In
computational intelligence
The expression computational intelligence (CI) usually refers to the ability of a computer to learn a specific task from data or experimental observation. Even though it is commonly considered a synonym of soft computing, there is still no c ...
(CI), an evolutionary algorithm (EA) is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of
evolutionary computation
In computer science, evolutionary computation is a family of algorithms for global optimization inspired by biological evolution, and the subfield of artificial intelligence and soft computing studying these algorithms. In technical terms, they ...
,
a generic population-based
metaheuristic
In computer science and mathematical optimization, a metaheuristic is a higher-level procedure or heuristic designed to find, generate, or select a heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimizati ...
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
. An EA uses mechanisms inspired by
biological evolution
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation t ...
, such as
reproduction
Reproduction (or procreation or breeding) is the biological process by which new individual organisms – "offspring" – are produced from their "parent" or parents. Reproduction is a fundamental feature of all known life; each individual or ...
,
mutation
In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, mi ...
,
recombination, and
selection
Selection may refer to:
Science
* Selection (biology), also called natural selection, selection in evolution
** Sex selection, in genetics
** Mate selection, in mating
** Sexual selection in humans, in human sexuality
** Human mating strategie ...
.
Candidate solution
In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potent ...
s to the
optimization problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
play the role of individuals in a population, and the
fitness function {{no footnotes, date=May 2015
A fitness function is a particular type of objective function that is used to summarise, as a single figure of merit, how close a given design solution is to achieving the set aims. Fitness functions are used in genetic ...
determines the quality of the solutions (see also
loss function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
).
Evolution
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
of the population then takes place after the repeated application of the above operators.
Evolutionary algorithms often perform well approximating solutions to all types of problems because they ideally do not make any assumption about the underlying
fitness landscape
Fitness may refer to:
* Physical fitness, a state of health and well-being of the body
* Fitness (biology), an individual's ability to propagate its genes
* Fitness (cereal), a brand of breakfast cereals and granola bars
* ''Fitness'' (magazine), ...
. Techniques from evolutionary algorithms applied to the modeling of biological evolution are generally limited to explorations of
microevolutionary processes and planning models based upon cellular processes. In most real applications of EAs, computational complexity is a prohibiting factor.
In fact, this computational complexity is due to fitness function evaluation.
Fitness approximation
Fitness approximationY. JinA comprehensive survey of fitness approximation in evolutionary computation ''Soft Computing'', 9:3–12, 2005 aims to approximate the objective or fitness functions in evolutionary optimization by building up machine l ...
is one of the solutions to overcome this difficulty. However, seemingly simple EA can solve often complex problems;
therefore, there may be no direct link between algorithm complexity and problem complexity.
Implementation
The following is an example of a generic single-objective
genetic algorithm
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gene ...
.
Step One: Generate the initial
population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
of
individual
An individual is that which exists as a distinct entity. Individuality (or self-hood) is the state or quality of being an individual; particularly (in the case of humans) of being a person unique from other people and possessing one's own Maslow ...
s randomly. (First generation)
Step Two: Repeat the following regenerational steps until termination:
#Evaluate the
fitness of each individual in the population (time limit, sufficient fitness achieved, etc.)
#Select the fittest individuals for
reproduction
Reproduction (or procreation or breeding) is the biological process by which new individual organisms – "offspring" – are produced from their "parent" or parents. Reproduction is a fundamental feature of all known life; each individual or ...
. (Parents)
# Breed new individuals through
crossover
Crossover may refer to:
Entertainment
Albums and songs
* ''Cross Over'' (Dan Peek album)
* ''Crossover'' (Dirty Rotten Imbeciles album), 1987
* ''Crossover'' (Intrigue album)
* ''Crossover'' (Hitomi Shimatani album)
* ''Crossover'' (Yoshino ...
and
mutation
In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, mi ...
operations to give birth to
offspring
In biology, offspring are the young creation of living organisms, produced either by a single organism or, in the case of sexual reproduction, two organisms. Collective offspring may be known as a brood or progeny in a more general way. This ca ...
.
# Replace the least-fit individuals of the population with new individuals.
Types
Similar techniques differ in
genetic representation
In computer programming, genetic representation is a way of presenting solutions/individuals in evolutionary computation methods. Genetic representation can encode appearance, behavior, physical qualities of individuals. Designing a good genetic r ...
and other implementation details, and the nature of the particular applied problem.
*
Genetic algorithm
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gene ...
– This is the most popular type of EA. One seeks the solution of a problem in the form of strings of numbers (traditionally binary, although the best representations are usually those that reflect something about the problem being solved),
[ by applying operators such as recombination and mutation (sometimes one, sometimes both). This type of EA is often used in ]optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problems.
*Genetic programming
In artificial intelligence, genetic programming (GP) is a technique of evolving programs, starting from a population of unfit (usually random) programs, fit for a particular task by applying operations analogous to natural genetic processes to t ...
– Here the solutions are in the form of computer programs, and their fitness is determined by their ability to solve a computational problem. There are many variants of Genetic Programming, including Cartesian genetic programming
Cartesian genetic programming is a form of genetic programming that uses a graph representation to encode computer programs. It grew from a method of evolving digital circuits developed by Julian F. Miller and Peter Thomson in 1997. The term ‘Ca ...
, gene expression programming
In computer programming, gene expression programming (GEP) is an evolutionary algorithm that creates computer programs or models. These computer programs are complex tree structures that learn and adapt by changing their sizes, shapes, and compos ...
, grammatical evolution, linear genetic programming
:''"Linear genetic programming" is unrelated to "linear programming".''
Linear genetic programming (LGP) is a particular subset of genetic programming wherein computer programs in a population are represented as a sequence of instructions from i ...
, multi expression programming
Multi Expression Programming (MEP) is an evolutionary algorithm for generating mathematical functions describing a given set of data. MEP is a Genetic Programming variant encoding multiple solutions in the same chromosome. MEP representation is not ...
etc.
*Evolutionary programming Evolutionary programming is one of the four major evolutionary algorithm paradigms. It is similar to genetic programming, but the structure of the program to be optimized is fixed, while its numerical parameters are allowed to evolve.
It was first ...
– Similar to genetic programming, but the structure of the program is fixed and its numerical parameters are allowed to evolve.
*Evolution strategy
In computer science, an evolution strategy (ES) is an optimization technique based on ideas of evolution. It belongs to the general class of evolutionary computation or artificial evolution methodologies.
History
The 'evolution strategy' optimizat ...
– Works with vectors of real numbers as representations of solutions, and typically uses self-adaptive mutation rates. The method is mainly used for numerical optimization, although there are also variants for combinatorial tasks.
*Differential evolution
In evolutionary computation, differential evolution (DE) is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Such methods are commonly known as metaheuristics as ...
– Based on vector differences and is therefore primarily suited for numerical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problems.
* Neuroevolution – Similar to genetic programming but the genomes represent artificial neural networks by describing structure and connection weights. The genome encoding can be direct or indirect.
*Learning classifier system
Learning classifier systems, or LCS, are a paradigm of rule-based machine learning methods that combine a discovery component (e.g. typically a genetic algorithm) with a learning component (performing either supervised learning, reinforcement lear ...
– Here the solution is a set of classifiers (rules or conditions). A Michigan-LCS evolves at the level of individual classifiers whereas a Pittsburgh-LCS uses populations of classifier-sets. Initially, classifiers were only binary, but now include real, neural net, or S-expression
In computer programming, an S-expression (or symbolic expression, abbreviated as sexpr or sexp) is an expression in a like-named notation for nested list (tree-structured) data. S-expressions were invented for and popularized by the programming la ...
types. Fitness is typically determined with either a strength or accuracy based reinforcement learning
Reinforcement learning (RL) is an area of machine learning concerned with how intelligent agents ought to take actions in an environment in order to maximize the notion of cumulative reward. Reinforcement learning is one of three basic machine ...
or supervised learning
Supervised learning (SL) is a machine learning paradigm for problems where the available data consists of labelled examples, meaning that each data point contains features (covariates) and an associated label. The goal of supervised learning alg ...
approach.
Theoretical background
The following theoretical principles apply to all or almost all EAs.
No free lunch theorem
The no free lunch theorem
In mathematical folklore, the "no free lunch" (NFL) theorem (sometimes pluralized) of David Wolpert and William Macready appears in the 1997 "No Free Lunch Theorems for Optimization".Wolpert, D.H., Macready, W.G. (1997),No Free Lunch Theorems f ...
of optimization states that all optimization strategies are equally effective when the set of all optimization problems is considered. Under the same condition, no evolutionary algorithm is fundamentally better than another. This can only be the case if the set of all problems is restricted. This is exactly what is inevitably done in practice. Therefore, to improve an EA, it must exploit problem knowledge in some form (e.g. by choosing a certain mutation strength or a problem-adapted coding). Thus, if two EAs are compared, this constraint is implied. In addition, an EA can use problem specific knowledge by, for example, not randomly generating the entire start population, but creating some individuals through heuristics
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
or other procedures. Another possibility to tailor an EA to a given problem domain is to involve suitable heuristics, local search procedures or other problem-related procedures in the process of generating the offspring. This form of extension of an EA is also known as a memetic algorithm
A memetic algorithm (MA) in computer science and operations research, is an extension of the traditional genetic algorithm. It may provide a sufficiently good solution to an optimization problem. It uses a local search technique to reduce the like ...
. Both extensions play a major role in practical applications, as they can speed up the search process and make it more robust.
Convergence
For EAs in which, in addition to the offspring, at least the best individual of the parent generation is used to form the subsequent generation (so-called elitist EAs), there is a general proof of convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
under the condition that an optimum
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
exists. Without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate ...
, a maximum search is assumed for the proof:
From the property of elitist offspring acceptance and the existence of the optimum it follows that per generation an improvement of the fitness of the respective best individual will occur with a probability . Thus:
:
I.e., the fitness values represent a monotonically non-decreasing sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
, which is bounded due to the existence of the optimum. From this follows the convergence of the sequence against the optimum.
Since the proof makes no statement about the speed of convergence, it is of little help in practical applications of EAs. But it does justify the recommendation to use elitist EAs. However, when using the usual panmictic
Panmixia (or panmixis) means random mating. A panmictic population is one where all individuals are potential partners. This assumes that there are no mating restrictions, neither genetic nor behavioural, upon the population and that therefore all ...
population model, elitist EAs tend to converge prematurely more than non-elitist ones. In a panmictic population model, mate selection (step 2 of the section about implementation) is such that every individual in the entire population is eligible as a mate. In non-panmictic populations, selection is suitably restricted, so that the dispersal speed of better individuals is reduced compared to panmictic ones. Thus, the general risk of premature convergence of elitist EAs can be significantly reduced by suitable population models that restrict mate selection.
Virtual alphabets
With the theory of virtual alphabets, David E. Goldberg
David Edward Goldberg (born September 26, 1953) is an American computer scientist, civil engineer, and former professor. Until 2010, he was a professor in the department of Industrial and Enterprise Systems Engineering (IESE) at the Universi ...
showed in 1990 that by using a representation with real numbers, an EA that uses classical recombination operators (e.g. uniform or n-point crossover) cannot reach certain areas of the search space, in contrast to a coding with binary numbers. This results in the recommendation for EAs with real representation to use arithmetic operators for recombination (e.g. arithmetic mean or intermediate recombination). With suitable operators, real-valued representations are more effective than binary ones, contrary to earlier opinion.
Comparison to biological processes
A possible limitation of many evolutionary algorithms is their lack of a clear genotype–phenotype distinction
The genotype–phenotype distinction is drawn in genetics. "Genotype" is an organism's full hereditary information. "Phenotype" is an organism's actual observed properties, such as morphology, development, or behavior. This distinction is fundam ...
. In nature, the fertilized egg cell undergoes a complex process known as embryogenesis
An embryo is an initial stage of development of a multicellular organism. In organisms that reproduce sexually, embryonic development is the part of the life cycle that begins just after fertilization of the female egg cell by the male sperm ...
to become a mature phenotype
In genetics, the phenotype () is the set of observable characteristics or traits of an organism. The term covers the organism's morphology or physical form and structure, its developmental processes, its biochemical and physiological proper ...
. This indirect encoding
In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communication ...
is believed to make the genetic search more robust (i.e. reduce the probability of fatal mutations), and also may improve the evolvability
Evolvability is defined as the capacity of a system for adaptive evolution. Evolvability is the ability of a population of organisms to not merely generate genetic diversity, but to generate ''adaptive'' genetic diversity, and thereby evolve throu ...
of the organism. Such indirect (also known as generative or developmental) encodings also enable evolution to exploit the regularity in the environment. Recent work in the field of artificial embryogeny, or artificial developmental systems, seeks to address these concerns. And gene expression programming
In computer programming, gene expression programming (GEP) is an evolutionary algorithm that creates computer programs or models. These computer programs are complex tree structures that learn and adapt by changing their sizes, shapes, and compos ...
successfully explores a genotype–phenotype system, where the genotype consists of linear multigenic chromosomes of fixed length and the phenotype consists of multiple expression trees or computer programs of different sizes and shapes.
Applications
The areas in which evolutionary algorithms are practically used are almost unlimited and range from industry, engineering, complex scheduling, agriculture and finance to research and art
Art is a diverse range of human activity, and resulting product, that involves creative or imaginative talent expressive of technical proficiency, beauty, emotional power, or conceptual ideas.
There is no generally agreed definition of wha ...
. The application of an evolutionary algorithm requires some rethinking from the inexperienced user, as the approach to a task using an EA is different from conventional exact methods and this is usually not part of the curriculum of engineers or other disciplines. For example, the fitness calculation must not only formulate the goal but also support the evolutionary search process towards it, e.g. by rewarding improvements that do not yet lead to a better evaluation of the original quality criteria. For example, if peak utilisation of resources such as personnel deployment or energy consumption is to be avoided in a scheduling task, it is not sufficient to assess the maximum utilisation. Rather, the number and duration of exceedances of a still acceptable level should also be recorded in order to reward reductions below the actual maximum peak value. There are therefore some publications that are aimed at the beginner and want to help avoiding beginner's mistakes as well as leading an application project to success. This includes clarifying the fundamental question of when an EA should be used to solve a problem and when it is better not to.
Related techniques
Swarm algorithms include:
* Ant colony optimization
In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial ants stand for multi ...
is based on the ideas of ant foraging by pheromone communication to form paths. Primarily suited for combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combi ...
and graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
problems.
* The runner-root algorithm (RRA) is inspired by the function of runners and roots of plants in nature.
* Artificial bee colony algorithm
In computer science and operations research, the artificial bee colony algorithm (ABC) is an optimization algorithm based on the intelligent foraging behaviour of honey bee swarm, proposed by Derviş Karaboğa (Erciyes University) in 2005.
Alg ...
is based on the honeybee foraging behaviour. Primarily proposed for numerical optimization and extended to solve combinatorial, constrained and multi-objective optimization problems.
* Bees algorithm
In computer science and operations research, the bees algorithm is a population-based search algorithm which was developed by Pham, Ghanbarzadeh et al. in 2005.Pham DT, Ghanbarzadeh A, Koc E, Otri S, Rahim S and Zaidi M. The Bees Algorithm. Technic ...
is based on the foraging behaviour of honeybees. It has been applied in many applications such as routing and scheduling.
* Cuckoo search
In operations research, cuckoo search is an optimization algorithm developed by Xin-She Yang and Suash Deb
in 2009. It was inspired by the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of host birds of other s ...
is inspired by the brooding parasitism of the cuckoo
Cuckoos are birds in the Cuculidae family, the sole taxon in the order Cuculiformes . The cuckoo family includes the common or European cuckoo, roadrunners, koels, malkohas, couas, coucals and anis. The coucals and anis are sometimes separ ...
species. It also uses Lévy flight
A Lévy flight is a random walk in which the step-lengths have a Lévy distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random direct ...
s, and thus it suits for global optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problems.
* Particle swarm optimization is based on the ideas of animal flocking behaviour. Also primarily suited for numerical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problems.
Other population-based metaheuristic methods
* Hunting Search – A method inspired by the group hunting of some animals such as wolves that organize their position to surround the prey, each of them relative to the position of the others and especially that of their leader. It is a continuous optimization method adapted as a combinatorial optimization method.
* Adaptive dimensional search – Unlike nature-inspired metaheuristic techniques, an adaptive dimensional search algorithm does not implement any metaphor as an underlying principle. Rather it uses a simple performance-oriented method, based on the update of the search dimensionality ratio (SDR) parameter at each iteration.
*Firefly algorithm In mathematical optimization, the firefly algorithm is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behavior of fireflies.
Algorithm
In pseudocode the algorithm can be stated as:
Begin
1) Objective function:
...
is inspired by the behavior of fireflies, attracting each other by flashing light. This is especially useful for multimodal optimization.
*Harmony search
This is a chronologically ordered list of metaphor-based metaheuristics and swarm intelligence algorithms, sorted by decade of proposal.
Algorithms
1980s-1990s
Simulated annealing (Kirkpatrick et al., 1983)
Simulated annealing is a pr ...
– Based on the ideas of musicians' behavior in searching for better harmonies. This algorithm is suitable for combinatorial optimization as well as parameter optimization.
* Gaussian adaptation – Based on information theory. Used for maximization of manufacturing yield, mean fitness
Fitness (often denoted w or ω in population genetics models) is the quantitative representation of individual reproductive success. It is also equal to the average contribution to the gene pool of the next generation, made by the same individu ...
or average information
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
. See for instance Entropy in thermodynamics and information theory The mathematical expressions for thermodynamic entropy in the statistical thermodynamics formulation established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s are similar to the information entropy by Claude Shannon and Ralph Hartley, develo ...
.
*Memetic algorithm
A memetic algorithm (MA) in computer science and operations research, is an extension of the traditional genetic algorithm. It may provide a sufficiently good solution to an optimization problem. It uses a local search technique to reduce the like ...
– A hybrid method, inspired by Richard Dawkins
Richard Dawkins (born 26 March 1941) is a British evolutionary biologist and author. He is an emeritus fellow of New College, Oxford and was Professor for Public Understanding of Science in the University of Oxford from 1995 to 2008. An ath ...
's notion of a meme, it commonly takes the form of a population-based algorithm coupled with individual learning procedures capable of performing local refinements. Emphasizes the exploitation of problem-specific knowledge and tries to orchestrate local and global search in a synergistic way.
Examples
In 2020, Google
Google LLC () is an American multinational technology company focusing on search engine technology, online advertising, cloud computing, computer software, quantum computing, e-commerce, artificial intelligence, and consumer electronics. ...
stated that their AutoML-Zero can successfully rediscover classic algorithms such as the concept of neural networks.
The computer simulations ''Tierra
Tierra may refer to:
Astronomy
*Earth in the Spanish and Asturian language
Computing and games
* Tierra (computer simulation), a computer simulation of life by the ecologist Thomas S. Ray
* Tierra Entertainment, now known as AGD Interactive, a ...
'' and ''Avida
Avida is an artificial life software platform to study the evolutionary biology of self-replicating and evolving computer programs ( digital organisms). Avida is under active development by Charles Ofria's Digital Evolution Lab at Michigan Stat ...
'' attempt to model macroevolution
Macroevolution usually means the evolution of large-scale structures and traits that go significantly beyond the intraspecific variation found in microevolution (including speciation). In other words, macroevolution is the evolution of taxa abov ...
ary dynamics.
Gallery
File:Two-population EA search (2).gif, A two-population EA search over a constrained Rosenbrock function
In mathematical optimization, the Rosenbrock function is a non- convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Ros ...
with bounded global optimum
File:Two-population EA search (3).gif, A two-population EA search over a constrained Rosenbrock function
In mathematical optimization, the Rosenbrock function is a non- convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Ros ...
. Global optimum is not bounded.
File:Estimation of Distribution Algorithm animation.gif, Estimation of distribution algorithm
''Estimation of distribution algorithms'' (EDAs), sometimes called ''probabilistic model-building genetic algorithms'' (PMBGAs), are stochastic optimization methods that guide the search for the optimum by building and sampling explicit probabilis ...
over Keane's function
File:Two population EA animation.gif, A two-population EA search of a bounded optima of Simionescu's function
References
External links
An Overview of the History and Flavors of Evolutionary Algorithms
Bibliography
* Ashlock, D. (2006), ''Evolutionary Computation for Modeling and Optimization'', Springer, .
* Bäck, T. (1996),
Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms
', Oxford Univ. Press.
* Bäck, T., Fogel, D., Michalewicz, Z. (1997), ''Handbook of Evolutionary Computation'', Oxford Univ. Press. doi:10.1201/9780367802486
* Banzhaf, W., Nordin, P., Keller, R., Francone, F. (1998), ''Genetic Programming - An Introduction'', Morgan Kaufmann, San Francisco.
* Eiben, A.E., Smith, J.E. (2003), ''Introduction to Evolutionary Computing'', Springer. doi:10.1007/978-3-662-44874-8
* Holland, J. H. (1992),
Adaptation in Natural and Artificial Systems
', The University of Michigan Press, Ann Arbor
* Michalewicz, Z.; Fogel, D.B. (2004), ''How To Solve It: Modern Heuristics''. Springer, Berlin, Heidelberg, , doi:10.1007/978-3-662-07807-5.
*
*
* Price, K., Storn, R.M., Lampinen, J.A., (2005)
''Differential Evolution: A Practical Approach to Global Optimization''
Springer, Berlin, Heidelberg, , doi:10.1007/3-540-31306-0.
* Ingo Rechenberg
Ingo Rechenberg (November 20, 1934 - September 25, 2021) was a German researcher and professor in the field of bionics. Rechenberg was a pioneer of the fields of evolutionary computation and artificial evolution. In the 1960s and 1970s he invente ...
(1971), ''Evolutionsstrategie - Optimierung technischer Systeme nach Prinzipien der biologischen Evolution'' (PhD thesis). Reprinted by Fromman-Holzboog (1973).
* Hans-Paul Schwefel
Hans-Paul Schwefel (born December 4, 1940) is a German computer scientist and professor emeritus at University of Dortmund (now Dortmund University of Technology), where he held the chair of systems analysis from 1985 until 2006. He is one of the ...
(1974), ''Numerische Optimierung von Computer-Modellen'' (PhD thesis). Reprinted by Birkhäuser (1977).
* Hans-Paul Schwefel
Hans-Paul Schwefel (born December 4, 1940) is a German computer scientist and professor emeritus at University of Dortmund (now Dortmund University of Technology), where he held the chair of systems analysis from 1985 until 2006. He is one of the ...
(1995), ''Evolution and Optimum Seeking''. Wiley & Sons, New York.
* Simon, D. (2013)
''Evolutionary Optimization Algorithms''
Wiley.
* Kruse, Rudolf; Borgelt, Christian; Klawonn, Frank; Moewes, Christian; Steinbrecher, Matthias; Held, Pascal (2013)
''Computational Intelligence: A Methodological Introduction''
Springer, London. , doi:10.1007/978-1-4471-5013-8.
*
{{DEFAULTSORT:Evolutionary Algorithm
Cybernetics
Evolution
Optimization algorithms and methods