In root finding and optimization, there are many cases where there is a closed set $A$ one likes that the sequence constructed by one's favourite method will not 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 two new methods aiming to achieve this. In the first method, 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. In the second method, which is more suitable for constrained optimization, we redefine the value of the function to be a big constant on $A$. We also propose, based on this, an algorithm to escape the basin of attraction of a component of positive dimension to reach another component. As an application, we prove a rigorous guarantee for finding roots of a meromorphic function of 1 complex variable in a given domain. Along the way, we compare with main existing relevant methods in the current literature. We provide several examples in various different settings to illustrate the usefulness of the new approach.

翻译：在求根和优化问题中，常存在一个闭集$A$，我们希望利用某种方法构造的序列不会收敛到该集合$A$（此处，我们不对$A$施加如凸性或连通性等额外假设）。例如，若欲寻找根，且初始点恰好落在某个根$x^*$的吸引域中（此特性事先未知），则算法将始终收敛于该根。此时，人们希望在下一次算法运行中能规避该点$z^*$。本文提出两种新方法以实现这一目标。第一种方法将代价函数除以到集合$A$的距离函数的适当幂次，该思路受单变量函数求所有根问题的启发。第二种方法更适用于约束优化，通过将函数在集合$A$上的值重新定义为一个较大的常数。基于此，我们还提出一种算法，用于逃离正维数分量的吸引域以到达另一分量。作为应用，我们严格证明了在给定复域内寻找单变量亚纯函数根的保证性。同时，本文与现有主流相关方法进行了对比，并通过多个不同场景的算例验证了新方法的有用性。