信赖域算法matlab程序

以下是一份MATLAB实现的信赖域算法程序：

function [xopt, fopt, iter] = trust_region(f, g, H, x0, delta, tol, itmax)
% TRUST_REGION Trust region algorithm for unconstrained optimization
%   [XOPT, FOPT, ITER] = TRUST_REGION(F, G, H, X0, DELTA, TOL, ITMAX)
%   finds a local minimizer of the function F using the trust region
%   method with initial point X0, trust region radius DELTA, and tolerance
%   TOL. The maximum number of iterations is ITMAX. The gradient of F
%   is given by G and the Hessian matrix of F is given by H.
%
%   XOPT is the optimal point, FOPT is the optimal function value, and
%   ITER is the number of iterations until convergence.

% Initialization
x = x0;
fval = f(x);
gval = g(x);
Hval = H(x);
iter = 0;

% Main loop
while norm(gval) > tol && iter < itmax
    % Solve trust region subproblem
    p = trust_region_subproblem(gval, Hval, delta);
    xnew = x + p;
    fnew = f(xnew);
    actual_reduction = fval - fnew;
    predicted_reduction = -gval' * p - 0.5 * p' * Hval * p;
    rho = actual_reduction / predicted_reduction;
    
    % Update trust region radius
    if rho < 0.25
        delta = 0.25 * delta;
    elseif rho > 0.75 && norm(p) == delta
        delta = min(2 * delta, 1e10);
    end
    
    % Update x, fval, gval, Hval
    if rho > 0
        x = xnew;
        fval = fnew;
        gval = g(x);
        Hval = H(x);
    end
    
    iter = iter + 1;
end

xopt = x;
fopt = fval;

end

function p = trust_region_subproblem(g, H, delta)
% TRUST_REGION_SUBPROBLEM Solve the trust region subproblem
%   P = TRUST_REGION_SUBPROBLEM(G, H, DELTA) solves the trust region
%   subproblem for a given gradient G, Hessian matrix H, and trust region
%   radius DELTA. The solution P is the minimizer of the quadratic model
%   m(p) = g'p + 0.5 * p'Hp subject to ||p|| <= DELTA.

% Solve unconstrained subproblem
p = -H \ g;

% Calculate norm of p
pnorm = norm(p);

% Check if p satisfies trust region constraint
if pnorm <= delta
    return;
end

% Solve constrained subproblem
p = -((g' * p) / (p' * H * p)) * p + sqrt(delta^2 - pnorm^2) * (p / pnorm);

end

在此程序中，我们首先定义了`trust_region`函数，该函数使用信赖域算法来最小化给定的函数。输入参数包括函数本身，函数的梯度和黑塞矩阵，初始点，信赖域半径，收敛容差和最大迭代次数。输出参数是最优点，最优函数值和迭代次数。

该程序的主要部分是`while`循环，该循环持续进行直到梯度的范数小于收敛容差或达到最大迭代次数。在每次迭代中，我们首先解决信赖域子问题，然后根据实际和预测的减少率更新信赖域半径。最后，我们更新$x$，$f$值，梯度和黑塞矩阵，并增加迭代计数器。

信赖域子问题由`trust_region_subproblem`函数解决。该函数首先求解无约束子问题，然后检查$p$是否满足信赖域约束。如果不是，则求解有约束子问题。

此程序仅为信赖域算法的一个基本实现。根据应用程序的具体要求，可能需要进行修改和优化。