Mathematical Interpretation between Genotype and Phenotype Spaces and Induced Geometric Crossovers

In this paper, we present that genotype-phenotype mapping can be theoretically interpreted using the concept of quotient space in mathematics. Quotient space can be considered as mathematically-defined phenotype space in the evolutionary computation theory. The quotient geometric crossover has the effect of reducing the search space actually searched by geometric crossover, and it introduces problem knowledge in the search by using a distance better tailored to the specific solution interpretation. Quotient geometric crossovers are directly applied to the genotype space but they have the effect of the crossovers performed on phenotype space. We give many example applications of the quotient geometric crossover

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

CSM-467: Quotient Geometric Crossovers

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. In previous work we have started studying how metric transformations of the distance associated with geometric crossover affect the original geometric crossover. In particular, we focused on the product of metric spaces. This metric transformation gives rise to the notion of product geometric crossover that allows to build new geometric crossovers combining pre-existing geometric crossovers in a simple way. In this paper, we study another 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 examples of application of the quotient geometric crossover

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 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

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

Searching the landscape of flux vacua with genetic algorithms

In this paper, we employ genetic algorithms to explore the landscape of type IIB flux vacua. We show that genetic algorithms can efficiently scan the landscape for viable solutions satisfying various criteria. More specifically, we consider a symmetric $T^{6}$ as well as the conifold region of a Calabi-Yau hypersurface. We argue that in both cases genetic algorithms are powerful tools for finding flux vacua with interesting phenomenological properties. We also compare genetic algorithms to algorithms based on different breeding mechanisms as well as random walk approaches