function [x, fval] = rosen_gradient_projection(f, x0, A, b, lb, ub, maxiter, tol) 各参数的含义
时间: 2023-10-18 13:04:34 浏览: 35
- `f`: 待优化的目标函数
- `x0`: 初始解向量
- `A`: 约束矩阵
- `b`: 约束向量
- `lb`: 变量下界向量
- `ub`: 变量上界向量
- `maxiter`: 最大迭代次数
- `tol`: 迭代收敛阈值
- `x`: 最优解向量
- `fval`: 最优解对应的目标函数值
相关问题
function [x, fval] = bisection(f, a, b, tol, maxit)
该实现了二分法求解函数 $f(x)$ 在区间 $[a,b]$ 内的根。
输入参数:
- f:函数句柄,即用于计算 $f(x)$ 的函数
- a,b:区间 $[a,b]$ 的端点
- tol:容差,即当 $|f(x)|<tol$ 时,认为找到了一个根
- maxit:最大迭代次数,防止算法陷入死循环
输出参数:
- x:找到的根
- fval:在根处的函数值
代码实现如下:
```
function [x, fval] = bisection(f, a, b, tol, maxit)
fa = f(a);
fb = f(b);
if sign(fa) == sign(fb)
error('f(a) 和 f(b) 的符号相同,无法使用二分法求解');
end
for k = 1:maxit
x = (a + b) / 2;
fx = f(x);
if abs(fx) < tol
fval = fx;
return;
end
if sign(fx) == sign(fa)
a = x;
fa = fx;
else
b = x;
fb = fx;
end
end
fval = fx;
warning('超过最大迭代次数,可能没有找到合适的根');
end
```
这个函数首先检查 $f(a)$ 和 $f(b)$ 的符号是否相同,如果相同则抛出错误。否则,进入迭代过程,每次迭代计算区间的中点 $x=(a+b)/2$,并计算 $f(x)$ 的值。如果 $|f(x)|<tol$,则认为找到了一个根,返回 $x$ 和 $f(x)$ 的值。如果 $f(x)$ 和 $f(a)$ 的符号相同,则根位于区间 $[x,b]$ 中,将 $a$ 更新为 $x$,并将 $f(a)$ 更新为 $f(x)$。反之,则根位于区间 $[a,x]$ 中,将 $b$ 更新为 $x$,并将 $f(b)$ 更新为 $f(x)$。如果超过最大迭代次数仍未找到合适的根,则抛出警告。
将这段代码转换为伪代码:def levenberg_marquardt(fun, grad, jacobian, x0, iterations, tol): """ Minimization of scalar function of one or more variables using the Levenberg-Marquardt algorithm. Parameters ---------- fun : function Objective function. grad : function Gradient function of objective function. jacobian :function function of objective function. x0 : numpy.array, size=9 Initial value of the parameters to be estimated. iterations : int Maximum iterations of optimization algorithms. tol : float Tolerance of optimization algorithms. Returns ------- xk : numpy.array, size=9 Parameters wstimated by optimization algorithms. fval : float Objective function value at xk. grad_val : float Gradient value of objective function at xk. grad_log : numpy.array The record of gradient of objective function of each iteration. """ fval = None # y的最小值 grad_val = None # 梯度的最后一次下降的值 x_log = [] # x的迭代值的数组,n*9,9个参数 y_log = [] # y的迭代值的数组,一维 grad_log = [] # 梯度下降的迭代值的数组 x0 = asarray(x0).flatten() if x0.ndim == 0: x0.shape = (1,) # iterations = len(x0) * 200 k = 1 xk = x0 updateJ = 1 lamda = 0.01 old_fval = fun(x0) gfk = grad(x0) gnorm = np.amax(np.abs(gfk)) J = [None] H = [None] while (gnorm > tol) and (k < iterations): if updateJ == 1: x_log = np.append(x_log, xk.T) yk = fun(xk) y_log = np.append(y_log, yk) J = jacobian(x0) H = np.dot(J.T, J) H_lm = H + (lamda * np.eye(9)) gfk = grad(xk) pk = - np.linalg.inv(H_lm).dot(gfk) pk = pk.A.reshape(1, -1)[0] # 二维变一维 xk1 = xk + pk fval = fun(xk1) if fval < old_fval: lamda = lamda / 10 xk = xk1 old_fval = fval updateJ = 1 else: updateJ = 0 lamda = lamda * 10 gnorm = np.amax(np.abs(gfk)) k = k + 1 grad_log = np.append(grad_log, np.linalg.norm(xk - x_log[-1:])) fval = old_fval grad_val = grad_log[-1] return xk, fval, grad_val, x_log, y_log, grad_log
伪代码如下:
function levenberg_marquardt(fun, grad, jacobian, x0, iterations, tol):
fval = None
grad_val = None
x_log = []
y_log = []
grad_log = []
x0 = asarray(x0).flatten()
if x0.ndim == 0:
x0.shape = (1,)
k = 1
xk = x0
updateJ = 1
lamda = 0.01
old_fval = fun(x0)
gfk = grad(x0)
gnorm = np.amax(np.abs(gfk))
J = None
H = None
while (gnorm > tol) and (k < iterations):
if updateJ == 1:
x_log = np.append(x_log, xk.T)
yk = fun(xk)
y_log = np.append(y_log, yk)
J = jacobian(x0)
H = np.dot(J.T, J)
H_lm = H + (lamda * np.eye(9))
gfk = grad(xk)
pk = - np.linalg.inv(H_lm).dot(gfk)
pk = pk.A.reshape(1, -1)[0]
xk1 = xk + pk
fval = fun(xk1)
if fval < old_fval:
lamda = lamda / 10
xk = xk1
old_fval = fval
updateJ = 1
else:
updateJ = 0
lamda = lamda * 10
gnorm = np.amax(np.abs(gfk))
k = k + 1
grad_log = np.append(grad_log, np.linalg.norm(xk - x_log[-1:]))
fval = old_fval
grad_val = grad_log[-1]
return xk, fval, grad_val, x_log, y_log, grad_log