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optMethods.py
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from numpy import *
# TODO: convergence check
def newton(start_point, obj_fun, modified=0, iterations=5):
# with second order information
x = start_point
track = x
k = 0
while k < iterations:
if modified > 0:
vI = modified * matrix([[1, 0], [0, 1]])
G_ = linalg.inv(obj_fun.G_x(x) + vI) # inverse?
else:
G_ = linalg.inv(obj_fun.G_x(x)) # inverse?
g_ = obj_fun.g_x(x)
delta = -1.0 * dot(G_, g_)
x = x + delta
track = concatenate((track, x), axis=1)
k += 1
return track
def BFGS(start_point, obj_fun, iteration=10, interpolation=0):
# with first order information
x = start_point
track = x
k = 0
#H = dot(obj_fun.g_x(x), obj_fun.g_x(x).T)
H = 1.0*eye(2)
gamma = 1.0
while k < iteration and linalg.norm(gamma) > 1e-10:
p = -dot(H, obj_fun.g_x(x))
if interpolation > 0:
alpha_k = _backtracking_line_search(obj_fun, x, p)
else:
alpha_k = _armijo_line_search(obj_fun, x, p)
s = alpha_k * p
g_k = obj_fun.g_x(x)
x = x + alpha_k * p
g_k_1 = obj_fun.g_x(x)
y = g_k_1 - g_k
z = dot(H, y)
sTy = dot(s.T, y)
if sTy > 0:
H += outer(s, s) * (sTy + dot(y.T, z))[0,0]/(sTy**2) \
- (outer(z, s) + outer(s, z))/sTy
track = concatenate((track, x), axis=1)
k += 1
return track
def quasi_newton(start_point, obj_fun, iteration=10, interpolation=0):
# with first order information
x = start_point
track = x
k = 0
#H = matrix([[.01, 0], [0, .01]])
H = 1.0*eye(2)
while k < iteration:
s = -1.0 * dot(H, obj_fun.g_x(x))
if interpolation > 0:
alpha_k = _backtracking_line_search(obj_fun, x, s)
else:
alpha_k = _armijo_line_search(obj_fun, x, s)
delta = alpha_k * s
x_k_1 = x + delta
gamma = obj_fun.g_x(x_k_1) - obj_fun.g_x(x)
u = delta - dot(H, gamma)
scale_a = dot(u.T, gamma)
if scale_a == 0: # :(
scale_a = 0.000001
H = H + outer(u, u) / scale_a
track = concatenate((track, x), axis=1)
x = x_k_1
k += 1
return track
def fletcher_reeves(start_point, obj_fun, iteration=10, alpha=0.1):
x = start_point
track = x
k = 0
while k < iteration:
if k < 1:
g = obj_fun.g_x(x)
beta = 0.0
s = -g
else:
g = obj_fun.g_x(x)
beta = dot(g.T, g)/dot(g_old.T, g_old)
beta = beta[0,0]
s = -g + beta * s
alpha_k = _armijo_line_search(obj_fun, x, s, alpha0=alpha)
g_old = obj_fun.g_x(x)
x = x + alpha_k * s
track = concatenate((track, x), axis=1)
k += 1
return track
def _backtracking_line_search(obj_fun, x, s, c=1e-4):
# dummy
alpha = 1.0
p = 0.5 # magic number
c = 0.0001
diff = 1
f_alpha = obj_fun.about_alpha(x, s)
g_alpha = obj_fun.about_alpha_prime(x, s)
i = 0
while (diff > 0 and i < 50):
f_x = obj_fun.f_x(x)
diff = f_alpha(alpha) - f_alpha(0) - c * alpha * g_alpha(0)
alpha *= p
i += 1
return alpha
def _armijo_line_search(obj_fun, x, s, c=1e-4, alpha0=1.0):
# adapted from scipy.optimisation
amin = 0.0
f_alpha = obj_fun.about_alpha(x, s)
g_alpha = obj_fun.about_alpha_prime(x, s)
if(f_alpha(alpha0) <= f_alpha(0) + c * alpha0 * g_alpha(0)):
return alpha0
alpha1 = -(g_alpha(0)) * alpha0**2 / \
2.0 / (f_alpha(alpha0) - f_alpha(0) - g_alpha(0) * alpha0)
if(f_alpha(alpha1) <= f_alpha(0) + c * alpha1 * g_alpha(0)):
return alpha1
while alpha1 > amin:
factor = alpha0**2 * alpha1**2 * (alpha1 - alpha0)
a = alpha0**2 * (f_alpha(alpha1) - f_alpha(0) - g_alpha(0) * alpha1) - \
alpha1**2 * (f_alpha(alpha0) - f_alpha(0) - g_alpha(0) * alpha0)
a = a / factor
b = -alpha0**3 * (f_alpha(alpha1) - f_alpha(0) - g_alpha(0) * alpha1)+ \
alpha1**3 * (f_alpha(alpha0) - f_alpha(0) - g_alpha(0) * alpha0)
b = b / factor
alpha2 = (-b + sqrt(abs(b**2 - 3 * a * g_alpha(0)))) / (3.0 * a)
if(f_alpha(alpha2) <= f_alpha(0) + c * alpha2 * g_alpha(0)):
return alpha2
if(alpha1 - alpha2) > alpha1 / 2.0 or (1 - alpha2/alpha1) < 0.96:
alpha2 = alpha1 / 2.0
alpha0 = alpha1
alpha1 = alpha2
return 0.0001