We study the computation of equilibrium points of electrostatic potentials: locations in space where the electrostatic force arising from a collection of charged particles vanishes. This is a novel scenario of optimization in which solutions are guaranteed to exist due to a nonconstructive argument, but gradient descent is unreliable due to the presence of singularities. We present an algorithm based on piecewise approximation of the potential function by Taylor series. The main insight is to divide the domain into a grid with variable coarseness, where grid cells are exponentially smaller in regions where the function changes rapidly compared to regions where it changes slowly. Our algorithm finds approximate equilibrium points in time poly-logarithmic in the approximation parameter, but these points are not guaranteed to be close to exact solutions. Nevertheless, we show that such points can be computed efficiently under a mild assumption that we call "strong non-degeneracy". We complement these algorithmic results by studying a generalization of this problem and showing that it is CLS-hard and in PPAD, leaving its precise classification as an intriguing open problem.

翻译：本文研究静电势能平衡点的计算问题：即在空间中，由一组带电粒子产生的静电力为零的位置。这是一种新颖的优化场景，其中由于非构造性论证保证了解的存在性，但梯度下降法因奇点的存在而不可靠。我们提出一种基于泰勒级数对势函数进行分段逼近的算法。核心思路是将定义域划分为具有可变粗糙度的网格，在函数变化剧烈的区域网格单元呈指数级缩小，而在变化平缓的区域则保持较大尺寸。该算法能在近似参数的多对数时间内找到近似平衡点，但这些点不能保证接近精确解。然而，我们证明在称为“强非退化性”的温和假设下，此类点可被高效计算。我们通过研究该问题的泛化形式来补充这些算法结果，证明其属于CLS-hard且包含于PPAD类中，其精确分类仍是一个引人深思的开放问题。