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CHAPTER 9 E VOLUTIONARY C OMPUTATION I : G ENETIC A LGORITHMS

Slides for Introduction to Stochastic Search and Optimization ( ISSO ) by J. C. Spall. CHAPTER 9 E VOLUTIONARY C OMPUTATION I : G ENETIC A LGORITHMS. Organization of chapter in ISSO Introduction and history Coding of  Standard GA operations Steps of basic GA algorithm

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CHAPTER 9 E VOLUTIONARY C OMPUTATION I : G ENETIC A LGORITHMS

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  1. Slides for Introduction to Stochastic Search and Optimization (ISSO)by J. C. Spall CHAPTER 9EVOLUTIONARYCOMPUTATIONI: GENETIC ALGORITHMS Organization of chapter in ISSO Introduction and history Coding of  Standard GA operations Steps of basic GA algorithm Extensions to basic GA algorithm Numerical examples

  2. Background: Evolutionary Computation • Evolutionary computation (EC) methods are based on population of solutions • Each iteration involves propagating all elements of the population • Each member of population (“chromosome”) corresponds to one value of  • Genetic algorithms (GAs) are most popular form of EC • Early work in 1950s and 1960s; influential 1975 book by John Holland laid foundation for modern implementations • Dramatic increase in activity in mid-1980s • Population-based structure well suited to parallel processing • But infeasible in some real-time applications

  3. Prototype EC Method Initial Population Selection NextIteration (Generation) Reproduction Mutation Background: EC (cont’d) • Motivation for EC: Evolution seems to work well in nature.…perhaps it can be used in optimization • Three main types of EC • Genetic Algorithms (Chap. 9 of ISSO) • Evolution Strategies (Chap. 10) • Evolutionary Programming (Chap. 10) • Many other types of EC exist (ant colony, particle swarm, differential evolution, etc.)

  4. Desirable Performance for GA with Population of 12 Candidate Solutions 9-4

  5. Role of GAs in Global Problems • GA largely motivated for global search/optimization problems • Global problem generally very difficult • GAs (and related) have long history of success in global problems • Some global problems essentially impossible to solve • Much solid research and applications with GAs • Unfortunately, more misrepresentations, dubious claims, and “hype” than other methods. For example, GA software ads: • “…can handle the most complex problems, including problems unsolvable by any other method.” • “…uses GAs to solve any optimization problem!” • No clear indication of problem class(es) for which GAs are superior to other methods

  6. Fitness Function and Coding of  • Need to define “fitness function” to be maximized • Fitness function may depend on  or on coded form of  • Common choice is L() when algorithm works directly with  • Other choices desirable in some applications • Add positive constant to ensure fitness always  0 • Rescaling to avoid extreme values that can cause instabilities due to selection step • Coding of : Binary bit and real-number (floating point) representation of  are most common forms • Bit representation applies to each element of  for each of the members of the population (e.g.,   [0 1 1 0…1 0]) • Real-number “coding” (i.e., no coding of ) becoming popular due to effectiveness in applications

  7. Standard GA Operations • Selection is the mechanism by which the “parents” are chosen for producing offspring to be passed into next generation • Selection tends to pick best population elements as parents • Elitism passes best chromosome(s) to next generation intact • Elite chromosomes also eligible for selection as parents • Inclusion of elitism critical to practical performance of GA (Holland’s original formulation did not include elitism) • Crossover takes parent-pairs from selection step and creates offspring • Mutation makes “slight” random modifications to some or all of the offspring in next generation • Selection, crossover, and mutation discussed below, followed by the basic steps of a GA….

  8. Selection • Parent selection methods based on probability of selection being increasing function of fitness • Roulette-wheel selectioniscommon method • Probability an individual is selected is equal to its fitness divided by the total fitness in the population • Problem: Selection probability highly dependent on units and scaling for fitness function • May cause premature convergence to local optima • Rank selection andtournament selection methods reduce sensitivity to choice of fitness function • More robust: Only compare which chromosomes are better, not relative magnitudes of fitness functions • Often produce better practical results (see, e.g., examples in Sect. 9.7 of ISSO)

  9. Crossover • Crossover provides means of “mixing” two parents (from selection step) to provide two offspring • For given parents, crossover occurs with specified probability  1 • Else, parents placed directly into next generation (clones) • Examples of crossover operator: A. One-point crossoverB. Two-point crossover

  10. Mutation • Mutation operator introduces spontaneous variability (as in random search algorithms) • Mutation generally makes only small changes to solution • Bit-based coding and real (floating point) coding require different type of mutation • Bit-based mutation generally involves “flipping” bit(s) • Real-based mutation often involves adding small (Monte Carlo) random vector to chromosomes • Example below shows mutation on one element in chromosome in bit-based coding:

  11. Essential Steps of Basic GA(Noise-Free Measurements) Step 0 (initialization) Randomly generate initial population of N (say) chromosomes and evaluate fitness function. Step 1 (parent selection) Set Ne = 0 if elitism strategy is not used; 0 < Ne < N otherwise. Select with replacement NNe parents from full population. Step 2 (crossover) For each pair of parents identified in step 1, perform crossover on parents at a randomly chosen splice point (or points if using multi-point crossover) with probability Pc.

  12. Essential Steps of GA (cont’d) Step 3 (replacement and mutation) Replace the non-elite N Ne chromosomes with the current population of offspring from step 2. Perform mutation on the bits with a small probability Pm. Step 4 (fitness and end test) Compute the fitness values for the new population of N chromosomes. Terminate the algorithm if stopping criterion or budget of fitness function evaluations is met; else return to step 1.

  13. Examples of GAs in ISSO • Section 9.7 of ISSO includes several numerical examples • Usual caveats regarding caution in drawing general inferences; summary of examples below • Example 9.4: unimodal function with p = 10 • GA with real coding outperforms GA with bit coding w/ noise-free loss measurements; real and bit coding similar with noisy measurements • Example 9.5: multimodal function with p = 2 and noise-free loss measurements • Plot of function on next slide • GA outperforms SA with injected randomness • Example 9.6: issues with traveling salesperson problem • Fundamental issue is to ensure legal tours

  14. Loss Function for Example 9.5

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