Розміщення прямокутних об’єктів з критерієм рівноваги в обмежених кордонах з використанням фрагментарного та еволюційного алгоритмів

В статті розглянуто задачу розміщення прямокутних об’єктів у обмежених кордонах з критерієм рівноваги. Показано, ця задача має фрагментарну структуру. Для пошуку наближеного розв’язку задачі запропоновано гібридний алгоритм на основі фрагментарного алгоритму і модифікації еволюційного алгоритму на перестановках. Запропоновано методи порівняльної оцінки алгоритму.In the article the problem of placement of rectangular objects in limited borders criterion balance. It is shown that this problem has a fragmented structure To search for an approximate solution of the problem proposed hybrid algorithm based on fragmentary algorithm and evolutionary algorithm modifications on permutations. The methods of comparative evaluation algorithm

Фрагментарная модель и эволюционный алгоритм 2D упаковки объектов

Рассмотрена задача двумерной упаковки в прямоугольник объектов сложной формы. Показано, что задача упаковки имеет фрагментарную структуру. Для поиска приближенного решения задачи предложена модификация эволюционного алгоритма на перестановках с геометрическим оператором кроссовера. Приводятся результаты численного эксперимента.The problem of two-dimensional packing in rectangle of objects of complex shape. It is shown that the packing problem has fragmentary structure. To find an approximate solution proposed modification of the evolutionary algorithm on permutations with geometric crossover operator. The results of numerical experiment

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

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Paper to be presented at the Fifteenth International Conference on Parallel Problem Solving from Nature (PPSN XV), Coimbra, Portugal on 8-12 September.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. 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 landscapes structure. This work begins linking both theories to better understand how such method could be devised using elementary landscapes. Examples on well known One Max, Leading Ones, Not-All-Equal Satisfiability and Weight Partitioning problems illustrate the fundamental concepts supporting this approach

A Self-Adaptive Approach to Exploit Topological Properties of Different GAs’ Crossover Operators

Evolutionary algorithms (EAs) are a family of optimization algorithms inspired by the Darwinian theory of evolution, and Genetic Algorithm (GA) is a popular technique among EAs. Similar to other EAs, common limitations of GAs have geometrical origins, like premature convergence, where the final population’s convex hull might not include the global optimum. Population diversity maintenance is a central idea to tackle this problem but is often performed through methods that constantly diminish the search space’s area. This work presents a self- adaptive approach, where the non-geometric crossover is strategically employed with geometric crossover to maintain diversity from a geometrical/topological perspective. To evaluate the performance of the proposed method, the experimental phase compares it against well-known diversity maintenance methods over well-known benchmarks. Experimental results clearly demonstrate the suitability of the proposed self-adaptive approach and the possibility of applying it to different types of crossover and EAs

A Unifying View on Recombination Spaces and Abstract Convex Evolutionary Search

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Proceedings of EvoCOP 2019 - 19th European Conference on Evolutionary Computation, 24-26 April 2019, Leipzig, GermanyPrevious work proposed to unify an algebraic theory of fitness landscapes and a geometric framework of evolutionary algorithms (EAs). One of the main goals behind this unification is to develop an analytical method that verifies if a problem's landscape belongs to certain abstract convex landscapes classes, where certain recombination-based EAs (without mutation) have polynomial runtime performance. This paper advances such unification by showing that: (a) crossovers can be formally classified according to geometric or algebraic axiomatic properties; and (b) the population behaviour induced by certain crossovers in recombination-based EAs can be formalised in the geometric and algebraic theories. These results make a significant contribution to the basis of an integrated geometric-algebraic framework with which analyse recombination spaces and recombination-based EAs

Inbreeding properties of geometric crossover and non-geometric recombinations

Abstract. Geometric crossover is a representation-independent generalization of traditional crossover for binary strings. It is defined using the distance associated to the search space in a simple geometric way. Many interesting recombination operators for the most frequently used representations are geometric crossovers under some suitable distance. Being a geometric crossover is useful because there is a growing number of theoretical results that apply to this class of operators. To show that a given recombination operator is a geometric crossover, it is sufficient to find a distance for which offspring are in the metric segment between parents associated with this distance. However, proving that a recombination operator is not a geometric crossover requires to prove that such an operator is not a geometric crossover under any distance. In this paper we develop some theoretical tools to prove non-geometricity results and show that some well-known operators are not geometric.

Unifying a Geometric Framework of Evolutionary Algorithms and Elementary Landscapes Theory

Evolutionary algorithms (EAs) are randomised general-purpose strategies, inspired by natural evolution, often used for finding (near) optimal solutions to problems in combinatorial optimisation. Over the last 50 years, many theoretical approaches in evolutionary computation have been developed to analyse the performance of EAs, design EAs or measure problem difficulty via fitness landscape analysis. An open challenge is to formally explain why a general class of EAs perform better, or worse, than others on a class of combinatorial problems across representations. However, the lack of a general unified theory of EAs and fitness landscapes, across problems and representations, makes it harder to characterise pairs of general classes of EAs and combinatorial problems where good performance can be guaranteed provably. This thesis explores a unification between a geometric framework of EAs and elementary landscapes theory, not tied to a specific representation nor problem, with complementary strengths in the analysis of population-based EAs and combinatorial landscapes. This unification organises around three essential aspects: search space structure induced by crossovers, search behaviour of population-based EAs and structure of fitness landscapes. First, this thesis builds a crossover classification to systematically compare crossovers in the geometric framework and elementary landscapes theory, revealing a shared general subclass of crossovers: geometric recombination P-structures, which covers well-known crossovers. The crossover classification is then extended to a general framework for axiomatically analysing the population behaviour induced by crossover classes on associated EAs. This shows the shared general class of all EAs using geometric recombination P-structures, but no mutation, always do the same abstract form of convex evolutionary search. Finally, this thesis characterises a class of globally convex combinatorial landscapes shared by the geometric framework and elementary landscapes theory: abstract convex elementary landscapes. It is formally explained why geometric recombination P-structure EAs expectedly can outperform random search on abstract convex elementary landscapes related to low-order graph Laplacian eigenvalues. Altogether, this thesis paves a way towards a general unified theory of EAs and combinatorial fitness landscapes