信赖域算法matlab程序

信赖域算法是一种非线性优化算法，用于求解无约束或有约束的非线性优化问题。以下是一个使用Matlab实现的信赖域算法的简单示例程序：

function [x, fval, exitflag] = trustregion(fun, x0, lb, ub)

% Initialization
x = x0;
fval = fun(x);
delta = 0.1;
tol = 1e-6;
maxiter = 100;
k = 0;

while k < maxiter
    % Compute gradient and Hessian
    [grad, hess] = gradient_hessian(fun, x);
    
    % Solve trust region subproblem
    [p, subfval, subexitflag] = trustregionsubproblem(grad, hess, delta, lb-x, ub-x);
    
    % Update x and fval
    xnew = x + p;
    fvalnew = fun(xnew);
    
    % Update trust region radius
    rho = (fval - fvalnew) / (subfval - fval);
    if rho < 0.25
        delta = 0.25 * delta;
    elseif rho > 0.75 && abs(norm(p) - delta) < tol
        delta = min(2 * delta, norm(p));
    end
    
    % Check convergence criteria
    if norm(grad) < tol
        exitflag = 1;
        break;
    elseif abs(fvalnew - fval) < tol
        exitflag = 2;
        break;
    end
    
    % Update x and fval
    x = xnew;
    fval = fvalnew;
    k = k + 1;
end

if k == maxiter
    exitflag = -1;
end

function [grad, hess] = gradient_hessian(fun, x)
% Compute gradient and Hessian
fval = fun(x);
grad = gradient(fun, x);
n = numel(x);
hess = zeros(n);
for i=1:n
    for j=i:n
        hess(i,j) = diff(diff(fun,x(i)),x(j));
        hess(j,i) = hess(i,j);
    end
end

function [p, subfval, subexitflag] = trustregionsubproblem(grad, hess, delta, lb, ub)
% Solve trust region subproblem
n = numel(grad);
opts = optimoptions('quadprog', 'Algorithm', 'interior-point-convex', 'Display', 'off');
p = quadprog(hess, grad, [], [], [], [], lb, ub, [], opts);
subfval = 0.5 * p' * hess * p + grad' * p;
if norm(p) > delta
    p = delta * p / norm(p);
    subfval = 0.5 * delta^2;
    subexitflag = 0;
else
    subexitflag = 1;
end

在这个示例程序中，`fun`是要最小化的目标函数，`x0`是初始点，`lb`和`ub`是可选的下界和上界。程序使用了Matlab自带的`gradient`和`diff`函数来计算梯度和Hessian矩阵，使用了Matlab自带的`quadprog`函数来求解信赖域子问题。在每次迭代中，程序计算当前点的梯度和Hessian矩阵，然后使用信赖域子问题求解器来求解信赖域子问题，更新当前点和目标函数值，并根据收敛标准调整信赖域半径。如果达到最大迭代次数而没有收敛，则算法停止并返回失败标志。