问题描述
我正在尝试最小化鲍威尔函数 (x1+10x2)^2+5(x3-x4)^2+(x2-2x3)^4+10(x1-x4)^4 在 matlab 上使用拟牛顿法更新 通过秩一校正、DFP 算法和 BFGS 算法的 Hessian-inverse 近似。但在这 函数，我对每种方法都得到相同的结果。正常吗？ 我的代码如下。
function [x_vals,fnc_vals] = quasi_newton(start_vec,termination,rank1,DFP1,BFGS1)
% the vector to be iterated is initialized to the initial guess.
x = start_vec;
% 2 matrices are initialized as matrices of zeros with very large
% sizes to avoid results exceeding the dimensions of the matrices.
% they will be used to store the vectors and the values of the function
% at each iteration.
% a variable to count number of iterations done when doing line search
% to determine the optimal step size is declared and initialized to 0.
vec= zeros(4,100000);
vals = zeros(1,100000);
% the initial hessian-inverse approximation is taken as the identity
% matrix.
H_k = eye(4);
grad = grad_eval_powell(x(1),x(2),x(3),x(4));
% a vector named grad is declared and initialized to the value of the gradient of
% the powell function at the point x. It will be changed in each
% iteration.
counter = 1;
% a counter is declared and initialized to 1.
while dot(grad,grad) > termination % when the termination criteria is not
% satisfied.
vec(:,counter) = vec(:,counter) + x; % the vector at the given iteration is
% stored in the corresponding index of the vec.
vals(counter) = vals(counter) + eval_powell(x);
% the value of the function at the given iteration is stored in the
% corresponding index of vals.
direction = -H_k*grad;
% to optimize the step size,the golden section method is used.
% for given iteration and vector,function bracketing is called
% to determine the initial interval in which the optimal step size
% value that makes the function value at the next iteration the
% minimum.
[a0,b0] = bracketing(@fi_gd,x,direction);
% then by using the function golden_section,the minimizer
% that makes the function value at the next iteration the
% minimum is determined and stored in the alpha_k variable.
alpha_k = golden_section(@fi_gd,a0,b0,direction);
% the next point for the iteration to be done is determined
% according to the quasi_newton method.
x1 = x + alpha_k*direction;
grad1 = grad_eval_powell(x1(1),x1(2),x1(3),x1(4));
delta_x = x1-x;
delta_grad = grad1-grad;
% if statement below is satisfied when we want to use the rank-one
% correction formula to determine U_k.
if (rank1 && not(DFP1) && not(BFGS1))
U_k = rank_one(delta_x,delta_grad,H_k);
end
% if statement below is satisfied when we want to use the DFP
% algorithm to determine U_k.
if (DFP1 && not(rank1) && not(BFGS1))
U_k = DFP(delta_x,H_k);
end
% if statement below is satisfied when we want to use the BFGS
% algorithm to determine U_k.
if (BFGS1 && not(rank1) && not(DFP1))
U_k = BFGS(delta_x,H_k);
end
% the approximation of hessian-inverse H_k is updated according to
% the quasi-Newton method.
H_k = H_k + U_k;
% the grad variable is assigned to be the grad1 for the next
% iteration.
grad = grad1;
% the x variable is assigned to be the x1 for the next iteration.
x = x1;
% counter is iterated by +1 for storing the next iteration.
counter = counter + 1;
end
vec = vec(:,1:counter-1); % the matrices vec and vals are shrinked to
% the size of the iterations.
vals = vals(1:counter-1);
% outputs are assigned by corresponding values.
x_vals = vec;
fnc_vals = vals;
end
哪里
function u_k = rank_one (delta_x,H)
% the vector parameter z is calculated according to the rank-one formula.
z = delta_x-H*delta_grad;
% the scalar parameter a is calculated according to the rank-one
% formula.
a = 1 / dot(z,delta_grad);
% the matrix u_k is calculated according to the rank-one formula.
u_k = a*(z*z');
end
function u_k = DFP (delta_x,H)
% the matrix u_k is calculated according to the DFP algorithm.
u_k = (delta_x*delta_x') / dot(delta_x,delta_grad) - ...
((H*delta_grad)*(H*delta_grad)') / dot(H*delta_grad,delta_grad) ;
end
function u_k = BFGS (delta_x,H)
% H*delta_grad and dot(delta_x,delta_grad) are assigned in Hg and xg
% respectively for the simplicity of the function that determines u_k.
Hg = H*delta_grad;
xg = dot(delta_x,delta_grad);
% u_k is calculated according to the BFGS algorithm.
u_k = (1 + dot(Hg,delta_grad)/xg)*((delta_x*delta_x')/xg) - ...
(Hg*delta_x' + delta_x*Hg')/xg ;
end
能否请您至少在您的计算机上运行代码并告诉我您的结果？ 我得到 0.0045，分 8 步计算。
顺便说一句，你可以使用
x = [3;-1;0;1];
term = 0.01;
[x1,y1] = quasi_newton (x,term,true,false,false);
[x2,y2] = quasi_newton (x,false);
[x3,y3] = quasi_newton (x,true);
以上代码为主要代码。
提前致谢。
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