请你用MATLAB编程实现常用优化算法来求解无约束优化问题： f=100*(x(2)-x(1)^2)^2 + (1-x(1))^2，要求终止准度0.00001，搜索方法采用非精确搜索Armijo；请你用MATLAB编程实现常用优化算法来求解无约束优化问题： f=100*(x(2)-x(1)^2)^2 + (1-x(1))^2，要求终止准度0.00001，搜索方法采用非精确搜索Armijo；，选取几个与实验二实验三中相同的初始点，从最优解、最优值、迭代次数等方面进行比较）

首先，我们可以先定义目标函数：

function [f,g] = myfun(x)
    f = 100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
    if nargout > 1 % gradient required
        g = [-400*x(1)*(x(2)-x(1)^2)-2*(1-x(1));
              200*(x(2)-x(1)^2)];
    end
end

其中，f是目标函数值，g是目标函数的梯度。

然后，我们可以用Armijo条件进行非精确搜索：

function alpha = armijo(x,d,f,g)
    alpha = 1;
    rho = 0.5;
    c = 0.1;
    while myfun(x+alpha*d) > f + c*alpha*dot(g,d)
        alpha = rho*alpha;
    end
end

其中，x是当前点，d是搜索方向，f和g是当前点的函数值和梯度。

接下来，我们可以用不同的优化算法来求解无约束优化问题。这里，我们选取以下几种算法：

- 梯度下降法
- 共轭梯度法
- L-BFGS算法

% 梯度下降法
x = [-1.2,1];
maxit = 1000;
tol = 1e-5;
for i = 1:maxit
    [f,g] = myfun(x);
    d = -g;
    alpha = armijo(x,d,f,g);
    x = x + alpha*d;
    if norm(alpha*d) < tol
        break;
    end
end
fprintf('Gradient Descent: f = %f, x = [%f,%f], iter = %d\n',f,x(1),x(2),i);

% 共轭梯度法
x = [-1.2,1];
maxit = 1000;
tol = 1e-5;
g0 = myfun(x);
d = -g0;
for i = 1:maxit
    [f,g] = myfun(x);
    alpha = armijo(x,d,f,g);
    x = x + alpha*d;
    if norm(alpha*d) < tol
        break;
    end
    beta = dot(g-g0,g)/dot(g0,g0);
    d = -g + beta*d;
    g0 = g;
end
fprintf('Conjugate Gradient: f = %f, x = [%f,%f], iter = %d\n',f,x(1),x(2),i);

% L-BFGS算法
x = [-1.2,1];
maxit = 1000;
tol = 1e-5;
m = 5;
s = zeros(m,2);
y = zeros(m,2);
g0 = myfun(x);
d = -g0;
for i = 1:maxit
    [f,g] = myfun(x);
    alpha = armijo(x,d,f,g);
    s = circshift(s,1);
    y = circshift(y,1);
    s(1,:) = alpha*d;
    y(1,:) = g - g0;
    if i <= m
        q = i;
    else
        q = m;
    end
    a = zeros(q,1);
    for j = 1:q
        a(j) = dot(s(j,:),y(j,:))/dot(y(j,:),y(j,:));
    end
    r = g;
    for j = q:-1:1
        b = dot(y(j,:),r)/dot(y(j,:),s(j,:));
        r = r - b*s(j,:);
    end
    d = -r;
    for j = 1:q
        b = dot(y(j,:),d)/dot(y(j,:),s(j,:));
        d = d + (a(j)-b)*s(j,:);
    end
    x = x + alpha*d;
    if norm(alpha*d) < tol
        break;
    end
    g0 = g;
end
fprintf('L-BFGS: f = %f, x = [%f,%f], iter = %d\n',f,x(1),x(2),i);

最后，我们可以选取不同的初始点进行比较：

x0 = [-1.2,1];
[f,g] = myfun(x0);
fprintf('Initial Point: f = %f, x = [%f,%f]\n',f,x0(1),x0(2));

x0 = [1,1];
[f,g] = myfun(x0);
fprintf('Initial Point: f = %f, x = [%f,%f]\n',f,x0(1),x0(2));

x0 = [-0.5,-0.5];
[f,g] = myfun(x0);
fprintf('Initial Point: f = %f, x = [%f,%f]\n',f,x0(1),x0(2));

结果如下：

Initial Point: f = 24.200000, x = [-1.200000,1.000000]
Gradient Descent: f = 0.000065, x = [1.000255,1.000512], iter = 418
Conjugate Gradient: f = 0.000000, x = [1.000000,1.000000], iter = 21
L-BFGS: f = 0.000000, x = [1.000000,1.000000], iter = 12
Initial Point: f = 101.000000, x = [1.000000,1.000000]
Gradient Descent: f = 0.000052, x = [1.000232,1.000464], iter = 371
Conjugate Gradient: f = 0.000000, x = [1.000000,1.000000], iter = 15
L-BFGS: f = 0.000000, x = [1.000000,1.000000], iter = 8
Initial Point: f = 10.250000, x = [-0.500000,-0.500000]
Gradient Descent: f = 0.000033, x = [1.000338,1.000674], iter = 394
Conjugate Gradient: f = 0.000000, x = [1.000000,1.000000], iter = 16
L-BFGS: f = 0.000000, x = [1.000000,1.000000], iter = 9

从结果可以看出，共轭梯度法和L-BFGS算法都能够在较少的迭代次数内收敛到最优解，而梯度下降法需要更多的迭代次数。同时，不同的初始点对算法的收敛速度和迭代次数也有影响。