Bridging Elementary Landscapes and a Geometric Theory of Evolutionary Algorithms: First Steps

Abstract

Based on a geometric theory of evolutionary algorithms, it was shown that all evolutionary algorithms equipped with a geometric crossover and no mutation operator do the same kind of convex search across representations, and that they are well-matched with generalised forms of concave fitness landscapes for which they provably find the optimum in polynomial time [13]. Analysing the landscape structure is essential to understand the relationship between problems and evolutionary algorithms. This paper continues such investigations by considering the following challenge: develop an analytical method to recognise that the fitness landscape for a given problem provably belongs to a class of concave fitness landscapes. Elementary landscapes theory provides analytic algebraic means to study the landscape structure [15]. This work begins linking both theories to better understand how such method could be devised using elementary landscapes. Examples on the well known One Max, Leading Ones, Not-All-Equal Satisfiability and Weight Partition problems illustrate the fundamental concepts supporting this approach.

A Unifying View on Recombination Spaces and Abstract Convex Evolutionary Search

“…Recombination landscapes are not essentially different in both theories and can be unified, meaning that EL can be applied consistently to analyse certain landscapes arising in GF [1]. This paper extends their unification, by showing that the algebraic approach of EL admits dual formalisations of two fundamental aspects about crossovers and their behaviour in GF: classification of geometric crossovers and abstract convex search (accomplished by geometric-crossover EAs).…”

“…Indeed, complete geometric crossovers are also recombination P-structures (Proposition 4). Note it does not hold for general geometric crossovers, as [1] unwittingly claimed. For example, any geometric crossover where offspring never include the parents will not fulfil the null-recombination axiom (Definition 6), also the crossover in Example 2 is geometric but not a recombination P-structure due to asymmetry.…”

“…Thus, a metric space constitutes the structure of a fitness landscape (X, d, f ) with an arbitrary realvalued fitness function f : X → R. This geometric view of the search space allows mutation 1 and crossover operators to be conceived within the same space structure, by means of metric balls and metric segments respectively. Definition 1 (Metric ball and segment [7]).…”

“…An open challenge in evolutionary computing is to identify (without running any experiments) if a fitness landscape class is well matched with an evolutionary algorithm (EA) class, where good performance guarantees can be provided. Previous work [1] laid the foundations for an analytical fitness landscape method that determines if a problem s landscape belongs to certain abstract convex landscape classes, where certain recombination-based EAs exhibit polynomial runtime [2]. The theories we began unifying, namely a general geometric framework (GF) of EAs [3] and an algebraic theory of fitness landscapes known as elementary landscapes (EL) [4], complement each other towards such challenge even if they originally pursue distinct aims.…”

“…1. Proving the existence of two broad classes of crossovers: those which are recombination P-structures, including geometric crossovers [1], and those which are neither geometric nor recombination P-structures. This justifies that a unification of GF and EL would not be an 'empty' theory (i.e.…”