拟牛顿法 matlab
时间: 2023-10-16 15:06:27 浏览: 185
拟牛顿法(Quasi-Newton method)是一种无需计算目标函数的二阶导数(Hessian矩阵)的优化算法。它通过逼近目标函数的Hessian矩阵来进行优化。
在Matlab中,你可以使用`fminunc`函数来实现拟牛顿法。以下是一个简单的示例:
```matlab
% 定义目标函数
fun = @(x) 100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;
% 设置初始点
x0 = [-1.2, 1];
% 使用fminunc函数进行优化
options = optimoptions('fminunc', 'Algorithm', 'quasi-newton');
[x, fval] = fminunc(fun, x0, options);
% 输出结果
disp('最优解:');
disp(x);
disp('最优值:');
disp(fval);
```
在上述示例中,我们定义了一个目标函数`fun`,然后设置了初始点`x0`,最后使用`fminunc`函数进行拟牛顿法的优化。我们使用了`optimoptions`函数来设置优化选项,其中将算法设置为了拟牛顿法。
请注意,拟牛顿法有多种变体,比如BFGS、DFP等。在Matlab的`fminunc`函数中,默认使用的是DFP算法。你可以根据需要选择不同的算法。
希望这个示例能够帮助你理解如何在Matlab中使用拟牛顿法进行优化。如有任何疑问,请随时提出。
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拟牛顿法matlab
以下是一个基本的拟牛顿法的 MATLAB 代码示例:
```
function [x, fval, exitflag, output] = myquasinewton(fun,x0,options,varargin)
% MYQUASINEWTON finds the minimum of a function of several variables.
% X = MYQUASINEWTON(FUN,X0) starts at X0 and finds a minimum of the function
% described in FUN. FUN is a function handle. The syntax for FUN is
% Y = FUN(X) where X is a vector and Y is a scalar. FUN can also accept a
% second input, the value returned by OPTIMSET.
%
% X = MYQUASINEWTON(FUN,X0,OPTIONS) minimizes with the default optimization
% parameters replaced by values in the structure OPTIONS, created with the
% OPTIMSET function. See OPTIMSET for details.
%
% [X, FVAL] = MYQUASINEWTON(FUN,X0,OPTIONS) returns the value of the objective
% function at X.
%
% [X, FVAL, EXITFLAG] = MYQUASINEWTON(FUN,X0,OPTIONS) returns an EXITFLAG that
% describes the exit condition. Possible values of EXITFLAG and the
% corresponding exit conditions are
%
% 1 First-order optimality measure falls below tolerance specified
% in OPTIONS.TolFun.
% 2 Change in x is smaller than the tolerance specified in
% OPTIONS.TolX.
% 0 Maximum number of function evaluations or iterations reached.
% -1 Algorithm was terminated by the output function.
%
% [X, FVAL, EXITFLAG, OUTPUT] = MYQUASINEWTON(FUN,X0,OPTIONS) returns a
% structure OUTPUT with the following fields:
%
% iterations Number of iterations.
% funccount Number of function evaluations.
% message Exit message.
%
% EXAMPLES:
%
% FUN can be specified using @:
% X = myquasinewton(@myfun,x0)
% where myfun is a MATLAB function such as:
% function y = myfun(x)
% y = x(1)^2 + x(2)^2;
%
% FUN can also be an anonymous function:
% X = myquasinewton(@(x) sin(x(1))+cos(x(2)),x0)
%
% FUN can include additional arguments through the VARARGIN parameter:
% X = myquasinewton(@(x) myfun2(x,c),x0,options,c,d,e)
% where c,d,e, etc. are additional arguments.
%
% OPTIONS can be created with the OPTIMSET function:
% options = optimset('Param1',value1,'Param2',value2,...)
% See OPTIMSET for details. Commonly used options are:
% Display Level of display [ off | iter | notify | final ]
% MaxIter Maximum number of iterations allowed.
% TolFun Termination tolerance on the function value.
% TolX Termination tolerance on X.
%
% See also FMINSEARCH, FZERO, FUNCTION_HANDLE, OPTIMSET.
% Parse inputs
if nargin < 2
error('MYQUASINEWTON requires at least two input arguments.');
end
if nargin < 3 || isempty(options)
options = optimset('GradObj','on');
end
if nargin < 4
varargin = {};
end
% Set algorithm parameters
DisplayIter = strcmp(options.Display,'iter');
TolFun = options.TolFun;
TolX = options.TolX;
MaxIter = options.MaxIter;
GradObj = strcmp(options.GradObj,'on');
DerivStep = options.DerivativeCheck;
% Initialize variables
x = x0(:); % Make sure x is a column vector
n = length(x);
fval = feval(fun,x,varargin{:});
funccount = 1;
iterations = 0;
delta = 1e-8; % Initial step size
H = eye(n); % Initial Hessian approximation
% Main loop
while iterations < MaxIter
% Compute gradient and Hessian approximation
if GradObj
[fval,grad] = feval(fun,x,varargin{:});
funccount = funccount + 1;
else
grad = zeros(n,1);
for i = 1:n
xplus = x;
xplus(i) = xplus(i) + DerivStep;
xminus = x;
xminus(i) = xminus(i) - DerivStep;
grad(i) = (feval(fun,xplus,varargin{:}) - ...
feval(fun,xminus,varargin{:})) / (2*DerivStep);
funccount = funccount + 2;
end
end
p = -H * grad;
pk = p' * grad;
% Check for convergence
if norm(p) < TolX
exitflag = 2;
message = sprintf('Change in x too small: %e',norm(p));
break;
elseif norm(grad) < TolFun
exitflag = 1;
message = sprintf('First-order optimality measure falls below tolerance: %e',norm(grad));
break;
elseif funccount >= options.MaxFunEvals
exitflag = 0;
message = 'Maximum number of function evaluations reached.';
break;
elseif iterations >= MaxIter
exitflag = 0;
message = 'Maximum number of iterations reached.';
break;
end
% Perform line search
t = 1;
while feval(fun,x+t*p,varargin{:}) > fval + t*delta*pk
t = t/2;
end
% Update x and Hessian approximation
xnew = x + t*p;
s = xnew - x;
y = (feval(fun,xnew,varargin{:}) - fval) / t - grad' * s;
if s' * y >= eps * norm(s) * norm(y) % Positive curvature test
H = H + (s-H*y)*(s-H*y)'/(s'*y) + ...
(y-H*s)*(y-H*s)'/(y'*s);
end
x = xnew;
iterations = iterations + 1;
% Display status
if DisplayIter
fprintf('Iteration %d: fval=%g\n',iterations,fval);
end
end
% Output results
output.iterations = iterations;
output.funccount = funccount;
output.message = message;
```
该算法使用了 BFGS 方法来近似 Hessian 矩阵的逆。要使用该代码,您需要提供一个函数句柄来计算目标函数及其梯度。您还可以通过选项结构来指定优化参数,例如最大迭代次数、函数评估次数等。在运行算法后,该代码将返回一个近似最优解、目标函数的值、退出标志和一个输出结构,其中包含有关迭代次数、函数评估次数和退出消息的信息。
bfgs拟牛顿法matlab
BFGS拟牛顿法是一种数值优化算法,在MATLAB中有相应的函数实现。该函数为“fminunc”,可以通过传入目标函数及初始猜测值等参数来使用。
使用BFGS拟牛顿法求解优化问题的过程如下:首先需要定义目标函数及其梯度函数,并将其传入“fminunc”函数中。接着,需要指定初始猜测值,可以使用随机数生成器或手动指定。然后,在“fminunc”函数中设定算法参数,例如最大迭代次数、收敛精度等。
BFGS拟牛顿法的主要思路是使用近似的Hessian矩阵来更新参数,在每次迭代中,通过计算梯度和Hessian矩阵的乘积,得到搜索方向并进一步更新参数。BFGS拟牛顿法相对于其他优化算法具有较快的收敛速度和高的精度。
总的来说,在MATLAB中使用BFGS拟牛顿法求解优化问题比较简单,只需要了解其基本原理及相关参数设定即可。同时,对于不同的优化问题,可能需要针对具体情况进行选择和调整优化算法及其参数。
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