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<html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <!--This HTML is auto-generated from an M-file.To make changes, update the M-file and republish this document. --> <title>Hybrid Scheme in the Genetic Algorithm</title> <meta name="generator" content="MATLAB 7.0.0.4032 (R14) Prerelease"> <meta name="date" content="2004-03-29"> <meta name="m-file" content="hybriddemo"> <meta name="title" content="Hybrid Scheme in the Genetic Algorithm"> <meta name="description" content=""><style>body { background-color: white;}h1 { color: #990000; font-size: x-large;}h2 { color: #990000; font-size: medium;}p.footer { text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;}pre.codeinput { margin-left: 30px;}span.keyword {color: #0000FF}span.comment {color: #228B22}span.string {color: #A020F0}span.untermstring {color: #B20000}span.syscmd {color: #B28C00}pre.showbuttons { margin-left: 30px; border: solid black 2px; padding: 4px; background: #EBEFF3;}pre.codeoutput { color: gray; font-style: italic;}pre.error { color: red;} </style></head> <body> <h1>Hybrid Scheme in the Genetic Algorithm</h1> <h2>Contents</h2> <ul> <li><a href="#1">Introduction</a></li> <li><a href="#2">Rosenbrock's function</a></li> <li><a href="#4">Genetic Algorithm Solution</a></li> <li><a href="#8">Adding a hybrid function</a></li> </ul> <h2>Introduction<a name="1"></a></h2> <p>This demo uses a hybrid scheme to optimize a function using the Genetic Algorithm and another optimization method. GA can reach the region near an optimum point relatively quickly, but it can take many function evaluations to achieve convergence. A commonly used technique is to run GA for a small number of generations to get near an optimum point. Then the solution from GA is used as an initial point for another optimization solver that is faster and more efficient for local search. </p> <h2>Rosenbrock's function<a name="2"></a></h2> <p>For the demo we will optimize Rosenbrock's function (also known as Dejong's second function):</p><pre> f(x)= 100*(x(2)-x(1)^2)^2+(1-x(1))^2</pre><p>This function is notorious in optimization because of the slow convergence most methods exhibit when trying to minimize this function. This function has a unique minimum at the point x* = (1,1) where it has a function value f(x*) = 0. </p> <p>We can view the code for this fitness function.</p><pre class="codeinput">type <span class="string">dejong2fcn.m</span></pre><pre class="codeoutput">function scores = dejong2fcn(pop)%DEJONG2FCN Compute DeJongs second function.%This function is also known as Rosenbrock's function% Copyright 2004 The MathWorks, Inc.% $Revision: 1.1.4.1 $ $Date: 2004/04/04 03:24:35 $scores = zeros(size(pop,1),1);for i = 1:size(pop,1) p = pop(i,:); scores(i) = 100 * (p(1)^2 - p(2)) ^2 + (1 - p(1))^2;end</pre><p>We can use the function PLOTOBJECTIVE in the toolbox to plot the function DEJONG2FCN over the range = [-2 2;-2 2].</p><pre class="codeinput">plotobjective(@dejong2fcn,[-2 2;-2 2]);</pre><img vspace="5" hspace="5" src="hybriddemo_01.png"><h2>Genetic Algorithm Solution<a name="4"></a></h2> <p>To start, we will use the genetic algorithm, GA, alone to find the minimum of Rosenbrock's function. We need to supply GA with a function handle to the fitness function dejong2fcn.m. Also, GA needs to know the how many variables are in the problem, which is two for this function. </p><pre class="codeinput">FitnessFcn = @dejong2fcn;numberOfVariables = 2;</pre><p>Some plot functions can be selected to monitor the performance of the solver.</p><pre class="codeinput">options = gaoptimset(<span class="string">'PlotFcns'</span>, {@gaplotbestf,@gaplotstopping});</pre><p>We run GA with the above inputs.</p><pre class="codeinput">[x,fval] = ga(FitnessFcn,numberOfVariables,options)</pre><pre class="codeoutput">Optimization terminated: maximum number of generations exceeded.x = 1.0335 1.0711fval = 0.0021</pre><img vspace="5" hspace="5" src="hybriddemo_02.png"><p>The global optimum is at x* = (1,1). GA found a point near the optimum, but could not get a more accurate answer with the default stopping criteria. By changing the stopping criteria, we might find a more accurate solution, but it may take many more function evaluations to reach x* = (1,1). Instead, we can use a more efficient local search that starts where GA left off. The hybrid function field in GA provides this feature automatically. </p> <h2>Adding a hybrid function<a name="8"></a></h2> <p>We will use a hybrid function to solve the optimization problem, i.e., when GA stops (or you ask it to stop) this hybrid function will start from the final point returned by GA. Our choices are FMINSEARCH, PATTERNSEARCH, or FMINUNC. Since this optimization example is smooth, i.e., continuously differentiable, we can use the FMINUNC function from the Optimization toolbox as our hybrid function. Since FMINUNC has its own options structure, we provide it as an additional argument when specifying the hybrid function. </p><pre class="codeinput">fminuncOptions = optimset(<span class="string">'Display'</span>,<span class="string">'iter'</span>, <span class="string">'LargeScale'</span>,<span class="string">'off'</span>);options = gaoptimset(options,<span class="string">'HybridFcn'</span>,{@fminunc, fminuncOptions});</pre><p>Run GA solver again with FMINUNC as the hybrid function.</p><pre class="codeinput">[x,fval] = ga(@dejong2fcn,numberOfVariables,options)</pre><pre class="codeoutput">Optimization terminated: stall generations limit exceeded.Switching to the hybrid optimization algorithm (FMINUNC). Gradient's Iteration Func-count f(x) Step-size infinity-norm 0 3 0.00669628 2.3 1 15 0.0029569 0.00108747 0.0445 2 27 0.00112911 820 0.125 3 30 0.000249404 1 0.61 4 33 2.28441e-005 1 0.018 5 36 1.16621e-006 1 0.00299 6 39 2.80076e-009 1 0.00196 7 42 2.58118e-011 1 1.86e-005 8 45 2.00284e-011 1 4.3e-008 Optimization terminated: relative infinity-norm of gradient less than options.TolFun.x = 1.0000 1.0000fval = 2.0028e-011</pre><img vspace="5" hspace="5" src="hybriddemo_03.png"><p>The first plot shows the best and mean values of the population in every generation. The best value found by GA when it terminated is also shown in the plot title. When GA terminated, FMINUNC (the hybrid function) was automatically called with the best point found by GA so far. The solution 'x' and 'fval' is the result of using GA and FMINUNC together. As shown here, using the hybrid function can improve the accuracy of the solution efficiently. </p> <p class="footer">Copyright 2004 The MathWorks, Inc.<br></p> <!--##### SOURCE BEGIN #####%% Hybrid Scheme in the Genetic Algorithm% Copyright 2004 The MathWorks, Inc.% $Revision: 1.1.4.1 $ $Date: 2004/04/04 03:24:35 $%% Introduction% This demo uses a hybrid scheme to optimize a function using the Genetic% Algorithm and another optimization method. GA can reach the region near% an optimum point relatively quickly, but it can take many function% evaluations to achieve convergence. A commonly used technique is to run% GA for a small number of generations to get near an optimum point. Then% the solution from GA is used as an initial point for another optimization% solver that is faster and more efficient for local search.%% Rosenbrock's function% For the demo we will optimize Rosenbrock's function (also known as% Dejong's second function):% % f(x)= 100*(x(2)-x(1)^2)^2+(1-x(1))^2 % % This function is notorious in optimization because of the slow% convergence most methods exhibit when trying to minimize this function.% This function has a unique minimum at the point x* = (1,1) where it has a% function value f(x*) = 0. %% We can view the code for this fitness function.type dejong2fcn.m%%% We can use the function PLOTOBJECTIVE in the toolbox to plot the % function DEJONG2FCN over the range = [-2 2;-2 2].plotobjective(@dejong2fcn,[-2 2;-2 2]);%% Genetic Algorithm Solution% To start, we will use the genetic algorithm, GA, alone to find the% minimum of Rosenbrock's function. We need to supply GA with a function% handle to the fitness function dejong2fcn.m. Also, GA needs to know the% how many variables are in the problem, which is two for this function.FitnessFcn = @dejong2fcn;numberOfVariables = 2;%% % Some plot functions can be selected to monitor the performance of the% solver.options = gaoptimset('PlotFcns', {@gaplotbestf,@gaplotstopping});%%% We run GA with the above inputs.[x,fval] = ga(FitnessFcn,numberOfVariables,options)%% % The global optimum is at x* = (1,1). GA found a point near the optimum,% but could not get a more accurate answer with the default stopping% criteria. By changing the stopping criteria, we might find a more% accurate solution, but it may take many more function evaluations to% reach x* = (1,1). Instead, we can use a more efficient local search that% starts where GA left off. The hybrid function field in GA provides this% feature automatically.%% Adding a hybrid function% We will use a hybrid function to solve the optimization problem, i.e.,% when GA stops (or you ask it to stop) this hybrid function will start% from the final point returned by GA. Our choices are FMINSEARCH,% PATTERNSEARCH, or FMINUNC. Since this optimization example is smooth,% i.e., continuously differentiable, we can use the FMINUNC function from% the Optimization toolbox as our hybrid function. Since FMINUNC has its % own options structure, we provide it as an additional argument when% specifying the hybrid function. fminuncOptions = optimset('Display','iter', 'LargeScale','off');options = gaoptimset(options,'HybridFcn',{@fminunc, fminuncOptions});%%% Run GA solver again with FMINUNC as the hybrid function.[x,fval] = ga(@dejong2fcn,numberOfVariables,options)%%% The first plot shows the best and mean values of the population in every% generation. The best value found by GA when it terminated is also shown% in the plot title. When GA terminated, FMINUNC (the hybrid function) was% automatically called with the best point found by GA so far. The solution% 'x' and 'fval' is the result of using GA and FMINUNC together. As shown% here, using the hybrid function can improve the accuracy of the% solution efficiently.##### SOURCE END #####--> </body></html>⌨️ 快捷键说明
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