def Fletcher_Reeves_conj():
f = conj_f3
c,b,A = f()
x0 = np.matrix('10.;-5.')
g0 = A * x0 + b
v0 = -g0
#pdb.set_trace()
lamb0, f_x0 = golden_section_search(lambda k:f_value(f, x0 + k*v0), [0,2], 0.001)
x1 = x0 + lamb0 * v0
g1 = A*x1 + b
v1 = -g1 + np.dot(g1.T, g1)[0,0]/np.dot(g0.T, g0)[0,0] * v0
lamb1, f_x1 = golden_section_search(lambda k:f_value(f, x1 + k*v1), [0,2], 0.001)
x2 = x1 + lamb1 * v1
return x2, f_x1
def powell_conj():
'''
u1=[-11.14, -24.46]
u2=[-1.8, -0.28]
'''
x0 = np.matrix('20.;20.')
#v1,v2线性无关
v = np.matrix('1.,1.;-1.,1.')
c, b, A = f_powell()
u = np.matrix('0.,0.;0.,0.')
lamb = np.matrix('0.;0.')
id = 0
total = 0
while total < 3:
k, min_fk = golden_section_search(
lambda k: f_value(f_powell, x0 + k * v[:, 0]), [-100., 100], 0.001)
x1 = x0 + k * v[:, 0]
k, min_fk = golden_section_search(
lambda k: f_value(f_powell, x1 + k * v[:, 1]), [-100., 100], 0.001)
x2 = x1 + k * v[:, 1]
#找到u向量
u[:, id] = x2 - x0
k, min_fk = golden_section_search(
lambda k: f_value(f_powell, x2 + k * u[:, id]), [-100., 100],
0.001)
x0 = x2 + k * u[:, id]
v[:, 0] = v[:, 1]
v[:, 1] = u[:, id]
id = (id + 1) % len(lamb)
total += 1
conj = u[:, 0].T * A * u[:, 1]
#print "conj: ", conj
x_star = x0
return x_star, f_value(f_powell, x_star)
def conj_grandient_method_for_f2():
u1 = np.matrix('1.;0.')
u2 = np.matrix('0.;1.')
x0 = np.matrix('0.;0.')
def_field = [-1,1]
esplison = 0.005
c,b, A = f2()
'''
线性搜索用的一次函数, 参数为k
f = f(xi + kui)
'''
k1 = golden_section_search(lambda k:f_value(f2, x0 + k*u1), def_field, esplison)
x1 = x0 + k1[0] * x0
k2 = golden_section_search(lambda k:f_value(f2, x1 + k*u2), def_field, esplison)
x2 = x0 + k1[0] * u1 + k2[0] * u2
return x2, f_value(f2, x2)
def conj_grandient_method_for_f2_direct():
u1 = np.matrix('1.;0.')
u2 = np.matrix('0.;1.')
x0 = np.matrix('0.;0.')
c, b, A = f2()
lamb1 = -1. * (u1.T * (A * x0 + b)) / (u1.T * (A * u1))
lamb2 = -1. * (u2.T * (A * x0 + b)) / (u2.T * (A * u2))
x2 = x0 + lamb1[0, 0] * u1 + lamb2[0, 0] * u2
return x2, f_value(f2, x2)
def optimal_grandient_for_f1(f, x0, epsilon):
c, b, A = f()
x = x0
AA = A * A
AAA = A * A * A
f_deriv = A * x
while True:
#对纯二次函数，直接求解k的值，而不是用线性搜索方法。推导公式见上
k = -1. * (x.T * AA * x) / (x.T * AAA * x)
k = np.sum(k)
x_n = x + k * f_deriv
f_deriv_n = A * x_n
if np.sum(np.abs(f_deriv_n)) < epsilon:
break
x = x_n
f_deriv = f_deriv_n
return x, f_value(f, x)