Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Creating walls to avoid unwanted points in root finding and optimization

Tuyen Trung Truong

TL;DR

The paper addresses avoiding unwanted points in root finding and optimization by introducing two wall-building techniques that modify the objective without destroying descent properties. It proves that, under suitable conditions, iterative methods with strong convergence guarantees continue to converge to internal minimizers or roots while avoiding a prescribed closed set $A$. The methods are validated across meromorphic-root finding and constrained optimization scenarios, demonstrating improved basin control and the ability to escape from large basins of attraction. The results have practical implications for exploring multiple solution components and for solving constrained problems where global minima lie interior to the domain. Collectively, the work offers a practical, theoretically grounded toolkit for steering iterative schemes away from known points while preserving convergence behavior.

Abstract

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 $z^*$ (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. Assume that one already has a method IM for optimization (and root finding) for non-constrained optimization. We provide a simple modification IM1 of the method to treat the situation discussed in the previous paragraph. If the method IM has strong theoretical guarantees, then so is IM1. As applications, we prove two theoretical applications: one concerns finding roots of a meromorphic function in an open subset of a Riemann surface, and the other concerns finding local minima of a function in an open subset of a Euclidean space inside it the function has at most countably many critical points. 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.

Creating walls to avoid unwanted points in root finding and optimization

TL;DR

The paper addresses avoiding unwanted points in root finding and optimization by introducing two wall-building techniques that modify the objective without destroying descent properties. It proves that, under suitable conditions, iterative methods with strong convergence guarantees continue to converge to internal minimizers or roots while avoiding a prescribed closed set . The methods are validated across meromorphic-root finding and constrained optimization scenarios, demonstrating improved basin control and the ability to escape from large basins of attraction. The results have practical implications for exploring multiple solution components and for solving constrained problems where global minima lie interior to the domain. Collectively, the work offers a practical, theoretically grounded toolkit for steering iterative schemes away from known points while preserving convergence behavior.

Abstract

In root finding and optimization, there are many cases where there is a closed set one likes that the sequence constructed by one's favourite method will not converge to A (here, we do not assume extra properties on 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 (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 in the next runs of one's algorithm. Assume that one already has a method IM for optimization (and root finding) for non-constrained optimization. We provide a simple modification IM1 of the method to treat the situation discussed in the previous paragraph. If the method IM has strong theoretical guarantees, then so is IM1. As applications, we prove two theoretical applications: one concerns finding roots of a meromorphic function in an open subset of a Riemann surface, and the other concerns finding local minima of a function in an open subset of a Euclidean space inside it the function has at most countably many critical points. 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.
Paper Structure (22 sections, 5 theorems, 2 equations, 12 figures)

This paper contains 22 sections, 5 theorems, 2 equations, 12 figures.

Table of Contents

  1. The problem, motivation and main result
  2. The problem and motivation
  3. Criteria for an optimization method to have strong theoretical guarantees
  4. A brief survey of main existing relevant methods in the current literature
  5. Application 1: finding roots of a meromorphic function in a given domain
  6. Application 2: Finding local minima of a function with countably many critical points in a domain
  7. Acknowledgements and the plan of the paper
  8. Two new methods: Creating walls to avoid unwanted points
  9. Multiplying poles at $A$
  10. Making the function to be a big constant on $A$
  11. Convergence issue
  12. Experimental results
  13. Example 1
  14. Example 2
  15. Example 3
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $g(z)$ be a non-constant meromorphic function in 1 complex variable $z$. Assume that $g$ is generic, in the sense that $\{z\in \mathbf{C}:~g(z)g"(z)=g'(z)=0\}=\emptyset$. Let $f(x,y)=|g(x+iy)|^2/2$. Let $U\subset \mathbf{C}$ be an open subset. There is a set $\mathcal{E}$ with Lebesgue measure 0

Figures (12)

  • Figure 1: Basins of attraction for the function G(x,y) in Example 1, using Gradient Descent with learning rate 0.1. Points are chosen on a square grid, with centre at the point $(0.5,-0.5003)$. Cyan: initial points which converge to $p_2$. Yellow: initial points which converge to $p_3$.
  • Figure 2: Basins of attraction for the function G(x,y) in Example 1, using Backtracking Gradient Descent. Points are chosen on a square grid, with centre at the point $(0.5,-0.5003)$. Cyan: initial points which converge to $p_2$. Yellow: initial points which converge to $p_3$.
  • Figure 3: Basins of attraction for the function G(x,y) in Example 1, using Backtracking New Q-Newton's method. Points are chosen on a square grid, with centre at the point $(0.5,-0.5003)$. Cyan: initial points which converge to $p_2$. Yellow: initial points which converge to $p_3$.
  • Figure 4: Basins of attraction for the function $H_1(x,y)$ in Example 1, with $\epsilon =0.001$ using Backtracking New Q-Newton's method. Points are chosen on a square grid, with centre at the point $(0.5,-0.5003)$. Cyan: initial points which converge to $p_2$. Yellow: initial points which converge to $p_3$.
  • Figure 5: Basins of attraction for the function $H_1(x,y)$ in Example 1, with $\epsilon =0.0001$ using Backtracking New Q-Newton's method. Points are chosen on a square grid, with centre at the point $(0.5,-0.5003)$. Cyan: initial points which converge to $p_2$. Yellow: initial points which converge to $p_3$.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof