Correlated and uncorrelated fitness landscapes and how to tell the difference

Abstract

Abstract. The properties of multi-peaked "fitness landscapes" have attracted attention in a wide variety of fields, including evolutionary biology. However, relatively little attention has been paid to the properties of the landscapes themselves. Herein, we suggest a framework for the mathematical treatment of such landscapes, including an explicit mathematical model. A central role in this discussion is played by the autocorrelation of fitnesses obtained from a random walk on the landscape. Our ideas about average autocorrelations allow us to formulate a condition (satisfied by a wide class of landscapes we call AR(1) landscapes) under which the average autocorrelation approximates a decaying exponential. We then show how our mathematical model can be used to estimate both the globally optimal fitnesses of AR(1) landscapes and their local structure. We illustrate some aspects of our method with computer experiments based on a single family of landscapes (Kauffman's "N-k model"), that is shown to be a generic AR(I) landscape. We close by discussing how these ideas might be useful in the "tuning" of combinatorial optimization algorithms, and in modelling in the experimental sciences.

Optimal execution with non-linear transient market impact

“…Alternatively, the landscape may be rugged, with many local minima separated by local peaks. Rugged landscapes have attracted attention in physics [54], evolutionary biology [53], and computer science. An example of a rugged landscape is Kauffman's N-k fitness landscape [54].…”

The linear ordering problem: Instances, search space analysis and algorithms

“…) [26,27,33]; the lower is the value of ¥ , the more rugged is the landscape. Interestingly, in landscape analysis literature general intuitions and some results suggest that there is a negative correlation between ¥ and the hardness of the problem [2].…”

“…First, we give a detailed analysis of the search space characteristics of all the instance classes introduced in the major algorithmic contributions to the LOP. This includes a structural analysis, where standard statistical information is gathered as well as an analysis of the main search space characteristics such as autocorrelation [27,33] and fitness-distance analysis [13]. A second contribution is the detailed analysis of two metaheuristics, an iterated local search algorithm [18] and a memetic algorithm [23].…”

Defining locality as a problem difficulty measure in genetic programming

“…Several other approaches to studying landscapes and problem difficulty have also been proposed, generally in a non-GP context, including: other measures of landscape correlation [68], autocorrelation [39]; epistasis, which measures the degree of interaction between genes and is a component of deception [20,21,41]; monotonicity, which is similar to fdc in that it measures how often fitness improves despite distance to the optimum increasing [41]; and distance distortion which relates overall distance in the genotype and phenotype spaces [52]. All of these measures are to some extent related.…”