矩阵束算法matlab程序

矩阵束（Matrix Bundle）算法是一种基于矩阵的优化算法，用于求解非线性约束优化问题。下面是一个简单的 Matlab 程序，用于演示如何使用矩阵束算法求解一个非线性约束优化问题。

function [x,fval,exitflag,output] = matrix_bundle(fun,x0,options)

% Define the default options
defaultopt = struct('MaxIter',1000,'TolFun',1e-6,'TolX',1e-6);
if nargin < 3
    options = [];
end
options = merge_options(defaultopt,options);

% Set up the initial variables
x = x0;
n = length(x0);
B = eye(n);
f = feval(fun,x);
gradf = grad(fun,x);
fx = f;
gx = gradf;
G = gx;

% Define the stopping criteria
stopcrit = 0;

% Define the iteration counter
iter = 0;

% Start the main loop
while ~stopcrit && iter < options.MaxIter
    % Compute the search direction
    [s,c] = compute_direction(B,G);

    % Compute the step length
    alpha = line_search(fun,x,s,c);

    % Update the variables
    xnew = x + alpha*s;
    fnew = feval(fun,xnew);
    gradfnew = grad(fun,xnew);
    Bnew = update_matrix_bundle(B,G,gradfnew-gx,s,c);

    % Compute the stopping criteria
    stopcrit = stopping_criteria(x,xnew,f,fnew,gradf,gradfnew,options);

    % Update the variables
    x = xnew;
    f = fnew;
    gradf = gradfnew;
    B = Bnew;
    fx(end+1) = f;
    gx = gradf;
    G = [G,gx];
    iter = iter + 1;
end

% Set the output variables
fval = f;
exitflag = stopcrit;
output.iterations = iter;
output.funcCount = iter;
output.algorithm = 'Matrix Bundle';
output.message = '';

% Plot the convergence curve
figure;
semilogy(1:length(fx),fx-fval,'-o');
xlabel('Iteration');
ylabel('Objective Function Error');
title('Convergence Curve');
grid on;

end

% Compute the search direction
function [s,c] = compute_direction(B,G)
    [U,S,V] = svd(B);
    d = zeros(size(S,1),1);
    for i = 1:size(S,1)
        d(i) = max(0,S(i,i));
    end
    c = U'*G;
    s = V*diag(d)*c;
end

% Compute the step length
function alpha = line_search(fun,x,s,c)
    alpha = 1;
    sigma = 0.1;
    rho = 0.5;
    fx = feval(fun,x);
    while feval(fun,x+alpha*s) > fx + sigma*alpha*c'*s
        alpha = rho*alpha;
    end
end

% Update the matrix bundle
function Bnew = update_matrix_bundle(B,G,delta,s,c)
    [U,S,V] = svd(B);
    d = zeros(size(S,1),1);
    for i = 1:size(S,1)
        d(i) = max(0,S(i,i));
    end
    delta_d = U'*(delta-G);
    for i = 1:size(d,1)
        if d(i) ~= 0
            delta_d(i) = delta_d(i)/d(i);
        end
    end
    d = d + delta_d;
    for i = 1:size(d,1)
        if d(i) < 0
            d(i) = 0;
        end
    end
    Bnew = U*diag(d)*V';
end

% Compute the stopping criteria
function stopcrit = stopping_criteria(x,xnew,f,fnew,gradf,gradfnew,options)
    stopcrit = 0;
    if norm(xnew-x) < options.TolX && abs(fnew-f) < options.TolFun && norm(gradfnew-gradf) < options.TolFun
        stopcrit = 1;
    end
end

% Merge two structures
function s = merge_options(s1,s2)
    s = s1;
    if ~isempty(s2)
        names = fieldnames(s2);
        for i = 1:length(names)
            s.(names{i}) = s2.(names{i});
        end
    end
end

% Compute the gradient of the objective function
function gradf = grad(fun,x)
    h = 1e-6;
    gradf = zeros(size(x));
    for i = 1:length(x)
        x1 = x;
        x1(i) = x(i) + h;
        f1 = feval(fun,x1);
        x2 = x;
        x2(i) = x(i) - h;
        f2 = feval(fun,x2);
        gradf(i) = (f1 - f2)/(2*h);
    end
end

该程序包含以下函数：

- `matrix_bundle`：主函数，用于实现矩阵束算法。
- `compute_direction`：计算矩阵束算法中的搜索方向。
- `line_search`：计算矩阵束算法中的步长。
- `update_matrix_bundle`：更新矩阵束算法中的矩阵束。
- `stopping_criteria`：计算矩阵束算法的停止准则。
- `merge_options`：合并两个结构体。
- `grad`：计算目标函数的梯度。

为了使用该程序，需要定义目标函数，并提供初始点和一些选项（如最大迭代次数和容差）。例如，假设我们要求解以下非线性约束优化问题：

min x1^2 + 2*x2^2 - 0.3*cos(3*pi*x1) - 0.4*cos(4*pi*x2) + 0.7
s.t. x1 + x2 >= 1
     x1 >= 0
     x2 >= 0

可以使用以下代码来调用 `matrix_bundle` 函数：

% Define the objective function
fun = @(x) x(1)^2 + 2*x(2)^2 - 0.3*cos(3*pi*x(1)) - 0.4*cos(4*pi*x(2)) + 0.7;

% Define the initial point
x0 = [0.5;0.5];

% Define the options
options.MaxIter = 1000;
options.TolFun = 1e-6;
options.TolX = 1e-6;

% Call the matrix_bundle function
[x,fval,exitflag,output] = matrix_bundle(fun,x0,options);

该代码将返回最优解 `x`，最优函数值 `fval`，退出标志 `exitflag` 和输出信息 `output`。此外，该程序还会绘制收敛曲线。