Creating walls to avoid unwanted points in root finding and optimization

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.