JPH08314883A - Optimization problem solving method and optimization device - Google Patents

Abstract

PURPOSE: To provide an optimization problem solving method and an optimization device capable of shortening the calculating time for finding a practical solution through the use of a genetic algorithm. CONSTITUTION: When a problem 1 of optimization is given, an optimizing article and a restraint condition are converted into the chromosome 2 to be used in a genetic algorithm 3. Next, the search of an optimum solution is performed by a geographical isolation model 3a and/or an environmental variation model 3b. In the geographical isolation model 3a, the whole group is divided into plural slave groups A, B,... and the genetic algorithm is applied to each slave group. The exchange of chromosome is permitted among slave groups and the excellent solution generated in a certain slave group can be effectively utilized for even other slave groups. In the environmental variation model 3a, a genetic operation is repeated till the solution satisfying a condition can be obtained while varying some or all of the parameters of the scaling of adaptability, selection technique and cross technique, etc., to find an optimal solution.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an optimization problem solving method and an optimization apparatus for finding a solution that minimizes (or maximizes) an objective function under given constraints. The present invention relates to a method for solving an optimization problem and an optimization device for obtaining an optimum solution using an algorithm.

[0002] A genetic algorithm (Geneti) which is attracting attention as a probabilistic search method and a new learning method.
c Algorithms (GA) is an engineered evolutionary model that simulates the evolutionary process of living organisms over many years.
According to the Darwinism evolution theory, which was derived from Charles Darwin, organisms have survived by being selected through natural selection by individuals who are better adapted to the environment due to changes in the environment surrounding them and interactions with other individuals in the population. Then, the individuals have been mated with each other to transmit superior traits to the next generation, and the mutations have been repeated to obtain new traits, resulting in adaptive evolution.

A genetic algorithm (hereinafter abbreviated as GA) repeatedly applies a genetic operation such as crossover, mutation, and selection to a population of individuals, so that individuals with higher adaptability to the environment may end up in competition for survival. It is a very simple one that utilizes the mechanism of biological evolution that survives as a result. The history of GA is surprisingly old and its origin is, of course, Darwinian evolution theory, and goes back to Charles Darwin's "origin of species." Holland used the findings from population genetics and cybernetics to 1975 “Adaptation in Natu
"Ral and Artificial Systems", and proposed a framework of adaptive system theory based on the genetics of living organisms and natural selection. In this book, Holland's "Basic Theorem of GA" is the source of current GA research.

The 1st International Conference on Genetic Algorithm in 1995
lgorithms (ICGA) was held at Carnegie Mellon University in the United States, and research into engineering applications of GA has been in full swing. At present, many studies aimed at practical application have been published, and there are also cases in which they are actually applied to industry, focusing on optimization problems such as gas pipeline control and jet engine design.

[0005]

2. Description of the Related Art A conventional GA is, for example, "DE Goldberg,
Genetic Algorithms in Search, Optimization and Mach
9 is a schematic drawing of a genetic algorithm. As shown in the figure, there are multiple individuals that are solution candidates in a population in a certain generation (k). The population of the next generation (k + 1) is obtained by performing genetic operations such as crossover, mutation, and selection on these individuals.By repeating this process, the fitness of the individuals in the population gradually increases. , And eventually reach the optimal solution.

GA basically performs the following processing using three types of genetic operations (selection, crossover, and mutation). 1. Generation of initial population An initial population consisting of multiple individuals having random genes is generated. 2. The following processing is repeated until the end condition is satisfied. (a) Selection: Select multiple sets of individuals for crossover. (b) Crossover: By exchanging a part of the individual, a new individual as a descendant is generated. (c) Mutation: A part of the offspring is replaced or mutated.

The conventional technique was to repeatedly perform the above-mentioned three gene manipulations until the termination condition was satisfied.

[0008]

Here, the GA is considered as an optimization problem solving means. The optimization problem is defined by the following (1) under the given constraint condition shown in the following (2).
The objective is to find a solution that minimizes (or maximizes) a certain objective function, and is given as follows. min f (x) or max f (x) (1) subject to g _i (x) ≦ 0 (i = 1, 2, ..., m) (2) In the optimization problem, x seems to be combinatorial The case is called a combinatorial optimization problem. In many combinatorial optimization problems, in order to obtain the optimum solution, it is necessary to enumerate all the solution candidates and find a solution that minimizes (or maximizes) the objective function among the solutions satisfying the constraints.

However, even if it is known that an optimal solution can be theoretically obtained by the method of all-retrieval search, the number of solution candidates exponentially increases as the problem scale increases, and even if it is known that an optimal solution can be theoretically obtained. It is impossible to execute in a short time. There are a heuristic search method and a probabilistic search method as approaches to find a suboptimal solution of a combinatorial optimization problem in a practical time.

The basic framework of GA can be regarded as a probabilistic search method. Therefore, Simulated
Like the conventional probabilistic search methods such as Annealing and Hopfield Model, it is a method that does not depend on the problem structure and can be easily applied to combinatorial optimization problems. Some probabilistic search methods guarantee convergence to the optimal solution. However, the probabilistic search method has a feature that the system development time is short and it can be used easily, but there is a problem that a huge amount of calculation time is required to obtain a suboptimal solution.

As described above, the enormous calculation time is required as the scale of the problem increases, which has been a major problem in applying GA to practical problems. The present invention has been made in view of the above-mentioned problems of GA, and an object of the present invention is to provide an optimization problem solving method and an optimization device capable of shortening the calculation time to reach a practical solution. It is to be.

[0012]

FIG. 1 is a diagram illustrating the principle of the present invention. In the figure, 1 is a given optimization problem consisting of an objective function and constraint conditions, 2 is a chromosome encoding the optimization problem, and 3 is a genetic algorithm. In order to solve the above-mentioned problems, the invention of claim 1 of the present invention is an optimization problem solving method using a genetic algorithm, which comprises evolving an individual population by selecting conditions that promote genetic changes. , The time for obtaining the optimal solution is shortened.

As shown in FIG. 1, the invention of claim 2 of the present invention comprises means for generating an initial population consisting of a plurality of individuals having random genes, and a solution satisfying the conditions for the population. In an optimizing device equipped with means for repeatedly performing genetic operations to obtain an optimal solution, the above-mentioned group is divided into a plurality of sub-populations, and the exchange of chromosomes between the sub-populations is allowed while the conditions are set in each sub-population. It is configured so that an optimal solution is obtained by repeating genetic operations until a satisfying solution is obtained.

The invention according to claim 3 of the present invention comprises means for generating an initial population consisting of a plurality of individuals having random genes, and an optimal solution for said population by repeating genetic operations until a solution satisfying the conditions is obtained. In the optimizing device including means for obtaining the condition, a means for varying the parameter that affects the state of the population in which the individual is placed, the state of the individual itself is provided, and while varying the parameter by the above means, the condition is satisfied. It is configured to repeat the genetic manipulation until a solution is obtained to obtain an optimal solution.

According to a fourth aspect of the present invention, means for generating an initial population consisting of a plurality of individuals having random genes, and genetic operations are repeated for the population until an optimal condition is obtained. And a means for varying the parameters that affect the state of the population in which the individual is placed, the state of the individual itself, and the above-mentioned population is divided into a plurality of child populations. While allowing the exchange of chromosomes between populations, the parameter is varied by the above means and the genetic operation is repeated until a solution satisfying the condition is obtained in each offspring population to obtain an optimal solution.

[0016]

[Operation] The present invention is a law of evolution of nature, Hardy
It is an engineering application of Weinberg's law. According to Hardy-Weinberg's law, a change in gene frequency does not occur when the following four conditions are met. Arbitrary mating. The population is large enough. Natural selection does not work. Mutations do not occur in the genes of individuals in the population.

That is, if a state contrary to the Hardy-Weinberg law is artificially generated in the GA simulation, as a result, the evolution is accelerated and the result can be obtained in a shorter time. . Arbitrary mating refers to the case where the mating rate between all two individuals in the population is equal, and the representative of an example in which arbitrary mating is not performed is inbreeding. In inbreeding, the frequency of homozygotes increases at all loci by mating individuals with genes from a common ancestor. Further, in the case of mating between individuals having similar phenotypic traits or mating between individuals having different phenotypic traits, the frequency of homozygotes increases or decreases at the locus expressing the trait.

The number of individuals in a population is generally finite. Therefore, the behavior of the group cannot be treated deterministically and must be considered as stochastic behavior. The stochastic behavior of a population that arises from the finite size of the population is called the size effect, and when considering the evolution of a finite population of organisms, changes in gene frequency due to the stochastic size effect are important. As the size of the group becomes smaller, as a result, it becomes difficult to carry out arbitrary crossing, so there is a deep relationship between size effect and arbitrary crossing. The change in gene frequency due to the difficulty of random mating due to the lack of random mating or the small size of the population is called genetic drift, and it is an evolutionary phenomenon that happens by chance regardless of natural selection.

Natural selection occurs whenever there is a difference in fitness between individuals within a population. The fitness of an individual is understood as the number of offspring that the individual retains in the next generation, that is, the reproductive rate, and is one of the traits determined by the genotype of the parent individual.
However, fitness is determined not only by such genetic factors, but also by both genetic factors and environmental factors that are greatly influenced by the influence of genotype at another locus and the surrounding environment.

What is the case when the fitness of an individual changes? When a mutation occurs in an individual due to mutation or genetic recombination (crossover), the fitness also changes, but the change has a greater effect on the environment. In the natural world, the environment surrounding an individual is rarely constant, and the environment is constantly changing. In a changing environment, even if no mutation occurs in an individual, the fitness changes and some genotypes may be advantageous or, conversely, disadvantageous.

Mutations in the genes of individuals in a population are often caused by mutation and genetic recombination (crossover).
Mutation means that when the gene of the parent is replicated and transmitted to the offspring, complete replication does not occur, and a replication error occurs according to a value called the mutation rate. Mutations that occur in an individual in a population are determined by natural selection to either disappear from the population or be spread and fixed throughout the population. If the mutation produced in the individual is favorable to survival, the probability of being fixed is high, and conversely, if the mutation is one that is disadvantageous to the survival of the individual, it is hardly fixed.

Genetic recombination (crossover) is different from mutation causing an entirely new allele in the gene,
Recombination between loci creates new gene combinations. Apart from recombination that occurs between loci, recombination that occurs within the same gene is called intragenic recombination and can give rise to new alleles similar to mutations.

[0023] Usually, mutations do not cause a great change in fitness for the individual. Most mutations are detrimental to the individual and are quickly eliminated from the population by natural selection. Even if an advantageous mutation occurs, it takes a very long time to spread and be fixed in the population. On the other hand, genetic recombination, when the fitness increases or decreases, is extremely large and it is considered that it plays a major role in evolution.

As described above, according to Hardy-Weinberg's law, the main factor of evolution is determined by the differences among individuals caused by mutation and genetic recombination that spread within the population due to size effect or natural selection. It is considered to have been done. The present invention is based on the above-mentioned Hardy
By artificially producing a state that goes against Weinberg's law, it is possible to obtain an optimal solution in a shorter time by accelerating the evolution, and particularly in the present invention,
As a case where the above size effect more affects the change of genetic frequency, the genetic algorithm is speeded up by the geographical isolation model and the genetic algorithm is speeded up by the environmental change model.

That is, as shown in FIG. 1, first, when an optimization problem 1 is given, the optimization terms and constraints are converted into the chromosome 2 used in the genetic algorithm. Next, the genetic algorithm 3 is used to search for an optimal solution. Here, the geographical algorithm 3a and / or the environmental variation model 3b described below are used to speed up the genetic algorithm. (1) Geographical isolation model Although a normal genetic algorithm tries to approach a more optimal solution by passing through the generations in one population,
In the geographical isolation model 3a according to the present invention, the entire population is divided into a plurality of child populations A, B, ... And the genetic algorithm is applied to each. In addition, the subgroups are allowed to exchange chromosomes with each other so that the excellent solution generated in another subgroup can be effectively used in another subgroup.

As a result, the size effect makes it possible to accelerate evolution and obtain an optimum solution in a short time. (2) Environmental variation model The "environment" in the genetic algorithm is the state of the population in which the solution candidates are placed and the state of the individuals themselves. A genetic algorithm, as the name implies, generally refers to a means for solving a problem, and its realization method is usually different for each user.

Therefore, the "environment" described here is not a defined one, but is only a commonly used one, and all the parameters used in the realization of a genetic algorithm are called "environment". it can. The following are the "Simple Genetic Algorithms" proposed by DE Goldberg, one of the proponents of genetic algorithms.
(SGA) ". Number of individuals in the population Fitness scaling selection method Crossover method Mutation method Generation model In the environmental variation model 3b of the present invention, some of the above parameters, or While varying everything, the genetic operation is repeated until a solution satisfying the conditions as shown in Fig. 1 is obtained, and the effective time of reaching the optimal solution is shortened by making the most effective use of natural selection. be able to.

[0028]

EXAMPLES Next, examples of the present invention will be described. (1) Geographical isolation model FIG. 2 is a diagram showing a geographical isolation model 3a according to an embodiment of the present invention. In the present embodiment, a population is divided into two child populations A and B, and each is genetically divided. The geographical isolation model which applied the algorithm is shown.

In the figure, first, a group of chromosomes that are candidates for a solution is generated in the groups A and B. Then, group A,
In each of B, a genetic operation such as crossover, mutation, and selection is performed to determine whether or not a solution satisfying the condition is obtained, and the genetic operation is repeated until a solution satisfying the condition is obtained. In addition, chromosomes are exchanged between child groups so that excellent solutions generated in other child groups can be effectively used in other child groups.

By dividing the population into child populations and applying the genetic algorithm to each population as described above, the size of the population can be reduced, and the speed of evolution can be increased as described above. (2) Environmental Change Model FIG. 3 is a diagram showing an environmental change model according to the embodiment of the present invention.

In the environment variation model 3b, as shown in the figure, first, a group of chromosomes that are solution candidates is generated, and genetic operations such as crossover, mutation, and selection are performed to obtain a solution satisfying the conditions. It is determined whether or not it has been performed, and the above genetic operation is repeated until a solution satisfying the conditions is obtained. that time,
Some or all of the above-mentioned parameters such as crossover method, mutation method, mutation rate, fitness scaling, and generation model are changed. As a result, the time required to reach the optimum solution can be shortened.

The environment variation model passes through generations while varying some or all of the parameters (1) to (3) as described above. The parameter variation from the above will be described in detail below. (a) Number of individuals in a population As described above, the genetic algorithm is a probabilistic method in which candidate solutions find the optimal solution in a population.

Here, what is different from the simple mountain climbing method is that not everyone in the group climbs toward the higher mountains around them, but rather the group climbs higher mountains.
In other words, some individuals may go in completely different directions due to mutations, etc., but this may eventually lead to the discovery of higher mountains, and escape from the local optimal solution. To help.

What influence does the number of individuals in the population have? Consider a very extreme case where there is only one individual. The quality of the solution at this time is greatly affected by stochastic parameters such as mutation rate and crossover probability, and the search strengthens the random search aspect. On the contrary, when the number of individuals in the population is very large, stochastic fluctuations that help escape from the local optimal solution cannot be expected, and even if the solution falls to the local optimal solution or the solution is obtained. It will take a very long time. (b) Scaling of fitness The fitness when the genetic algorithm is considered as a solution to the optimization problem is the value that indicates the competitiveness of each individual in the environment, that is, the solution value itself. Is. Now,
Even if the fitness is determined from the optimization problem, this value need not be reflected as it is in the probability at the time of selection, and some function may be introduced to expand or reduce the difference in fitness.

Introducing such a function is called scaling, and by varying the scaling, individuals with low fitness are more likely to survive in the next generation, which is also effective for escape from the local optimum solution. There is. The following table shows typical scaling methods. In the table below, f is the fitness function, a, b, c, k are constants, and σ
Is the variance and f above the bar is the average value of f.

[0036]

[Table 1]

(C) Selection method In selection, the selection pressure is changed. At this time, the important thing is
It depends on which individuals are mated. The following method has been proposed as a method for this purpose. Fitness proportional strategy It is also called a roulette model or Monte Carlo model, and it is a model that can leave offspring with a probability proportional to the fitness of each individual.

In this strategy, the probability p that an individual i is selected by each selection is expressed by the following equation (3). That is, the probability p i that an individual can leave offspring is a value obtained by dividing the fitness f i of the individual by the sum of the fitness of all the individuals.

[0039]

[Equation 1]

In this strategy, individuals having a high selection probability participate in a plurality of matings, so that the gene spreads throughout the population. Expected value strategy In the expected value strategy, the expected value (fi x p
i) is calculated. Then, when the individual is selected, 0.5 is subtracted from the expected value. As a result, even in the worst case, it is possible to leave offspring with a deviation of 0.5 from the expected value. Elite Conservation Strategy The Elite Conservation Strategy is a method of leaving the individual with the highest fitness in the population as it is to the next generation. This method has the advantage that the best solution at that time is not destroyed by crossover or mutation. (d) Crossover method Crossover is an operation in which two parent chromosomes are recombined to create a child chromosome, and the following crossover method can be adopted. Simple crossover This is the simplest crossover method, in which one crossover position is decided and the parental genotype is inherited before or after it. Multipoint crossover Multipoint crossover is a method in which there are multiple crossover positions. For example, if the crossover positions are 2 and 5, one of the new individuals is individual A.
Genes are created from the first to the second, the third to the fifth of the individual B, and the sixth to the last of the individual A.

At the same time, a gene of another new individual is created by the reverse combination. (e) Mutation method Mutation is an operation that changes a gene with a certain probability. If mutation is set to a too large mutation probability, the aspect of random search becomes strong, and on the contrary, if there is no mutation, it is not possible to search the space other than the combination of initial genes, and therefore the solution to be sought. There is a limit to the quality of. (f) Generation model In general, a simple GA adopts a discrete generation model in which all individuals simultaneously make descendants and make a next-generation set, but, conversely, it is based on a continuous generation model. There is also. In the continuous generation model, a parameter called the generation gap indicates how much of the current generation is replaced.
2 if the generation gap is 0.2 and the size of the group is 100
0 individuals will take turns. (3) Application examples of the present invention and evaluation of effectiveness of the present invention The effectiveness of the genetic algorithm based on the above-mentioned geographical isolation model and environment variation model was evaluated by a four-color painting problem. The four-color painting problem is a problem in which areas divided on a plane are painted by using only four colors. However, there is a restriction that adjacent sections cannot be painted in the same color.

Here, in addition to the optimization purpose of equalizing the areas of the sections painted in each color, the problem was formulated as in the following equations (4) and (5) and the genetic algorithm was applied.

[0043]

[Equation 2]

However, Sred, Sgreen, Sblue, Syellow: Sum of areas of red, green, blue, and yellow areas V (Sred, Sgreen, Sblue, Syellow): Variance of sum of areas of each color d _{i, j} : 1 when city i and city j are adjacent to each other in the N × N matrix, and 0 when they are not adjacent c _{i, j} : City i and city j are painted in the same color in the N × N matrix 1 when it is painted, 0 when it is painted in a different color The problem described above is shown in FIG.
= 55) and simulated as follows. (a) Encoding The genotype was encoded as shown in the following formulas (6) and (7).

[0045]

(Equation 3)

Where X is the total population of individuals and p is the collection
Represents the number of individuals in the group. x, which is the kth individual^kIs the length
Has a character string consisting of N (= 55) as a chromosome. Dyeing
Color x^kL (lowercase letter L) element x in^k _lIs
Take the value of {1, 2, 3, 4}, and fill each area with each value.
It corresponds to red, green, blue, and yellow which are the colors to divide. One
Mari, x^kElement x at^k _lTo each city in Tokyo
Therefore, the element x^k _lDepending on the value {1, 2, 3, 4} taken by
Then, the color of the different colors is determined. (b) Fitness function Fitness is a value that indicates the competitiveness of each individual in the environment.
It In this embodiment, the fitness function shown in the following equation (8) is
Was used to evaluate the individual.

However, α and β are constants between 0 and 1.

[0048]

[Equation 4]

(C) Crossover As shown in the following equations (9) to (12), two parent individuals were cut at the same position, and a one-point crossover was performed to connect the first half part and the second half part.

[0050]

(Equation 5)

(D) Mutation Mutation was carried out by selecting a certain individual in the population as shown in the following equations (13) and (14), and carrying out a single point mutation for changing one chromosomal element.

[0052]

(Equation 6)

(E) Selection The fitness function is not scaled, and the fitness proportional strategy according to the following equation (15) is adopted. It was decided whether to leave each individual in the next generation with the probability that p of the formula (15) was followed.

[0054]

(Equation 7)

(F) Simulation Results FIG. 5 shows the simulation results by the conventional exhaustive search method, in which the horizontal axis represents the number of generations and the vertical axis represents the fitness. Figure 6
Is a simulation result when the geographical isolation model shown in FIG. 2 is used. In the figure, as in FIG. 5, the horizontal axis represents the number of generations and the vertical axis represents the fitness.

FIG. 7 shows a simulation result when the environment variation model shown in FIG. 3 is used. In the figure, as in FIG. 5, the horizontal axis represents the number of generations and the vertical axis represents the fitness. FIG. 8 shows simulation results when both the geographical isolation model and the environmental change model are used at the same time. In the figure, the horizontal axis represents the number of generations, as in FIG.
The vertical axis is the fitness.

As is apparent from the figure, in the conventional method shown in FIG. 5, the solution having the fitness of about 0.4 is obtained by repeating the generation for about 2000 generations. When the geographical isolation model and environmental change model of 7 are used, the fitness is 1.0 by repeating about 400 generations.
The optimal solution of can be obtained. Further, when the geographical isolation model and the environment variation model are used at the same time, as shown in FIG. 8, an optimal solution with a fitness of 1.0 can be obtained by repeating 200 to 300 generations.

[0058]

As described above, according to the present invention, when solving the optimization problem using the genetic algorithm, the geographical isolation model and / or the environmental change model are used to accelerate the evolution. You can find a practical solution in time.

[Brief description of drawings]

FIG. 1 is a principle diagram of the present invention.

FIG. 2 is a diagram showing an embodiment of the present invention using a geographical isolation model.

FIG. 3 is a diagram showing an embodiment of the present invention using an environment variation model.

FIG. 4 is a diagram showing a map of Tokyo, which is a target of the simulation of the present invention.

FIG. 5 is a diagram showing results obtained by a conventional method.

FIG. 6 is a diagram showing a result of a geographical environment model.

FIG. 7 is a diagram showing a result of an environment variation model.

FIG. 8 is a diagram showing a result of a geographical environment model + environmental change model.

FIG. 9 is a diagram showing an outline of a genetic algorithm.

[Explanation of symbols]

 1 Given optimization problem 2 Coded chromosome 3 Genetic algorithm 3a Geographic isolation model 3b Environmental variation model

Claims (4)

[Claims] 1. A method for solving an optimization problem using a genetic algorithm, characterized in that the time for obtaining an optimal solution is shortened by evolving an individual population by selecting conditions that promote genetic changes. And the optimization problem solving method. 2. An optimum method comprising means for generating an initial population composed of a plurality of individuals having random genes, and means for obtaining an optimal solution by repeating genetic operations for the population until a solution satisfying a condition is obtained. In the optimization device, the population is divided into a plurality of subpopulations, and while allowing the exchange of chromosomes between the subpopulations, the genetic operation is repeated until a solution satisfying the condition is obtained in each subpopulation to obtain an optimal solution. An optimizing device characterized by the above. 3. An optimal method comprising means for generating an initial population consisting of a plurality of individuals having random genes, and means for obtaining an optimal solution by repeating genetic operations for the population until a solution satisfying a condition is obtained. In the optimization apparatus, a means for varying parameters that affect the state of the population in which the individuals are placed and the state of the individuals themselves is provided, and while varying the parameters by the above means, genetic manipulation is performed until a solution satisfying the conditions is obtained. An optimizing device characterized by repeatedly obtaining an optimal solution. 4. An optimum method comprising means for generating an initial population consisting of a plurality of individuals having random genes, and means for repeating the genetic operation for the population to obtain an optimal solution until a solution satisfying the conditions is obtained. In the apparatus, a means for varying the state of the population in which the individual is placed and the parameters that affect the state of the individual itself is provided, and the population is divided into multiple subpopulations, and the chromosomes are exchanged between the subpopulations. An optimizing apparatus characterized in that the parameter is varied by the above means while allowing, and the genetic operation is repeated until a solution satisfying a condition is obtained in each child group to obtain an optimal solution.