通过渗透梯度算法应用最深下降策略求解多项式系统
Solving Polynomial Systems by Penetrating Gradient Algorithm Applying Deepest Descent Strategy
AI总结：
本文提出一种在优化框架中求解多项式系统的算法和策略，通过渗透梯度算法和最深下降策略实现全局最优解的快速求解，实验验证其有效性并揭示与高斯-塞德尔方法的关联。
摘要：
本文提出了一种在优化框架中求解多项式系统的算法和策略，分别称为渗透梯度算法和最深下降策略。渗透梯度算法的核心特点是能够沿搜索方向穿透误差空间中的障碍物并直接跳转到全局最小值点。该算法能够找到任意方向上的最深点，无论该点距离多远或误差空间的地形如何，从而在成本函数最大化减少的方向上移动，而非局部最优方向（如最陡下降方向）。渗透梯度算法被推导并数学上证明了其性质，而最深下降策略的特征则通过比较模拟展示。广泛基准测试证实，所提出的算法和策略共同构成了有效的多项式系统求解器。此外，第五部分进一步的理论考虑揭示了所提出方法在求解线性系统时与高斯-塞德尔方法之间存在令人惊讶且有趣的关联。
英文摘要：
An algorithm and associated strategy for solving polynomial systems within the optimization framework is presented. The algorithm and strategy are named, respectively, the penetrating gradient algorithm and the deepest descent strategy. The most prominent feature of penetrating gradient algorithm, after which it was named, is its ability to see and penetrate through the obstacles in error space along the line of search direction and to jump to the global minimizer in a single step. The ability to find the deepest point in an arbitrary direction, no matter how distant the point is and regardless of the relief of error space between the current and the best point, motivates movements in directions in which cost function can be maximally reduced, rather than in directions that seem to be the best locally (like, for instance, the steepest descent, i.e., negative gradient direction). Therefore, the strategy is named the deepest descent, in contrast but alluding to the steepest descent. Penetrating gradient algorithm is derived and its properties are proven mathematically, while features of the deepest descent strategy are shown by comparative simulations. Extensive benchmark tests confirm that the proposed algorithm and strategy jointly form an effective solver of polynomial systems. In addition, further theoretical considerations in Section 5 about solving linear systems by the proposed method reveal a surprising and interesting relation of proposed and Gauss-Seidel method.
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