Category: genetic Algorithms

Genetic Algorithms – Further Readings The following books can be referred to further enhance the reader’s knowledge of Genetic Algorithms, and Evolutionary Computation in general − Genetic Algorithms in Search, Optimization and Machine Learning by David E. Goldberg. Genetic Algorithms + Data Structures = Evolutionary Programs by Zbigniew Michalewicz. Practical Genetic Algorithms by Randy L. Haupt and Sue Ellen Haupt. Multi Objective Optimization using Evolutionary Algorithms by Kalyanmoy Deb. Learning working make money

Effective Implementation GAs are very general in nature, and just applying them to any optimization problem wouldn’t give good results. In this section, we describe a few points which would help and assist a GA designer or GA implementer in their work. Introduce problem-specific domain knowledge It has been observed that the more problem-specific domain knowledge we incorporate into the GA; the better objective values we get. Adding problem specific information can be done by either using problem specific crossover or mutation operators, custom representations, etc. The following image shows Michalewicz’s (1990) view of the EA − Reduce Crowding Crowding happens when a highly fit chromosome gets to reproduce a lot, and in a few generations, the entire population is filled with similar solutions having similar fitness. This reduces diversity which is a very crucial element to ensure the success of a GA. There are numerous ways to limit crowding. Some of them are − Mutation to introduce diversity. Switching to rank selection and tournament selection which have more selection pressure than fitness proportionate selection for individuals with similar fitness. Fitness Sharing − In this an individual’s fitness is reduced if the population already contains similar individuals. Randomization Helps! It has been experimentally observed that the best solutions are driven by randomized chromosomes as they impart diversity to the population. The GA implementer should be careful to keep sufficient amount of randomization and diversity in the population for the best results. Hybridize GA with Local Search Local search refers to checking the solutions in the neighborhood of a given solution to look for better objective values. It may be sometimes useful to hybridize the GA with local search. The following image shows the various places in which local search can be introduced in a GA. Variation of parameters and techniques In genetic algorithms, there is no “one size fits all” or a magic formula which works for all problems. Even after the initial GA is ready, it takes a lot of time and effort to play around with the parameters like population size, mutation and crossover probability etc. to find the ones which suit the particular problem. Learning working make money

Genetic Algorithms – Fitness Function The fitness function simply defined is a function which takes a candidate solution to the problem as input and produces as output how “fit” our how “good” the solution is with respect to the problem in consideration. Calculation of fitness value is done repeatedly in a GA and therefore it should be sufficiently fast. A slow computation of the fitness value can adversely affect a GA and make it exceptionally slow. In most cases the fitness function and the objective function are the same as the objective is to either maximize or minimize the given objective function. However, for more complex problems with multiple objectives and constraints, an Algorithm Designer might choose to have a different fitness function. A fitness function should possess the following characteristics − The fitness function should be sufficiently fast to compute. It must quantitatively measure how fit a given solution is or how fit individuals can be produced from the given solution. In some cases, calculating the fitness function directly might not be possible due to the inherent complexities of the problem at hand. In such cases, we do fitness approximation to suit our needs. The following image shows the fitness calculation for a solution of the 0/1 Knapsack. It is a simple fitness function which just sums the profit values of the items being picked (which have a 1), scanning the elements from left to right till the knapsack is full. Learning working make money

Genetic Algorithms – Termination Condition The termination condition of a Genetic Algorithm is important in determining when a GA run will end. It has been observed that initially, the GA progresses very fast with better solutions coming in every few iterations, but this tends to saturate in the later stages where the improvements are very small. We usually want a termination condition such that our solution is close to the optimal, at the end of the run. Usually, we keep one of the following termination conditions − When there has been no improvement in the population for X iterations. When we reach an absolute number of generations. When the objective function value has reached a certain pre-defined value. For example, in a genetic algorithm we keep a counter which keeps track of the generations for which there has been no improvement in the population. Initially, we set this counter to zero. Each time we don’t generate off-springs which are better than the individuals in the population, we increment the counter. However, if the fitness any of the off-springs is better, then we reset the counter to zero. The algorithm terminates when the counter reaches a predetermined value. Like other parameters of a GA, the termination condition is also highly problem specific and the GA designer should try out various options to see what suits his particular problem the best. Learning working make money

Genetic Algorithms – Crossover In this chapter, we will discuss about what a Crossover Operator is along with its other modules, their uses and benefits. Introduction to Crossover The crossover operator is analogous to reproduction and biological crossover. In this more than one parent is selected and one or more off-springs are produced using the genetic material of the parents. Crossover is usually applied in a GA with a high probability – pc . Crossover Operators In this section we will discuss some of the most popularly used crossover operators. It is to be noted that these crossover operators are very generic and the GA Designer might choose to implement a problem-specific crossover operator as well. One Point Crossover In this one-point crossover, a random crossover point is selected and the tails of its two parents are swapped to get new off-springs. Multi Point Crossover Multi point crossover is a generalization of the one-point crossover wherein alternating segments are swapped to get new off-springs. Uniform Crossover In a uniform crossover, we don’t divide the chromosome into segments, rather we treat each gene separately. In this, we essentially flip a coin for each chromosome to decide whether or not it’ll be included in the off-spring. We can also bias the coin to one parent, to have more genetic material in the child from that parent. Whole Arithmetic Recombination This is commonly used for integer representations and works by taking the weighted average of the two parents by using the following formulae − Child1 = α.x + (1-α).y Child2 = α.x + (1-α).y Obviously, if α = 0.5, then both the children will be identical as shown in the following image. Davis’ Order Crossover (OX1) OX1 is used for permutation based crossovers with the intention of transmitting information about relative ordering to the off-springs. It works as follows − Create two random crossover points in the parent and copy the segment between them from the first parent to the first offspring. Now, starting from the second crossover point in the second parent, copy the remaining unused numbers from the second parent to the first child, wrapping around the list. Repeat for the second child with the parent’s role reversed. There exist a lot of other crossovers like Partially Mapped Crossover (PMX), Order based crossover (OX2), Shuffle Crossover, Ring Crossover, etc. Learning working make money

Genetic Algorithms – Parent Selection Parent Selection is the process of selecting parents which mate and recombine to create off-springs for the next generation. Parent selection is very crucial to the convergence rate of the GA as good parents drive individuals to a better and fitter solutions. However, care should be taken to prevent one extremely fit solution from taking over the entire population in a few generations, as this leads to the solutions being close to one another in the solution space thereby leading to a loss of diversity. Maintaining good diversity in the population is extremely crucial for the success of a GA. This taking up of the entire population by one extremely fit solution is known as premature convergence and is an undesirable condition in a GA. Fitness Proportionate Selection Fitness Proportionate Selection is one of the most popular ways of parent selection. In this every individual can become a parent with a probability which is proportional to its fitness. Therefore, fitter individuals have a higher chance of mating and propagating their features to the next generation. Therefore, such a selection strategy applies a selection pressure to the more fit individuals in the population, evolving better individuals over time. Consider a circular wheel. The wheel is divided into n pies, where n is the number of individuals in the population. Each individual gets a portion of the circle which is proportional to its fitness value. Two implementations of fitness proportionate selection are possible − Roulette Wheel Selection In a roulette wheel selection, the circular wheel is divided as described before. A fixed point is chosen on the wheel circumference as shown and the wheel is rotated. The region of the wheel which comes in front of the fixed point is chosen as the parent. For the second parent, the same process is repeated. It is clear that a fitter individual has a greater pie on the wheel and therefore a greater chance of landing in front of the fixed point when the wheel is rotated. Therefore, the probability of choosing an individual depends directly on its fitness. Implementation wise, we use the following steps − Calculate S = the sum of a finesses. Generate a random number between 0 and S. Starting from the top of the population, keep adding the finesses to the partial sum P, till P<S. The individual for which P exceeds S is the chosen individual. Stochastic Universal Sampling (SUS) Stochastic Universal Sampling is quite similar to Roulette wheel selection, however instead of having just one fixed point, we have multiple fixed points as shown in the following image. Therefore, all the parents are chosen in just one spin of the wheel. Also, such a setup encourages the highly fit individuals to be chosen at least once. It is to be noted that fitness proportionate selection methods don’t work for cases where the fitness can take a negative value. Tournament Selection In K-Way tournament selection, we select K individuals from the population at random and select the best out of these to become a parent. The same process is repeated for selecting the next parent. Tournament Selection is also extremely popular in literature as it can even work with negative fitness values. Rank Selection Rank Selection also works with negative fitness values and is mostly used when the individuals in the population have very close fitness values (this happens usually at the end of the run). This leads to each individual having an almost equal share of the pie (like in case of fitness proportionate selection) as shown in the following image and hence each individual no matter how fit relative to each other has an approximately same probability of getting selected as a parent. This in turn leads to a loss in the selection pressure towards fitter individuals, making the GA to make poor parent selections in such situations. In this, we remove the concept of a fitness value while selecting a parent. However, every individual in the population is ranked according to their fitness. The selection of the parents depends on the rank of each individual and not the fitness. The higher ranked individuals are preferred more than the lower ranked ones. Chromosome Fitness Value Rank A 8.1 1 B 8.0 4 C 8.05 2 D 7.95 6 E 8.02 3 F 7.99 5 Random Selection In this strategy we randomly select parents from the existing population. There is no selection pressure towards fitter individuals and therefore this strategy is usually avoided. Learning working make money

Genetic Algorithms – Introduction Genetic Algorithm (GA) is a search-based optimization technique based on the principles of Genetics and Natural Selection. It is frequently used to find optimal or near-optimal solutions to difficult problems which otherwise would take a lifetime to solve. It is frequently used to solve optimization problems, in research, and in machine learning. Introduction to Optimization Optimization is the process of making something better. In any process, we have a set of inputs and a set of outputs as shown in the following figure. Optimization refers to finding the values of inputs in such a way that we get the “best” output values. The definition of “best” varies from problem to problem, but in mathematical terms, it refers to maximizing or minimizing one or more objective functions, by varying the input parameters. The set of all possible solutions or values which the inputs can take make up the search space. In this search space, lies a point or a set of points which gives the optimal solution. The aim of optimization is to find that point or set of points in the search space. What are Genetic Algorithms? Nature has always been a great source of inspiration to all mankind. Genetic Algorithms (GAs) are search based algorithms based on the concepts of natural selection and genetics. GAs are a subset of a much larger branch of computation known as Evolutionary Computation. GAs were developed by John Holland and his students and colleagues at the University of Michigan, most notably David E. Goldberg and has since been tried on various optimization problems with a high degree of success. In GAs, we have a pool or a population of possible solutions to the given problem. These solutions then undergo recombination and mutation (like in natural genetics), producing new children, and the process is repeated over various generations. Each individual (or candidate solution) is assigned a fitness value (based on its objective function value) and the fitter individuals are given a higher chance to mate and yield more “fitter” individuals. This is in line with the Darwinian Theory of “Survival of the Fittest”. In this way we keep “evolving” better individuals or solutions over generations, till we reach a stopping criterion. Genetic Algorithms are sufficiently randomized in nature, but they perform much better than random local search (in which we just try various random solutions, keeping track of the best so far), as they exploit historical information as well. Advantages of GAs GAs have various advantages which have made them immensely popular. These include − Does not require any derivative information (which may not be available for many real-world problems). Is faster and more efficient as compared to the traditional methods. Has very good parallel capabilities. Optimizes both continuous and discrete functions and also multi-objective problems. Provides a list of “good” solutions and not just a single solution. Always gets an answer to the problem, which gets better over the time. Useful when the search space is very large and there are a large number of parameters involved. Limitations of GAs Like any technique, GAs also suffer from a few limitations. These include − GAs are not suited for all problems, especially problems which are simple and for which derivative information is available. Fitness value is calculated repeatedly which might be computationally expensive for some problems. Being stochastic, there are no guarantees on the optimality or the quality of the solution. If not implemented properly, the GA may not converge to the optimal solution. GA – Motivation Genetic Algorithms have the ability to deliver a “good-enough” solution “fast-enough”. This makes genetic algorithms attractive for use in solving optimization problems. The reasons why GAs are needed are as follows − Solving Difficult Problems In computer science, there is a large set of problems, which are NP-Hard. What this essentially means is that, even the most powerful computing systems take a very long time (even years!) to solve that problem. In such a scenario, GAs prove to be an efficient tool to provide usable near-optimal solutions in a short amount of time. Failure of Gradient Based Methods Traditional calculus based methods work by starting at a random point and by moving in the direction of the gradient, till we reach the top of the hill. This technique is efficient and works very well for single-peaked objective functions like the cost function in linear regression. But, in most real-world situations, we have a very complex problem called as landscapes, which are made of many peaks and many valleys, which causes such methods to fail, as they suffer from an inherent tendency of getting stuck at the local optima as shown in the following figure. Getting a Good Solution Fast Some difficult problems like the Travelling Salesperson Problem (TSP), have real-world applications like path finding and VLSI Design. Now imagine that you are using your GPS Navigation system, and it takes a few minutes (or even a few hours) to compute the “optimal” path from the source to destination. Delay in such real world applications is not acceptable and therefore a “good-enough” solution, which is delivered “fast” is what is required. Learning working make money

Genetic Algorithms – Population Population is a subset of solutions in the current generation. It can also be defined as a set of chromosomes. There are several things to be kept in mind when dealing with GA population − The diversity of the population should be maintained otherwise it might lead to premature convergence. The population size should not be kept very large as it can cause a GA to slow down, while a smaller population might not be enough for a good mating pool. Therefore, an optimal population size needs to be decided by trial and error. The population is usually defined as a two dimensional array of – size population, size x, chromosome size. Population Initialization There are two primary methods to initialize a population in a GA. They are − Random Initialization − Populate the initial population with completely random solutions. Heuristic initialization − Populate the initial population using a known heuristic for the problem. It has been observed that the entire population should not be initialized using a heuristic, as it can result in the population having similar solutions and very little diversity. It has been experimentally observed that the random solutions are the ones to drive the population to optimality. Therefore, with heuristic initialization, we just seed the population with a couple of good solutions, filling up the rest with random solutions rather than filling the entire population with heuristic based solutions. It has also been observed that heuristic initialization in some cases, only effects the initial fitness of the population, but in the end, it is the diversity of the solutions which lead to optimality. Population Models There are two population models widely in use − Steady State In steady state GA, we generate one or two off-springs in each iteration and they replace one or two individuals from the population. A steady state GA is also known as Incremental GA. Generational In a generational model, we generate ‘n’ off-springs, where n is the population size, and the entire population is replaced by the new one at the end of the iteration. Learning working make money