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Ranga Rodrigo April 6, 2014 Most of the sides are from the Matlab tutorial. 1 • Global Optimization Toolbox provides methods that search for global solutions to problems that contain multiple maxima or minima. • This includes the solvers: – global search – multistart – pattern search – genetic algorithm, and – simulated annealing 2 • Optimization is the process of finding the point that minimizes a function. More specifically: – A local minimum of a function is a point where the function value is smaller than or equal to the value at nearby points, but possibly greater than at a distant point. – A global minimum is a point where the function value is smaller than or equal to the value at all other feasible points. – Genetic algorithms is able to find global minima. 3 • The GA is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution. • The GA repeatedly modifies a population of individual solutions. • At each step, the GA selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. • Over successive generations, the population "evolves" toward an optimal solution. 4 • You can apply the genetic algorithm to solve a variety of optimization problems that are not well suited for standard optimization algorithms, including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear. • The genetic algorithm can address problems of mixed integer programming, where some components are restricted to be integer-valued. 5 • The GA uses three main types of rules at each step to create the next generation from the current population: – Selection rules select the individuals, called parents, that contribute to the population at the next generation. – Crossover rules combine two parents to form children for the next generation. – Mutation rules apply random changes to individual parents to form children. 6 7 • optimtool('ga') 8 9 • To use the Optimization Tool, wemust first enter the following information: • Fitness function: – The objective function you want to minimize. Enter the fitness function in the form @fitnessfun, where fitnessfun.m is a file that computes the fitness function. Computing Objective Functions explains how write this file. The @ sign creates a function handle to fitnessfun. • Number of variables: – The length of the input vector to the fitness function. 10 • You can enter constraints or a nonlinear constraint function for the problem in the Constraints pane. If the problem is unconstrained, leave these fields blank. • To run the genetic algorithm, click the Start button. The tool displays the results of the optimization in the Run solver and view results pane. • You can change the options for the genetic algorithm in the Options pane. • To view the options in one of the categories listed in the pane, click the + sign next to it. 11 • Lets go through example that shows how to find the minimum of Rastrigin's function, a function that is often used to test the genetic algorithm. • For two independent variables, Rastrigin's function: Ras( x) 20 x12 x22 10(cos2x1 cos2x2 ) 12 In the Fitness function field, enter @rastriginsfcn. In the Number of variables field, enter 2, the number of independent variables for Rastrigin's function 13 While the algorithm is running, the Current iteration field displays the number of the current generation. You can temporarily pause the algorithm by clicking the Pause button. When you do so, the button name changes to Resume. To resume the algorithm from the point at which you paused it, click Resume. When the algorithm is finished, the Run solver and view results pane appears as shown in the following figure. Your numerical results might differ from those in the figure, since ga is stochastic The display shows: The final value of the fitness function when the algorithm terminated: Objective function value: 0.15078442701228667 The reason the algorithm terminated. Optimization terminated: average change in the fitness value less than options.TolFun. The final point, which in this example is [-0.021 0.018]. 14 The Optimization Tool Plot functions pane enables you to display various plots that provide information about the genetic algorithm while it is running. This information can help you change options to improve the performance of the algorithm. For example, to plot the best and mean values of the fitness function at each generation, select the box next to Best fitness, as shown in the following figure 15