1. 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 –Extensions to basic GA algorithm –Numerical examples Slides for Introduction to Stochastic Search and Optimization (ISSO) by J. C. Spall

2. 9-2 Background: Evolutionary Computation populationEvolutionary 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. 9-3 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.) Initial Population Selection Reproduction Mutation Next Iteration (Generation) Prototype EC Method

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

5. 9-5 Role of GAs in Global Problems globalGA 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. 9-6 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. 9-7 Standard GA Operations SelectionSelection 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 ElitismElitism 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) CrossoverCrossover takes parent-pairs from selection step and creates offspring MutationMutation 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. 9-8 Selection Parent selection methods based on probability of selection being increasing function of fitness Roulette-wheel selectionRoulette-wheel selection is common 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 tournament selectionRank selection and tournament 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. 9-9 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 crossover B. Two-point crossover

10. 9-10 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. 9-11 Essential Steps of Basic GA (Noise-Free Measurements) Step 0 (initialization) Step 0 (initialization) Randomly generate initial population of N (say) chromosomes and evaluate fitness function. Step 1 (parent selection) Step 1 (parent selection) Set N e = 0 if elitism strategy is not used; 0 < N e < N otherwise. Select with replacement N  N e parents from full population. Step 2 (crossover) 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 P c .

12. 9-12 Essential Steps of GA (cont’d) Step 3 (replacement and mutation) Step 3 (replacement and mutation) Replace the non-elite N  N e chromosomes with the current population of offspring from step 2. Perform mutation on the bits with a small probability P m. Step 4 (fitness and end test) 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. 9-13 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. 9-14 Loss Function for Example 9.5