Sébastien Verel

Local Optima Networks of NK Landscapes With Neutrality

In previous work, we have introduced a network based model that abstracts many details of the underlying landscape and compresses the landscape information into a weighted, oriented graph which we call the local optima network. The vertices of this graph are the local optima of the given fitness landscape, while the arcs are transition probabilities between local optima basins. Here, we extend this formalism to neutral fitness landscapes, which are common in difficult combinatorial search spaces. The study is based on two neutral variants of the well-known NK family of landscapes (where N stands for the chromosome length, and K for the number of gene epistatic interactions within the chromosome). By using these two NK variants, probabilistic (NKp), and quantified NK (NKq), in which the amount of neutrality can be tuned by a parameter, we show that our new definitions of the optima networks and the associated basins are consistent with the previous definitions for the non-neutral case. Moreover, our empirical study and statistical analysis show that the features of neutral landscapes interpolate smoothly between landscapes with maximum neutrality and non-neutral ones. We found some unknown structural differences between the two studied families of neutral landscapes. But overall, the network features studied confirmed that neutrality, in landscapes with percolating neutral networks, may enhance heuristic search. Our current methodology requires the exhaustive enumeration of the underlying search space. Therefore, sampling techniques should be developed before this analysis can have practical implications. We argue, however, that the proposed model offers a new perspective into the problem difficulty of combinatorial optimization problems and may inspire the design of more effective search heuristics.

Fitness Clouds and Problem Hardness in Genetic Programming

This paper presents an investigation of genetic programming fitness landscapes. We propose a new indicator of problem hardness for tree-based ge- netic programming, called negative slope coefficient, based on the concept of fitness cloud. The negative slope coefficient is a predictive measure, i.e. it can be calculated without prior knowledge of the global optima. The fitness cloud is gener- ated via a sampling of individuals obtained with the Metropolis-Hastings method. The reliability of the negative slope coefficient is tested on a set of well known and representative genetic programming benchmarks, comprising the binomial-3 problem, the even parity problem and the artificial ant on the Santa Fe trail.

Negative slope coefficient: a measure to characterize genetic programming fitness landscapes

Negative slope coefficient has been recently introduced and empirically proven a suitable hardness indicator for some well known genetic programming benchmarks, such as the even parity problem, the binomial-3 and the artificial ant on the Santa Fe trail. Nevertheless, the original definition of this measure contains several limitations. This paper points out some of those limitations, presents a new and more relevant definition of the negative slope coefficient and empirically shows the suitability of this new definition as a hardness measure for some genetic programming benchmarks, including the multiplexer, the intertwined spirals problem and the royal trees.

Complex-network analysis of combinatorial spaces: the NK landscape case.

We propose a network characterization of combinatorial fitness landscapes by adapting the notion of inherent networks proposed for energy surfaces. We use the well-known family of NK landscapes as an example. In our case the inherent network is the graph whose vertices represent the local maxima in the landscape, and the edges account for the transition probabilities between their corresponding basins of attraction. We exhaustively extracted such networks on representative NK landscape instances, and performed a statistical characterization of their properties. We found that most of these network properties are related to the search difficulty on the underlying NK landscapes with varying values of K .

On the structure of multiobjective combinatorial search space: MNK-landscapes with correlated objectives

The structure of the search space explains the behavior of multiobjective search algorithms, and helps to design well-performing approaches. In this work, we analyze the properties of multiobjective combinatorial search spaces, and we pay a particular attention to the correlation between the objective functions. To do so, we extend the multiobjective NK-landscapes in order to take the objective correlation into account. We study the co-influence of the problem dimension, the degree of non-linearity, the number of objectives, and the objective correlation on the structure of the Pareto optimal set, in terms of cardinality and number of supported solutions, as well as on the number of Pareto local optima. This work concludes with guidelines for the design of multiobjective local search algorithms, based on the main fitness landscape features.

Where are bottlenecks in NK fitness landscapes

Usually the offspring-parent fitness correlations is used to visualize and analyze some characteristics of fitness landscapes such as evolvability. In this paper, we introduce a more general representation of this correlation, the fitness cloud (FC). We use the bottleneck metaphor to emphasis fitness levels in landscape that cause local search process to slow down. For a local search heuristic such as hill-climbing or simulated annealing, FC allows one to visualize the bottleneck and neutrality of landscapes. To confirm the relevance of the FC representation we show where the bottlenecks are in the well-known NK fitness landscape and also how to use neutrality information from the FC to combine some neutral operator with local search heuristic.

First-improvement vs. best-improvement local optima networks of NK landscapes

This paper extends a recently proposed model for combinatorial landscapes: Local Optima Networks (LON), to incorporate a first-improvement (greedy-ascent) hill-climbing algorithm, instead of a best-improvement (steepestascent) one, for the definition and extraction of the basins of attraction of the landscape optima. A statistical analysis comparing best and first improvement network models for a set of NK landscapes, is presented and discussed. Our results suggest structural differences between the two models with respect to both the network connectivity, and the nature of the basins of attraction. The impact of these differences in the behavior of search heuristics based on first and best improvement local search is thoroughly discussed.

Local Optima Networks of the Quadratic Assignment Problem

Using a recently proposed model for combinatorial landscapes, Local Optima Networks (LON), we conduct a thorough analysis of two types of instances of the Quadratic Assignment Problem (QAP). This network model is a reduction of the landscape in which the nodes correspond to the local optima, and the edges account for the notion of adjacency between their basins of attraction. The model was inspired by the notion of ‘inherent network’ of potential energy surfaces proposed in physical-chemistry. The local optima networks extracted from the so called uniform and real-like QAP instances, show features clearly distinguishing these two types of instances. Apart from a clear confirmation that the search difficulty increases with the problem dimension, the analysis provides new confirming evidence explaining why the real-like instances are easier to solve exactly using heuristic search, while the uniform instances are easier to solve approximately. Although the local optima network model is still under development, we argue that it provides a novel view of combinatorial landscapes, opening up the possibilities for new analytical tools and understanding of problem difficulty in combinatorial optimization.

Local Optima Networks: A New Model of Combinatorial Fitness Landscapes

This chapter overviews a recently introduced network-based model of combinatorial landscapes: Local Optima Networks (LON). The model compresses the information given by the whole search space into a smaller mathematical object that is a graph having as vertices the local optima and as edges the possible weighted transitions between them. Two definitions of edges have been proposed: basin-transition and escape-edges, which capture relevant topological features of the underlying search spaces. This network model brings a new set of metrics to characterize the structure of combinatorial landscapes, those associated with the science of complex networks. These metrics are described, and results are presented of local optima network extraction and analysis for two selected combinatorial landscapes: NK landscapes and the quadratic assignment problem. Network features are found to correlate with and even predict the performance of heuristic search algorithms operating on these problems.

ParadisEO-MO: from fitness landscape analysis to efficient local search algorithms

This paper presents a general-purpose software framework dedicated to the design, the analysis and the implementation of local search metaheuristics: ParadisEO-MO. A substantial number of single solution-based local search metaheuristics has been proposed so far, and an attempt of unifying existing approaches is here presented. Based on a fine-grained decomposition, a conceptual model is proposed and is validated by regarding a number of state-of-the-art methodologies as simple variants of the same structure. This model is then incorporated into the ParadisEO-MO software framework. This framework has proven its efficiency and high flexibility by enabling the resolution of many academic and real-world optimization problems from science and industry.