A Mathematical Unification of Geometric Crossovers Defined on Phenotype Space

Geometric crossover is a representation-independent definition of crossover based on the distance of the search space interpreted as a metric space. It generalizes the traditional crossover for binary strings and other important recombination operators for the most frequently used representations. Using a distance tailored to the problem at hand, the abstract definition of crossover can be used to design new problem specific crossovers that embed problem knowledge in the search. This paper is motivated by the fact that genotype-phenotype mapping can be theoretically interpreted using the concept of quotient space in mathematics. In this paper, we study a metric transformation, the quotient metric space, that gives rise to the notion of quotient geometric crossover. This turns out to be a very versatile notion. We give many example applications of the quotient geometric crossover

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

Geometric Semantic Grammatical Evolution

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Geometric Semantic Genetic Programming (GSGP) is a novel form of Genetic Programming (GP), based on a geometric theory of evolutionary algorithms, which directly searches the semantic space of programs. In this chapter, we extend this framework to Grammatical Evolution (GE) and refer to the new method as Geometric Semantic Grammatical Evolution (GSGE). We formally derive new mutation and crossover operators for GE which are guaranteed to see a simple unimodal fitness landscape. This surprising result shows that the GE genotypephenotype mapping does not necessarily imply low genotype-fitness locality. To complement the theory, we present extensive experimental results on three standard domains (Boolean, Arithmetic and Classifier)

Geometric Semantic Genetic Programming

Traditional Genetic Programming (GP) searches the space of functions/programs by using search operators that manipulate their syntactic representation, regardless of their actual semantics/behaviour. Recently, semantically aware search operators have been shown to outperform purely syntactic operators. In this work, using a formal geometric view on search operators and representations, we bring the semantic approach to its extreme consequences and introduce a novel form of GP – Geometric Semantic GP (GSGP) – that searches directly the space of the underlying semantics of the programs. This perspective provides new insights on the relation between program syntax and semantics, search operators and fitness landscape, and allows for principled formal design of semantic search operators for different classes of problems. We de- rive specific forms of GSGP for a number of classic GP domains and experimentally demonstrate their superiority to conventional operators

CSM429: Abstract Geometric Crossover for the Permutation Representation

Abstract crossover and abstract mutation are representation-independent operators that are well-defined once a notion of distance over the solution space is defined. They were obtained as generalization of genetic operators for binary strings and real vectors. In this paper we explore how the abstract geometric framework applies to the permutation representation. This representation is challenging for various reasons: because of the inherent difference between permutations and the representations that inspired the abstraction; because the whole notion of geometry over permutation spaces radically departs from traditional geometries and it is almost unexplored mathematical territory; because the many notions of distance available and their subtle interconnections make it hard to see the right distance to use, if any; because the various available interpretations of permutations make ambiguous what a permutation represents, hence, how to treat it; because of the existence of various permutation-like representations that are incorrectly confused with permutations; and finally because of the existence of many mutation and recombination operators and their many variations for the same representation. This article shows that the application of our geometric framework naturally clarifies and unifies an important domain,the permutation representation and the related operators, in which there was little or no hope to find order. In addition the abstract geometric framework is used to improve the design of crossover operators for well-known problems naturally connected with the permutation representation

Geometric Semantic Genetic Programming

Traditional Genetic Programming (GP) searches the space of functions/programs by using search operators that manipulate their syntactic representation, regardless of their actual semantics/behaviour. Recently, semantically aware search operators have been shown to outperform purely syntactic operators. In this work, using a formal geometric view on search operators and representations, we bring the semantic approach to its extreme consequences and introduce a novel form of GP – Geometric Semantic GP (GSGP) – that searches directly the space of the underlying semantics of the programs. This perspective provides new insights on the relation between program syntax and semantics, search operators and fitness landscape, and allows for principled formal design of semantic search operators for different classes of problems. We de- rive specific forms of GSGP for a number of classic GP domains and experimentally demonstrate their superiority to conventional operators

Geometric generalisation of surrogate model-based optimisation to combinatorial and program spaces

Open access journalSurrogate models (SMs) can profitably be employed, often in conjunction with evolutionary algorithms, in optimisation in which it is expensive to test candidate solutions. The spatial intuition behind SMs makes them naturally suited to continuous problems, and the only combinatorial problems that have been previously addressed are those with solutions that can be encoded as integer vectors. We show how radial basis functions can provide a generalised SM for combinatorial problems which have a geometric solution representation, through the conversion of that representation to a different metric space. This approach allows an SM to be cast in a natural way for the problem at hand, without ad hoc adaptation to a specific representation. We test this adaptation process on problems involving binary strings, permutations, and tree-based genetic programs. © 2014 Yong-Hyuk Kim et al

A flexible and efficient multi-purpose optimization library in python

Bakurov, I., Buzzelli, M., Castelli, M., Vanneschi, L., & Schettini, R. (2021). General purpose optimization library (Gpol): A flexible and efficient multi-purpose optimization library in python. Applied Sciences (Switzerland), 11(11), 1-34. [4774]. https://doi.org/10.3390/app11114774Several interesting libraries for optimization have been proposed. Some focus on individual optimization algorithms, or limited sets of them, and others focus on limited sets of problems. Frequently, the implementation of one of them does not precisely follow the formal definition, and they are difficult to personalize and compare. This makes it difficult to perform comparative studies and propose novel approaches. In this paper, we propose to solve these issues with the General Purpose Optimization Library (GPOL): a flexible and efficient multipurpose optimization library that covers a wide range of stochastic iterative search algorithms, through which flexible and modular implementation can allow for solving many different problem types from the fields of continuous and combinatorial optimization and supervised machine learning problem solving. Moreover, the library supports full-batch and mini-batch learning and allows carrying out computations on a CPU or GPU. The package is distributed under an MIT license. Source code, installation instructions, demos and tutorials are publicly available in our code hosting platform (the reference is provided in the Introduction).publishersversionpublishe

Automatic synthesis of sorting algorithms by gene expression programming + (geometric) semantic gene expression programming + encouraging phenotype variation with a new semantic operator: semantic conditional crossover

Gene Expression Programming (GEP) is an alternative to Genetic Programming (GP). Given its characteristics compared to GP, we question if GEP should be the standard choice for evolutionary program synthesis, both as base for research and practical application. We raise the question if such a shift could increase the rate of investigation, applicability and the quality of results obtained from evolutionary techniques for code optimization. We present three distinct and unprecedented studies using GEP in an attempt to develop understanding, investigate the potential and forward the branch. Each study has an individual contribution on its own involving GEP. As a whole, the three studies try to investigate di erent aspects that might be critical to answer the questions raised in the previous paragraph. In the rst individual contribution, we investigate GEP's applicability to automatically synthesize sorting algorithms. Performance is compared against GP under similar experimental conditions. GEP is shown to be capable of producing sorting algorithms and outperforms GP in doing so. As a second experiment, we enhanced GEP's evolutionary process with semantic awareness of candidate programs, originating Semantic Gene Expression Programming (SGEP), similarly to how Semantic Genetic Programming (SGP) builds over GP. Geometric semantic concepts are then introduced to SGEP, forming Geometric Semantic Gene Expression Programming (GSGEP). A comparative experiment between GP, GEP, SGP and SGEP is performed using di erent problems and setup combinations. Results were mixed when comparing SGEP and SGP, suggesting performance is signi cantly related to the problem addressed. By out-performing the alternatives in many of the benchmarks, SGEP demonstrates practical potential. The results are analyzed in di erent perspectives, also providing insight on the potential of di erent crossover variations when applied along GP/GEP. GEP' compatibility with innovation developed to work with GP is demonstrated possible without extensive adaptation. Considerations for integration of SGEP are discussed. In the last contribution, a new semantic operator is proposed, SCC, which applies crossover conditionally only when elements are semantically di erent enough, performing mutation otherwise. The strategy attempts to encourage semantic diversity and wider the portion of the semantic-solution space searched. A practical experiment was performed alternating the integration of SCC in the evolutionary process. When using the operator, the quality of obtained solutions alternated between slight improvements and declines. The results don't show a relevant indication of possible advantage from its employment and don't con rm what was expected in the theory. We discuss ways in which further work might investigate this concept and assess if it has practical potential under di erent circumstances. On the other hand, in regards to the basilar questions of this investigation, the process of development and testing of SCC is performed completely on a GEP/SGEP base, suggesting how the latest can be used as the base for future research on evolutionary program synthesis.Programa c~ao Gen etica por Express~oes (GEP) e uma alternativa recente a Programa c~ao Gen etica (GP). Neste estudo observamos o GEP e colocamos a quest~ao se este n~ao deveria ser tratado como primeira escolha quando se trata de sintetiza c~ao autom atica de programas atrav es de m etodos evolutivos. Dadas as caracteristicas do GEP perguntamonos se esta mudan ca de perspectiva poderia aumentar a investiga c~ao, aplicabilidade e qualidade dos resultados obtidos para a optimiza c~ao de c odigo por m etodos evolutivos. Neste estudo apresentamos tr^es contribui c~oes in editas e distintas usando o algoritmo GEP. Cada uma das contribui c~oes apresenta um avan co ou investiga c~ao no campo da GEP. Como um todo, estas contribui c~oes tentam obter cohecimento e informa c~oes para se abordar a quest~ao geral apresentada no p aragrafo anterior. Na primeira contribui c~ao, investiga-mos e testamos o GEP no problema da sintese autom atica de algoritmos de ordena c~ao. Para o melhor do nosso conhecimento, esta e a primeira vez que este problema e abordado com o GEP. A performance e comparada a do GP em condi c~oes semelhantes, de modo a isolar as caracteristicas de cada algoritmo como factor de distin c~ao. As a second experiment, we enhanced GEP's evolutionary process with semantic awareness of candidate programs, originating Semantic Gene Expression Programming (SGEP), similarly to how Semantic Genetic Programming (SGP) builds over GP. Geometric semantic concepts are then introduced to SGEP, forming Geometric Semantic Gene Expression Programming (GSGEP). A comparative experiment between GP, GEP, SGP and SGEP is performed using di erent problems and setup combinations. Results were mixed when comparing SGEP and SGP, suggesting performance is signi cantly related to the problem addressed. By out-performing the alternatives in many of the benchmarks, SGEP demonstrates practical potential. The results are analyzed in di erent perspectives, also providing insight on the potential of di erent crossover variations when applied along GP/GEP. GEP's compatibility with innovation developed to work with GP is demonstrated possible without extensive adaptation. Considerations for integration of SGEP are discussed. Na segunda contribui c~ao, adicionamos ao processo evolutivo do GEP a capacidade de medir o valor sem^antico dos programas que constituem a popula c~ao. A esta variante damos o nome de Programa c~ao Gen etica por Express~oes Sem^antica (SGEP). Esta variante tr as para o GEP as mesmas caracteristicas que a Programa c~ao Gen etica Sem^antica(SGP) trouxe para o GP convencional. Conceitos geom etricos s~ao tamb em apresentados para o SGEP, extendendo assim a variante e criando a Programa c~ao Gen etica por Express~oes Geom etrica Sem^antica (GSGEP). De forma a testar estas novas variantes, efectuamos uma experi^encia onde s~ao comparados o GP, GEP, SGP e SGEP entre diferentes problemas e combina c~oes de operadores de cruzamento. Os resultados mostraram que n~ao houve um algoritmo que se destaca-se em todas as experi^encias, sugerindo que a performance est a signi cativamente relacionada com o problema a ser abordado. De qualquer modo, o SGEP obteve vantagem em bastantes dos benchmarks, dando assim ind cios de pot^encial ter utilidade pr atica. De um modo geral, esta contribui c~ao demonstra que e possivel utilizar tecnologia desenvolvida a pensar em GP no GEP sem grande esfor co na adapta c~ao. No m da contribui c~ao, s~ao discutidas algumas considera c~oes sobre o SGEP. Na terceira contribui c~ao propomos um novo operador, o Cruzamento Sem^antico Condicional (SCC). Este operador, baseado na dist^ancia sem^antica entre dois elementos propostos, decide se os elementos s~ao propostos para cruzamento, ou se um deles e mutato e ambos re-introduzidos na popula c~ao. Esta estrat egia tem como objectivo aumentar a diversidade gen etica na popula c~ao em fases cruciais do processo evolutivo e alargar a por c~ao do espa co sem^antico pesquisado. Para avaliar o pot^encial deste operador, realizamos uma experi^encia pr atica e comparamos processos evolutivos semelhantes onde o uso ou n~ao uso do SCC e o factor de distin c~ao. Os resultados obtidos n~ao demonstraram vantagens no uso do SCC e n~ao con rmam o esperado em teoria. No entanto s~ao discutidas maneiras em que o conceito pode ser reaproveitado para novos testes em que possa ter pot^encial para demonstrar resultados possitivos. Em rela c~ao a quest~ao central da tese, visto este estudo ter sido desenvolvido com base em GEP/SGEP e visto a teoria do SCC ser compativel com GP, e demonstrado que um estudo geral a area da sintese de algoritmos por meios evolutivos, pode ser conduzido com base no GEP