On Pareto Local Optimal Solutions Networks

Abstract

Pareto local optimal solutions (PLOS) are believed to highly influence the dynamics and the performance of multi-objective optimization algorithms, especially those based on local search and Pareto dominance. A number of studies so far have investigated their impact on the difficulty of searching the landscape underlying a problem instance. However, the community still lacks knowledge on the structure of PLOS and the way it impacts the effectiveness of multi-objective algorithms. Inspired by the work on local optima networks in single-objective optimization, we introduce a PLOS network (PLOS-net) model as a step toward the fundamental understanding of multi-objective landscapes and search algorithms. Using a comprehensive set of ρmnk-landscapes, PLOS-nets are constructed by full enumeration, and selected network features are further extracted and analyzed with respect to instance characteristics. A correlation and regression analysis is then conducted to capture the importance of the PLOS-net features on the runtime and effectiveness of two prototypical Pareto-based heuristics. In particular, we are able to provide empirical evidence for the relevance of the PLOS-net model to explain algorithm performance. For instance, the degree of connectedness in the PLOS-net is shown to play an even more important role than the number of PLOS in the landscape.

A Survey of Advances in Landscape Analysis for Optimisation

“…A LON is a graph-based abstraction of the search space representing the global structure, where each node of the LON is a local optimum and edges between nodes represent adjacency of the basins of optima (the possibility of search transitioning from one local optimum to another). The LON model is also extended to multiobjective problems to form pareto local optimal solutions networks (PLOS-nets) [30,31]. More detail and resources on LONs can be found on the website: http://lonmaps.com.…”

“…The models of five of the algorithm variants achieved testing accuracy levels above 90%. • Liefooghe et al [30] used a random forest regression model to predict the performance of multiobjective optimisation algorithms in combinatorial optimisation based on a combination of landscape features and problem-specific features. They later developed a decision tree model for selecting the best performing algorithm out of three multiobjective algorithms [50].…”

“…There is still a gap in techniques for practically characterising global features of multiobjective search spaces. PLOS-nets [30,31] have been proposed for capturing the global structure of multiobjective landscapes, but it is still not clear how this approach can be scaled to large-size problems.…”

“…Technique 23:Local optima networks (LONs) by Ochoa et al[23] with extensions[24][25][26][27][28][29][30][31].…”

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]. 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].…”