flip approach is proven to achieve an expected iteration bound of $\mathcal{O}\bigl(n^3 \log(nL)/m\bigr)$ without requiring explicit diagonal preconditioning. As a corollary, GNM solves the AVE in $\mathcal{O}(n^2 \log(nL))$ iterations, improving on the prior result of finite termination under the stricter condition that $ρ(|A^{-1}|)$ is less than~$1/3$. Crucially, by equivalently reformulating linear complementarity problems (LCPs) as AVEs, we extend this DPO framework to yield GNM and pivot-type methods with polynomial iteration complexity for LCPs. Numerical experiments validate the practical efficiency of GNM and the structural robustness of the proposed sign-flip approaches.
Comments32 pages, 3 figures
