Anatomy of the Attraction Basins: Breaking with the Intuition

Abstract

Solving combinatorial optimization problems efficiently requires the development of algorithms that consider the specific properties of the problems. In this sense, local search algorithms are designed over a neighborhood structure that partially accounts for these properties. Considering a neighborhood, the space is usually interpreted as a natural landscape, with valleys and mountains. Under this perception, it is commonly believed that, if maximizing, the solutions located in the slopes of the same mountain belong to the same attraction basin, with the peaks of the mountains being the local optima. Unfortunately, this is a widespread erroneous visualization of a combinatorial landscape. Thus, our aim is to clarify this aspect, providing a detailed analysis of, first, the existence of plateaus where the local optima are involved, and second, the properties that define the topology of the attraction basins, picturing a reliable visualization of the landscapes. Some of the features explored in this article have never been examined before. Hence, new findings about the structure of the attraction basins are shown. The study is focused on instances of permutation-based combinatorial optimization problems considering the 2-exchange and the insert neighborhoods. As a consequence of this work, we break away from the extended belief about the anatomy of attraction basins.

On Stochastic Fitness Landscapes: Local Optimality and Fitness Landscape Analysis for Stochastic Search Operators

“…The number of local optima in the fitness landscape provides a first information about the difficulty of a combinatorial optimization problem, and about the performance of local and evolutionary search algorithms [8]. For large landscapes, different methods allow one to estimate the number of local optima using uniform random sampling, biased random sampling [1,7], or the length of an adaptive walk before being trapped [11].…”

“…For large landscapes, different methods allow one to estimate the number of local optima using uniform random sampling, biased random sampling [1,7], or the length of an adaptive walk before being trapped [11]. In addition to the number of local optima, the size, the distribution and the structure of local optima's basins of attraction is one major feature related to algorithm performance [5,8], including for problems from machine learning [3]. The basin of attraction of a local optimum x is defined as the set of solutions from which a hill-climbing algorithm h falls into:…”

Towards the landscape rotation as a perturbation strategy on the quadratic assignment problem

“…In brief, a basin of attraction of an optima is a subset of connected solutions that lead to a local optima when a steepest ascent hill-climbing algorithm is applied. Hernando et al in [6] study the shape of the fitness landscape and the attraction basins' anatomy on combinatorial problems, and suggest how to enhance the design of future algorithms. The authors mention that local search based algorithms get stuck on plateaus or a local optima, although these are generally connected to neighbouring solutions that belong to more favourable basins of attractions.…”

“…Literature presents a wide variety of perturbation strategies to avoid getting trapped in a given attraction basin [6]. In this paper, we use the fitness landscape rotation as a perturbation technique to redirect an algorithm that is stuck in a local optima.…”

Journey to the center of the linear ordering problem

“…While other heuristic techniques have been developed for particular permutation-based combinatorial optimization problems (Stützle, 2006;Schiavinotto and Stützle, 2005;Santucci and Ceberio, 2020). In the last few decades, researchers have seen the necessity to carry out landscape analyses more than to continue to propose new algorithms without knowledge about the problem at hand (Hernando et al, 2019a(Hernando et al, , 2018Albrecht et al, 2008;Alyahya and Rowe, 2014;Chicano et al, 2012;Humeau et al, 2013;Schiavinotto and Stützle, 2003).…”