In root finding and optimization, there are many cases where there is a closed set $A$ one does not the sequence constructed by one's favourite method will converge to A (here, we do not assume extra properties on $A$ such as being convex or connected). For example, if one wants to find roots, and one chooses initial points in the basin of attraction for 1 root $x^*$ (a fact which one may not know before hand), then one will always end up in that root. In this case, one would like to have a mechanism to avoid this point $z^*$ in the next runs of one's algorithm. In this paper, we propose a new method aiming to achieve this: we divide the cost function by an appropriate power of the distance function to $A$. This idea is inspired by how one would try to find all roots of a function in 1 variable. We first explain the heuristic for this method in the case where the minimum of the cost function is exactly 0, and then explain how to proceed if the minimum is non-zero (allowing both positive and negative values). The method is very suitable for iterative algorithms which have the descent property. We also propose, based on this, an algorithm to escape the basin of attraction of a component of positive dimension to reach another component. Along the way, we compare with main existing relevant methods in the current literature. We provide several examples to illustrate the usefulness of the new approach.

翻译：在根查找与优化问题中，常存在一个闭集 $A$，使得我们的优选方法构造的序列不会收敛到 $A$（此处不对 $A$ 附加凸性或连通性等额外性质）。例如，若欲寻找根，且初始点选择在某个根 $x^*$ 的吸引域内（这一事实可能事先未知），则算法将始终收敛至该根。此时，我们希望设计一种机制，在后续算法运行中避免该点 $z^*$。本文提出一种新方法：将代价函数除以到 $A$ 的距离函数的适当幂次。该思想源于单变量函数求根问题。我们首先阐释当代价函数最小值为零时该方法的启发式原理，继而讨论最小值非零（允许正负值）时的处理方法。该方法特别适用于具有下降性质的迭代算法。基于此，我们进一步提出一种算法，用于逃逸正维数分支的吸引域并抵达其他分支。本文还与现有主流相关方法进行了比较，并通过多个示例验证了新方法的有效性。