def _omorf(self, s_old, del_k, del_min, eta1, eta2, gam1, gam2, omega_s,
max_evals, random_initial, epsilon, d, subspace_method):
"""
Computes optimum using the ``omorf`` method
"""
self.n = s_old.size
self.s_old = self._apply_scaling(s_old)
if del_k is None:
if self.bounds is None:
self.del_k = 0.1 * max(np.linalg.norm(self.s_old, ord=np.inf),
1.0)
else:
self.del_k = 0.1
else:
self.del_k = del_k
self._update_bounds()
self.f_old = self._blackbox_evaluation(self.s_old)
self.d = d
self.q = int(comb(self.d + 2, 2))
self.p = self.n + 1
self.random_initial = random_initial
self.subspace_method = subspace_method
self.epsilon = epsilon
Base = Basis('total-order', orders=np.tile([2], self.d))
self.basis = Base.get_basis()[:, range(self.d - 1, -1, -1)]
itermax = 10000
# Construct the sample set
S_full, f_full = self._generate_initial_set()
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
for i in range(itermax):
# self._update_bounds()
if len(self.f) >= max_evals or self.del_k < del_min:
break
my_poly = self._build_model(S_red, f_red)
m_old = np.asscalar(my_poly.get_polyfit(np.dot(self.s_old,
self.U)))
s_new, m_new = self._compute_step(my_poly)
# Safety step implemented in BOBYQA
if np.linalg.norm(s_new - self.s_old,
ord=np.inf) < omega_s * self.del_k:
if max(np.linalg.norm(
S_full - self.s_old, axis=1,
ord=np.inf)) <= self.epsilon * self.del_k:
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
self.del_k *= gam1
elif max(np.linalg.norm(
S_red - self.s_old, axis=1,
ord=np.inf)) <= self.epsilon * self.del_k:
S_full, f_full = self._sample_set('improve',
S_full,
f_full,
full_space=True)
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
else:
S_red, f_red = self._sample_set('improve', S_red, f_red)
S_full, f_full = self._sample_set('improve',
S_full,
f_full,
full_space=True)
continue
if self.S.shape == np.unique(np.vstack((self.S, s_new)),
axis=0).shape:
ind_repeat = np.argmin(
np.linalg.norm(self.S - s_new, ord=np.inf, axis=1))
f_new = self.f[ind_repeat]
else:
f_new = self._blackbox_evaluation(s_new)
S_red = np.vstack((S_red, s_new))
f_red = np.vstack((f_red, f_new))
S_full = np.vstack((S_full, s_new))
f_full = np.vstack((f_full, f_new))
# Calculate trust-region factor
rho_k = (self.f_old - f_new) / (m_old - m_new)
self._choose_best(self.S, self.f)
self._update_bounds()
if len(self.f) >= max_evals or self.del_k < del_min:
break
if rho_k >= eta2:
S_red, f_red = self._sample_set('replace', S_red, f_red)
S_full, f_full = self._sample_set('replace', S_full, f_full)
self.del_k *= gam2
elif rho_k >= eta1:
S_red, f_red = self._sample_set('replace', S_red, f_red)
S_full, f_full = self._sample_set('replace', S_full, f_full)
else:
if max(np.linalg.norm(
S_full - self.s_old, axis=1,
ord=np.inf)) <= self.epsilon * self.del_k:
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
self.del_k *= gam1
elif max(np.linalg.norm(
S_red - self.s_old, axis=1,
ord=np.inf)) <= self.epsilon * self.del_k:
S_full, f_full = self._sample_set('improve',
S_full,
f_full,
full_space=True)
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
else:
S_red, f_red = self._sample_set('improve', S_red, f_red)
S_full, f_full = self._sample_set('improve',
S_full,
f_full,
full_space=True)
self.S = self._remove_scaling(self.S)
self._choose_best(self.S, self.f)
return self.s_old, self.f_old
def _well_poised_LU(self, S, f, S_hat, f_hat):
"""
Ensures the regression set is well-poised using the LU algorithm (proposed by Andrew Conn) for ``trust-region`` method
"""
# Poised constant of algorithm
psi = 1.0
# Generate natural monomial basis
Base = Basis('total-order', orders=np.tile([1], self.n))
basis = Base.get_basis()[:, range(self.n - 1, -1, -1)]
def natural_basis_function(x, basis):
phi = np.zeros(basis.shape[0])
for j in range(basis.shape[0]):
phi[j] = 1.0
for k in range(basis.shape[1]):
phi[j] *= (x[k]**basis[j, k]) / factorial(basis[j, k])
return phi
phi_function = lambda x: natural_basis_function(x, basis)
# Initialise U matrix of LU factorisation of M matrix (see Conn et al.)
U = np.zeros((self.p, self.p))
# Initialise the first row of U to the e1 basis vector which corresponds to solution with all zeros
U[0, 0] = 1.0
# Perform the LU factorisation algorithm for the rest of the points
for k in range(1, self.p):
v = np.zeros(self.p)
for j in range(k):
v[j] = -U[j, k] / U[j, j]
v[k] = 1.0
# If there are still points to choose from, find if points meet criterion. If so, use the index to choose
# point with given index to be next point in regression/interpolation set
if S_hat.size != 0:
M = self._natural_basis_matrix(S_hat, v, phi_function)
index2 = np.argmax(M)
if M[index2] < psi:
index2 = None
else:
index2 = None
# If index exists, choose the point with that index and delete it from possible choices
if index2 is not None:
s = S_hat[index2, :].flatten()
S = np.vstack((S, s))
f = np.vstack((f, f_hat[index2].flatten()))
S_hat = np.delete(S_hat, index2, 0)
f_hat = np.delete(f_hat, index2, 0)
phi = phi_function(s.flatten())
# If index doesn't exist, solve an optimisation point to find the point in the range which best satisfies criterion
else:
s = optimize.minimize(
lambda x: -abs(np.dot(v, phi_function(x.flatten()))),
np.zeros(self.n),
method='COBYLA',
constraints=[{
'type': 'ineq',
'fun': lambda x: 1.0 - x
}, {
'type': 'ineq',
'fun': lambda x: 1.0 + x
}],
options={'disp': False})['x'].flatten()
S = np.vstack((S, s))
f = np.vstack((f, np.array([np.inf])))
phi = phi_function(s.flatten())
# Update U factorisation in LU algorithm
U[k, k] = np.dot(v, phi)
for i in range(k + 1, self.p):
U[k, i] += phi[i]
for j in range(k):
U[k, i] -= (phi[j] * U[j, i]) / U[j, j]
return S, f, S_hat, f_hat
def _trust_region(self, s_old, del_k, del_min, eta1, eta2, gam1, gam2, omega_s, max_evals, random_initial, scale_bounds, epsilon):
"""
Computes optimum using the ``trust-region`` method
"""
itermax = 10000
self.n = s_old.size
self.q = int(comb(self.n+2, 2))
self.p = int(comb(self.n+2, 2))
self.random_initial = random_initial
self.scale_bounds = scale_bounds
self.epsilon = epsilon
Base = Basis('total-order', orders=np.tile([2], self.n))
self.basis = Base.get_basis()[:,range(self.n-1, -1, -1)]
self.s_old = self._apply_scaling(s_old)
self.f_old = self._blackbox_evaluation(self.s_old)
if del_k is None:
if self.bounds is None:
self.del_k = 0.1*max(np.linalg.norm(self.s_old, ord=np.inf), 1.0)
else:
self.del_k = 0.1
else:
self.del_k = del_k
self._update_bounds()
# Construct the sample set
S, f = self._generate_initial_set()
for i in range(itermax):
# print(self.s_old)
# print('-------------')
self._update_bounds()
if len(self.f) >= max_evals or self.del_k < del_min:
break
my_poly = self._build_model(S, f)
m_old = np.asscalar(my_poly.get_polyfit(self.s_old))
s_new, m_new = self._compute_step(my_poly)
# Safety step implemented in BOBYQA
if np.linalg.norm(s_new - self.s_old, ord=np.inf) < omega_s*self.del_k:
S, f = self._sample_set('improve', S, f)
if max(np.linalg.norm(S-self.s_old, axis=1, ord=np.inf)) <= self.epsilon*self.del_k:
self.del_k *= gam1
continue
elif self.S.shape == np.unique(np.vstack((self.S, s_new)), axis=0).shape:
ind_repeat = np.argmin(np.linalg.norm(self.S - s_new, ord=np.inf, axis=1))
f_new = self.f[ind_repeat]
else:
f_new = self._blackbox_evaluation(s_new)
S = np.vstack((S, s_new))
f = np.vstack((f, f_new))
# Calculate trust-region factor
rho_k = (self.f_old - f_new) / (m_old - m_new)
self._choose_best(self.S, self.f)
self._update_bounds()
if len(self.f) >= max_evals or self.del_k < del_min:
break
if rho_k >= eta2:
S, f = self._sample_set('replace', S, f)
self.del_k *= gam2
elif rho_k >= eta1:
S, f = self._sample_set('replace', S, f)
else:
if max(np.linalg.norm(S-self.s_old, axis=1, ord=np.inf)) <= self.epsilon*self.del_k:
S, f = self._sample_set('improve', S, f)
self.del_k *= gam1
else:
S, f = self._sample_set('improve', S, f)
self.S = self._remove_scaling(self.S)
self._choose_best(self.S, self.f)
return self.s_old, self.f_old