# Fast Re-Optimization via Structural Diversity

## TL;DR

This paper analyzes the limitations of evolutionary algorithms in re-optimization scenarios and introduces a diversity mechanism that significantly improves their efficiency on various problems.

## Contribution

It identifies the challenge of structural solution loss in re-optimization and proposes a simple diversity mechanism to overcome this, improving re-optimization times.

## Key findings

- Re-optimization with standard EA can take as long as random restart.
- Diversity mechanisms can reduce re-optimization time to linear in problem size.
- The approach applies to linear functions and minimum spanning tree problems.

## Abstract

When a problem instance is perturbed by a small modification, one would hope to find a good solution for the new instance by building on a known good solution for the previous one. Via a rigorous mathematical analysis, we show that evolutionary algorithms, despite usually being robust problem solvers, can have unexpected difficulties to solve such re-optimization problems. When started with a random Hamming neighbor of the optimum, the (1+1) evolutionary algorithm takes $\Omega(n^2)$ time to optimize the LeadingOnes benchmark function, which is the same asymptotic optimization time when started in a randomly chosen solution. There is hence no significant advantage from re-optimizing a structurally good solution. We then propose a way to overcome such difficulties. As our mathematical analysis reveals, the reason for this undesired behavior is that during the optimization structurally good solutions can easily be replaced by structurally worse solutions of equal or better fitness. We propose a simple diversity mechanism that prevents this behavior, thereby reducing the re-optimization time for LeadingOnes to $O(\gamma\delta n)$, where $\gamma$ is the population size used by the diversity mechanism and $\delta \le \gamma$ the Hamming distance of the new optimum from the previous solution. We show similarly fast re-optimization times for the optimization of linear functions with changing constraints and for the minimum spanning tree problem.