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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>Coding and minimizing an objective function using Pattern Search</title> <meta name="generator" content="MATLAB 7.0.0.4032 (R14) Prerelease"> <meta name="date" content="2004-03-29"> <meta name="m-file" content="psobjective"> <meta name="title" content="Coding and minimizing an objective function using Pattern Search"> <meta name="description" content="This is a demonstration of how to create and minimize an objective function using Pattern Search in the Genetic Algorithm and Direct Search Toolbox."><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>Coding and minimizing an objective function using Pattern Search</h1> <p>This is a demonstration of how to create and minimize an objective function using Pattern Search in the Genetic Algorithm and Direct Search Toolbox. </p> <h2>Contents</h2> <ul> <li><a href="#1">A simple objective function</a></li> <li><a href="#2">Coding the objective function</a></li> <li><a href="#3">Minimizing using PATTERNSEARCH</a></li> <li><a href="#4">A objective function with additional arguments</a></li> <li><a href="#5">Minimizing using additional arguments</a></li> <li><a href="#6">Vectorizing your objective function</a></li> </ul> <h2>A simple objective function<a name="1"></a></h2> <p>Here we want to minimize a simple function of two variables</p><pre> min f(x) = (4 - 2.1*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-4 + 4*x2^2)*x2^2; xThe above function is known as 'cam' and is taken from L.C.W. Dixon andG.P. Szegö (eds.), Towards Global Optimisation 2, North-Holland,Amsterdam 1978.</pre><h2>Coding the objective function<a name="2"></a></h2> <p>We create an M-file named simple_objective.m with the following code in it:</p><pre> function y = simple_objective(x) y = (4 - 2.1*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ... (-4 + 4*x(2)^2)*x(2)^2;</pre><p>The Pattern Search solver assumes the objective function will take one input x where x has as many elements as number of variables in the problem. The objective function computes the value of the function and returns that scalar value in its one return argument y. </p> <h2>Minimizing using PATTERNSEARCH<a name="3"></a></h2> <p>To minimize our objective function using the PATTERNSEARCH function, we need to pass in a function handle to the objective function as well as specifying a start point as the second argument. </p><pre class="codeinput"> ObjectiveFunction = @simple_objective; X0 = [0.5 0.5]; <span class="comment">% Starting point</span> [x,fval] = patternsearch(ObjectiveFunction,X0)</pre><pre class="codeoutput">Optimization terminated: current mesh size 9.5367e-007 is less than 'TolMesh'.x = -0.0898 0.7127fval = -1.0316</pre><h2>A objective function with additional arguments<a name="4"></a></h2> <p>Sometimes we want our objective function to be parameterized by extra arguments that act as constants during the optimization. For example, in the previous objective function, say we want to replace the constants 4, 2.1, and 4 with parameters that we can change to create a family of objective functions. We can re-write the above function to take three additional parameters to give the new minimization problem </p><pre> min f(x) = (a - b*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-c + c*x2^2)*x2^2; x</pre><p>a, b, and c are parameters to the objective function that act as constants during the optimization (they are not varied as part of the minimization). One can create an M-file called parameterized_objective.m containing the following code: </p><pre> function y = parameterized_objective(x,a,b,c) y = (a - b*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ... (-c + c*x(2)^2)*x(2)^2;</pre><h2>Minimizing using additional arguments<a name="5"></a></h2> <p>Again, we need to pass in a function handle to the objective function as well as a start point as the second argument.</p> <p>PATTERNSEARCH will call our objective function with just one argument 'x', but our objective function has four arguments: x, a, b, c. We can use an anonymous function to capture the values of the additional arguments, the constants a, b, and c. We create a function handle 'ObjectiveFunction' to an anonymous function that takes one input 'x', but calls 'parameterized_objective' with x, a, b and c. The variables a, b, and c have values when the function handle 'ObjectiveFunction' is created, so these values are captured by the anonymous function. </p><pre class="codeinput"> a = 4; b = 2.1; c = 4; <span class="comment">% define constant values</span> ObjectiveFunction = @(x) parameterized_objective(x,a,b,c); X0 = [0.5 0.5]; [x,fval] = patternsearch(ObjectiveFunction,X0)</pre><pre class="codeoutput">Optimization terminated: current mesh size 9.5367e-007 is less than 'TolMesh'.x = -0.0898 0.7127fval = -1.0316</pre><h2>Vectorizing your objective function<a name="6"></a></h2> <p>Consider the above function again:</p><pre> f(x) = (a - b*x1^2+x1^4/3)*x1^2+x1*x2+(-c + c*x2^2)*x2^2;</pre><p>By default, the PATTERNSEARCH solver only passes in one point at a time to the objective function. However, sometimes speed up can be achieved if the objective function is vectorized to take a set of points and return a set of function values. </p> <p>For example if the solver wants to evaluate a set of five points in one call to this objective function, then it will call the objective with a matrix of size 5-by-2, i.e., 5 rows and 2 columns (recall 2 is the number of variables). </p> <p>Create an M-file called vectorized_objective.m with the following code:</p><pre> function y = vectorized_objective(x,a,b,c) y = zeros(size(x,1),1); %Pre-allocate y for i = 1:size(x,1) % for the number of rows in x x1 = x(i,1); x2 = x(i,2); y(i) = (a - b*x1^2+x1^4/3)*x1^2+x1*x2+(-c + c*x2^2)*x2^2; end</pre><p>This vectorized version of the objective function takes a matrix x with an arbitrary number of points, the rows of x, and returns a column vector y of length the same as the number of rows of x. </p> <p>To take advantage of the vectorized objective function, we need to tell PATTERNSEARCH that the objective is vectorized using the options structure that is created using PSOPTIMSET, and is passed in as the ninth argument. </p><pre class="codeinput"> ObjectiveFunction = @(x) vectorized_objective(x,4,2.1,4); X0 = [0.5 0.5]; options = psoptimset(<span class="string">'Vectorized'</span>,<span class="string">'on'</span>); [x,fval] = patternsearch(ObjectiveFunction,X0,[],[],[],[],[],[],options)</pre><pre class="codeoutput">Optimization terminated: current mesh size 9.5367e-007 is less than 'TolMesh'.x = -0.0898 0.7127fval = -1.0316</pre><p class="footer">Copyright 2004 The MathWorks, Inc.<br></p> <!--##### SOURCE BEGIN #####%% Coding and minimizing an objective function using Pattern Search% This is a demonstration of how to create and minimize an objective% function using Pattern Search in the Genetic Algorithm and Direct Search% Toolbox.% Copyright 2004 The MathWorks, Inc. % $Revision: 1.1.4.1 $ $Date: 2004/04/04 03:24:37 $%% A simple objective function% Here we want to minimize a simple function of two variables%% min f(x) = (4 - 2.1*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-4 + 4*x2^2)*x2^2;% x% The above function is known as 'cam' and is taken from L.C.W. Dixon and% G.P. Szeg枚 (eds.), Towards Global Optimisation 2, North-Holland,% Amsterdam 1978. %% Coding the objective function% We create an M-file named simple_objective.m with the following% code in it:%% function y = simple_objective(x) % y = (4 - 2.1*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ...% (-4 + 4*x(2)^2)*x(2)^2;%% The Pattern Search solver assumes the objective function will take one% input x where x has as many elements as number of variables in the% problem. The objective function computes the value of the function and% returns that scalar value in its one return argument y.%% Minimizing using PATTERNSEARCH% To minimize our objective function using the PATTERNSEARCH function, % we need to pass in a function handle to the objective function as well % as specifying a start point as the second argument. ObjectiveFunction = @simple_objective; X0 = [0.5 0.5]; % Starting point [x,fval] = patternsearch(ObjectiveFunction,X0)%% A objective function with additional arguments% Sometimes we want our objective function to be parameterized by extra% arguments that act as constants during the optimization. For example,% in the previous objective function, say we want to replace the% constants 4, 2.1, and 4 with parameters that we can change to create a% family of objective functions. We can re-write the above function to % take three additional parameters to give the new minimization problem %% min f(x) = (a - b*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-c + c*x2^2)*x2^2;% x%% a, b, and c are parameters to the objective function that act as% constants during the optimization (they are not varied as part of the% minimization). One can create an M-file called parameterized_objective.m % containing the following code:%% function y = parameterized_objective(x,a,b,c)% y = (a - b*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ...% (-c + c*x(2)^2)*x(2)^2;%%% Minimizing using additional arguments% Again, we need to pass in a function handle to the objective function as% well as a start point as the second argument. %% PATTERNSEARCH will call our objective function with just one argument% 'x', but our objective function has four arguments: x, a, b, c. We can% use an anonymous function to capture the values of the additional% arguments, the constants a, b, and c. We create a function handle% 'ObjectiveFunction' to an anonymous function that takes one input 'x',% but calls 'parameterized_objective' with x, a, b and c. The variables a,% b, and c have values when the function handle 'ObjectiveFunction' is% created, so these values are captured by the anonymous function. a = 4; b = 2.1; c = 4; % define constant values ObjectiveFunction = @(x) parameterized_objective(x,a,b,c); X0 = [0.5 0.5]; [x,fval] = patternsearch(ObjectiveFunction,X0) %% Vectorizing your objective function% Consider the above function again:%% f(x) = (a - b*x1^2+x1^4/3)*x1^2+x1*x2+(-c + c*x2^2)*x2^2;%% By default, the PATTERNSEARCH solver only passes in one point at a time% to the objective function. However, sometimes speed up can be achieved% if the objective function is vectorized to take a set of points and% return a set of function values.%% For example if the solver wants to evaluate a set of five points in one% call to this objective function, then it will call the objective with% a matrix of size 5-by-2, i.e., 5 rows and 2 columns (recall 2 is the % number of variables). %% Create an M-file called vectorized_objective.m with the following code:%% function y = vectorized_objective(x,a,b,c)% y = zeros(size(x,1),1); %Pre-allocate y% for i = 1:size(x,1) % for the number of rows in x% x1 = x(i,1);% x2 = x(i,2);% y(i) = (a - b*x1^2+x1^4/3)*x1^2+x1*x2+(-c + c*x2^2)*x2^2;% end%% This vectorized version of the objective function takes a matrix x% with an arbitrary number of points, the rows of x, and returns% a column vector y of length the same as the number of rows of x.%% To take advantage of the vectorized objective function, we need to tell% PATTERNSEARCH that the objective is vectorized using the options% structure that is created using PSOPTIMSET, and is passed in as the ninth% argument. ObjectiveFunction = @(x) vectorized_objective(x,4,2.1,4); X0 = [0.5 0.5]; options = psoptimset('Vectorized','on'); [x,fval] = patternsearch(ObjectiveFunction,X0,[],[],[],[],[],[],options) ##### SOURCE END #####--> </body></html>⌨️ 快捷键说明
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