def tringulation_search_bound_constantK(inter_par, xi, K, ind_min):
'''
This function is the core of constant-K continuous search function.
:param inter_par: Contains interpolation information w, v.
:param xi: The union of xE(Evaluated points) and xU(Support points)
:param K: Tuning parameter for constant-K, K = K*K0. K0 is the range of yE.
:param ind_min: The correspoding index of minimum of yE.
:return: The minimizer, xc, and minimum, yc, of continuous search function.
'''
inf = 1e+20
n = xi.shape[0]
# Delaunay Triangulation
if n == 1:
sx = sorted(range(xi.shape[1]), key=lambda x: xi[:, x])
tri = np.zeros((xi.shape[1] - 1, 2))
tri[:, 0] = sx[:xi.shape[1] - 1]
tri[:, 1] = sx[1:]
tri = tri.astype(np.int32)
else:
options = 'Qt Qbb Qc' if n <= 3 else 'Qt Qbb Qc Qx'
tri = Delaunay(xi.T, qhull_options=options).simplices
keep = np.ones(len(tri), dtype=bool)
for i, t in enumerate(tri):
if abs(np.linalg.det(np.hstack(
(xi.T[t], np.ones([1, n + 1]).T)))) < 1E-15:
keep[i] = False # Point is coplanar, we don't want to keep it
tri = tri[keep]
# Search the minimum of the synthetic quadratic model
Sc = np.zeros([np.shape(tri)[0]])
Scl = np.zeros([np.shape(tri)[0]])
for ii in range(np.shape(tri)[0]):
# R2-circumradius, xc-circumcircle center
R2, xc = Utils.circhyp(xi[:, tri[ii, :]], n)
# x is the center of the current simplex
x = np.dot(xi[:, tri[ii, :]], np.ones([n + 1, 1]) / (n + 1))
Sc[ii] = interpolation.interpolate_val(
x, inter_par) - K * (R2 - np.linalg.norm(x - xc)**2)
if np.sum(ind_min == tri[ii, :]):
Scl[ii] = np.copy(Sc[ii])
else:
Scl[ii] = inf
# Global one
t = np.min(Sc)
ind = np.argmin(Sc)
R2, xc = Utils.circhyp(xi[:, tri[ind, :]], n)
x = np.dot(xi[:, tri[ind, :]], np.ones([n + 1, 1]) / (n + 1))
xm, ym = Constant_K_Search(x, inter_par, xc, R2, K)
# Local one
t = np.min(Scl)
ind = np.argmin(Scl)
R2, xc = Utils.circhyp(xi[:, tri[ind, :]], n)
# Notice!! ind_min may have a problem as an index
x = np.copy(xi[:, ind_min].reshape(-1, 1))
xml, yml = Constant_K_Search(x, inter_par, xc, R2, K)
if yml < ym:
xm = np.copy(xml)
ym = np.copy(yml)
xm = xm.reshape(-1, 1)
ym = ym[0, 0]
return xm, ym
def Constant_K_Search(simplex, inter_par, K, lb=[], ub=[]):
n = simplex.shape[0]
R2, xc = Utils.circhyp(simplex, n)
x = np.dot(simplex, np.ones([n + 1, 1]) / (n + 1))
costfun = lambda x: Continuous_search_cost(x, inter_par, xc, R2, K)[0]
costjac = lambda x: Continuous_search_cost(x, inter_par, xc, R2, K)[1]
opt = {'disp': False}
bnds = tuple([(0, 1) for i in range(int(n))])
res = optimize.minimize(costfun,
x,
jac=costjac,
method='TNC',
bounds=bnds,
options=opt)
x = res.x
y = res.fun
return x, y
def continuous_search_plot1D(self, adogs):
'''
Plot the continuous search function.
:return:
'''
xU = Utils.bounds(adogs.lb, adogs.ub, adogs.n)
if adogs.iter_type == 'scmin' and Utils.mindis(adogs.xc, xU)[0] > 1e-6:
xi = np.hstack((adogs.xE[:, :-1], adogs.xU))
else:
xi = np.hstack((adogs.xE, adogs.xU))
sx = sorted(range(xi.shape[1]), key=lambda x: xi[:, x])
tri = np.zeros((xi.shape[1] - 1, 2))
tri[:, 0] = sx[:xi.shape[1] - 1]
tri[:, 1] = sx[1:]
tri = tri.astype(np.int32)
num_plot_points = 2000
xe_plot = np.zeros((tri.shape[0], num_plot_points))
e_plot = np.zeros((tri.shape[0], num_plot_points))
sc_plot = np.zeros((tri.shape[0], num_plot_points))
for ii in range(len(tri)):
simplex_range = np.copy(xi[:, tri[ii, :]])
# Discretized mesh grid on x direction in one simplex
x = np.linspace(simplex_range[0, 0], simplex_range[0, 1],
num_plot_points)
# Circumradius and circumcenter for current simplex
R2, xc = Utils.circhyp(xi[:, tri[ii, :]], adogs.n)
for jj in range(len(x)):
# Interpolation p(x)
p = adogs.inter_par.inter_val(x[jj])
# Uncertainty function e(x)
e_plot[ii, jj] = (R2 - np.linalg.norm(x[jj] - xc)**2)
# Continuous search function s(x)
sc_plot[ii, jj] = p - adogs.K * adogs.K0 * e_plot[ii, jj]
xe_plot[ii, :] = np.copy(x)
for i in range(len(tri)):
# Plot the uncertainty function e(x)
# plt.plot(xe_plot[i, :], e_plot[i, :] - 5.5, c='g', label=r'$e(x)$')
# Plot the continuous search function sc(x)
plt.plot(xe_plot[i, :],
sc_plot[i, :],
'r--',
zorder=20,
label=r'$S_c(x)$')
yc_min = sc_plot.flat[np.abs(xe_plot - adogs.xc).argmin()]
plt.scatter(adogs.xc,
yc_min,
c=(1, 0.769, 0.122),
marker='D',
zorder=15,
label=r'min $S_c(x)$')
def safe_continuous_constantK_search_1d_plot(xE, xU, fun_eval, safe_eval,
L_safe, K, xc_min, Nm):
'''
Given evaluated points set xE, and the objective function. Plot the
interpolation, uncertainty function and continuous search function.
:param xE:
:param xU:
:param fun_eval:
:param safe_eval:
:param L_safe:
:param K:
:param xc_min:
:param Nm:
:return:
'''
N = xE.shape[1]
yE = np.zeros(N)
for i in range(N):
yE[i] = fun_eval(xE[:, i])
inter_par = interpolation.Inter_par(method='NPS')
inter_par, _ = interpolation.interpolateparameterization(xE, yE, inter_par)
x = np.linspace(0, 1, 1000)
y, yp = np.zeros(x.shape), np.zeros(x.shape)
for i in range(x.shape[0]):
y[i] = fun_eval(x[i])
yp[i] = interpolation.interpolate_val(x[i], inter_par)
xi = np.hstack((xE, xU))
sx = sorted(range(xi.shape[1]), key=lambda x: xi[:, x])
tri = np.zeros((xi.shape[1] - 1, 2))
tri[:, 0] = sx[:xi.shape[1] - 1]
tri[:, 1] = sx[1:]
tri = tri.astype(np.int32)
xe_plot = np.zeros((tri.shape[0], 2000))
e_plot = np.zeros((tri.shape[0], 2000))
sc_plot = np.zeros((tri.shape[0], 2000))
n = xE.shape[0]
for ii in range(len(tri)):
temp_x = np.copy(xi[:, tri[ii, :]])
simplex = xi[:, tri[ii, :]]
# Determine if the boundary corner exists or not in simplex
exist = 0
for i in range(simplex.shape[1]):
vertice = simplex[:, i].reshape(-1, 1)
val, _, _ = Utils.mindis(vertice, xE)
if val == 0:
pass
else:
exist = 1
break
x_ = np.linspace(temp_x[0, 0], temp_x[0, 1], 2000)
temp_Sc = np.zeros(len(x_))
temp_e = np.zeros(len(x_))
p = np.zeros(len(x_))
R2, xc = Utils.circhyp(xi[:, tri[ii, :]], n)
for jj in range(len(x_)):
p[jj] = interpolation.interpolate_val(x_[jj], inter_par)
if exist == 0:
temp_e[jj] = (R2 - np.linalg.norm(x_[jj] - xc)**2)
else:
val, _, _ = Utils.mindis(x_[jj], xE)
temp_e[jj] = val**(2)
# val, idx, vertex = Utils.mindis(x_[jj], xE)
# c = 0.1
# temp_e[jj] = (val + c) ** (1/2) - c ** (1/2)
temp_Sc[jj] = p[jj] - K * temp_e[jj]
e_plot[ii, :] = temp_e
xe_plot[ii, :] = x_
sc_plot[ii, :] = temp_Sc
# The minimizer of continuous search must be subject to safe constraints.
# sc_min = np.min(sc_plot, axis=1)
# index_r = np.argmin(sc_min)
# index_c = np.argmin(sc_plot[index_r, :])
# sc_min_x = xe_plot[index_r, index_c]
# sc_min = min(np.min(sc_plot, axis=1))
safe_plot = {}
## plot the safe region
for ii in range(xE.shape[1]):
y_safe = safe_eval(xE[:, ii])
safe_index = []
y_safe_plot = []
safe_eval_lip = lambda x: y_safe - L_safe * np.sqrt(
np.dot((x - xE[:, ii]).T, x - xE[:, ii]))
for i in range(x.shape[0]):
safe_val = safe_eval_lip(x[i])
y_safe_plot.append(safe_val[0])
if safe_val > 0:
safe_index.append(i)
name = str(ii)
safe_plot[name] = [safe_index, y_safe_plot]
# ================== First plot =================
fig = plt.figure()
plt.subplot(2, 1, 1)
# plot the essentials for DeltaDOGS
plt.plot(x, y, c='k')
plt.plot(x, yp, c='b')
for i in range(len(tri)):
if i == 0 or i == len(tri) - 1:
amplify_factor = 3
else:
amplify_factor = 50
plt.plot(xe_plot[i, :], sc_plot[i, :], c='r')
plt.plot(xe_plot[i, :], amplify_factor * e_plot[i, :] - 5.5, c='g')
plt.scatter(xE, yE, c='b', marker='s')
yc_min = sc_plot.flat[np.abs(xe_plot - xc_min).argmin()]
plt.scatter(xc_min, yc_min, c='r', marker='^')
# plot the safe region in cyan
xlow, xupp = safe_region_plot(x, safe_plot)
y_vertical = np.linspace(-2, 2, 100)
xlow_y_vertical = xlow * np.ones(100)
xupp_y_vertical = xupp * np.ones(100)
plt.plot(xlow_y_vertical, y_vertical, color='cyan', linestyle='--')
plt.plot(xupp_y_vertical, y_vertical, color='cyan', linestyle='--')
plt.ylim(-6.5, 3.5)
plt.gca().axes.get_yaxis().set_visible(False)
# ================== Second plot =================
plt.subplot(2, 1, 2)
y_safe_all = np.zeros(x.shape)
for i in range(x.shape[0]):
y_safe_all[i] = safe_eval(x[i])
plt.plot(x, y_safe_all)
zero_indicator = np.zeros(x.shape)
plt.plot(x, zero_indicator, c='k')
x_scatter = np.hstack((xE, xc_min))
y_scatter = np.zeros(x_scatter.shape[1])
for i in range(x_scatter.shape[1]):
ind = np.argmin(np.abs(x_scatter[:, i] - x))
y_scatter[i] = y_safe_all[ind]
plt.scatter(x_scatter[:, :-1][0], y_scatter[:-1], c='b', marker='s')
plt.scatter(x_scatter[:, -1], y_scatter[-1], c='r', marker='^')
# plot the safe region
xlow, xupp = safe_region_plot(x, safe_plot)
low_idx = np.argmin(np.abs(xlow - x))
upp_idx = np.argmin(np.abs(xupp - x))
ylow_vertical = np.linspace(y_safe_all[low_idx], 2, 100)
yupp_vertical = np.linspace(y_safe_all[upp_idx], 2, 100)
xlow_y_vertical = xlow * np.ones(100)
xupp_y_vertical = xupp * np.ones(100)
plt.plot(xlow_y_vertical, ylow_vertical, color='cyan', linestyle='--')
plt.plot(xupp_y_vertical, yupp_vertical, color='cyan', linestyle='--')
plt.ylim(-1, 2.2)
plt.gca().axes.get_yaxis().set_visible(False)
# plt.show()
current_path = os.path.dirname(
os.path.abspath(inspect.getfile(inspect.currentframe())))
plot_folder = current_path[:-5] + "/plot/DDOGS/0"
num_iter = xE.shape[1] - 1 + math.log(Nm / 8, 2)
plt.savefig(plot_folder + '/pic' + str(int(num_iter)) + '.png',
format='png',
dpi=250)
plt.close(fig)
return
def continuous_search_1d_plot(xE, fun_eval):
'''
Given evaluated points set xE, and the objective function. Plot the
interpolation, uncertainty function and continuous search function.
:param xE: evaluated points set xE
:param fun_eval: obj function
:return:
'''
N = xE.shape[1]
yE = np.zeros(N)
for i in range(N):
yE[i] = fun_eval(xE[:, i])
inter_par = interpolation.Inter_par(method='NPS')
inter_par, _ = interpolation.interpolateparameterization(xE, yE, inter_par)
x = np.linspace(0, 1, 1000)
y, yp = np.zeros(x.shape), np.zeros(x.shape)
for i in range(x.shape[0]):
y[i] = fun_eval(x[i])
yp[i] = interpolation.interpolate_val(x[i], inter_par)
xi = np.copy(xE)
sx = sorted(range(xi.shape[1]), key=lambda x: xi[:, x])
tri = np.zeros((xi.shape[1] - 1, 2))
tri[:, 0] = sx[:xi.shape[1] - 1]
tri[:, 1] = sx[1:]
tri = tri.astype(np.int32)
xe = np.copy(xi)
xe_plot = np.zeros((tri.shape[0], 2000))
e_plot = np.zeros((tri.shape[0], 2000))
sc_plot = np.zeros((tri.shape[0], 2000))
n = xE.shape[0]
K = 3
for ii in range(len(tri)):
temp_x = np.copy(xi[:, tri[ii, :]])
x_ = np.linspace(temp_x[0, 0], temp_x[0, 1], 2000)
temp_Sc = np.zeros(len(x_))
temp_e = np.zeros(len(x_))
p = np.zeros(len(x_))
R2, xc = Utils.circhyp(xi[:, tri[ii, :]], n)
for jj in range(len(x_)):
p[jj] = interpolation.interpolate_val(x_[jj], inter_par)
temp_e[jj] = (R2 - np.linalg.norm(x_[jj] - xc)**2)
temp_Sc[jj] = p[jj] - K * temp_e[jj]
e_plot[ii, :] = temp_e
xe_plot[ii, :] = x_
sc_plot[ii, :] = temp_Sc
sc_min = np.min(sc_plot, axis=1)
index_r = np.argmin(sc_min)
index_c = np.argmin(sc_plot[index_r, :])
sc_min_x = xe_plot[index_r, index_c]
sc_min = min(np.min(sc_plot, axis=1))
plt.figure()
plt.plot(x, y, c='k')
plt.plot(x, yp, c='b')
for i in range(len(tri)):
plt.plot(xe_plot[i, :], sc_plot[i, :], c='r')
plt.plot(xe_plot[i, :], 3 * e_plot[i, :] - 2.3, c='g')
plt.scatter(xE, yE, c='b', marker='s')
plt.scatter(sc_min_x, sc_min, c='r', marker='s')
plt.ylim(-2.5, 2)
# plt.gca().axes.get_yaxis().set_visible(False)
plt.show()
return
def triangulation_search_bound_snopt(inter_par, xi, K, ind_min, y_safe,
L_safe):
# reddir is a vector
inf = 1e+20
n = xi.shape[0] # The dimension of the reduced model.
xE = inter_par.xi
# 0: Build up the Delaunay triangulation based on reduced subspace.
if n == 1:
sx = sorted(range(xi.shape[1]), key=lambda x: xi[:, x])
tri = np.zeros((xi.shape[1] - 1, 2))
tri[:, 0] = sx[:xi.shape[1] - 1]
tri[:, 1] = sx[1:]
tri = tri.astype(np.int32)
else:
options = 'Qt Qbb Qc' if n <= 3 else 'Qt Qbb Qc Qx'
tri = Delaunay(xi.T, qhull_options=options).simplices
keep = np.ones(len(tri), dtype=bool)
for i, t in enumerate(tri):
if abs(np.linalg.det(np.hstack(
(xi.T[t], np.ones([1, n + 1]).T)))) < 1E-15:
keep[i] = False # Point is coplanar, we don't want to keep it
tri = tri[keep]
# Sc contains the continuous search function value of the center of each Delaunay simplex
# 1: Identify the minimizer of adaptive K continuous search function
Sc = np.zeros([np.shape(tri)[0]])
Scl = np.zeros([np.shape(tri)[0]])
for ii in range(np.shape(tri)[0]):
R2, xc = Utils.circhyp(xi[:, tri[ii, :]], n)
if R2 < inf:
# initialize with body center of each simplex
x = np.dot(xi[:, tri[ii, :]], np.ones([n + 1, 1]) / (n + 1))
exist = unevaluated_vertices_identification(xE, xi[:, tri[ii, :]])
if exist == 0:
Sc[ii] = interpolation.interpolate_val(
x, inter_par) - K * (R2 - np.linalg.norm(x - xc)**2)
else:
val, idx, x_nn = Utils.mindis(x, xE)
Sc[ii] = interpolation.interpolate_val(x,
inter_par) - K * val**2
# discrete min
# val, idx, vertex = Utils.mindis(x, xE)
# c = 0.1
# e = (val + c) ** (1/2) - c ** (1/2)
# Sc[ii] = interpolation.interpolate_val(x, inter_par) - K * e
if np.sum(ind_min == tri[ii, :]):
Scl[ii] = np.copy(Sc[ii])
else:
Scl[ii] = inf
else:
Scl[ii] = inf
Sc[ii] = inf
# Global one, the minimum of Sc has the minimum value of all circumcenters.
ind = np.argmin(Sc)
xm, ym = constantk_search_snopt_min(xi[:, tri[ind, :]], inter_par, K,
y_safe, L_safe)
# Local one
ind = np.argmin(Scl)
xml, yml = constantk_search_snopt_min(xi[:, tri[ind, :]], inter_par, K,
y_safe, L_safe)
if yml < ym:
xm = np.copy(xml)
ym = np.copy(yml)
result = 'local'
else:
result = 'glob'
return xm, ym, result
def constantk_search_snopt_min(simplex, inter_par, K, y_safe, L_safe):
'''
The function F is composed as: 1st - objective
2nd to nth - simplex bounds
n+1 th .. - safe constraints
:param x0 : The mass-center of delaunay simplex
:param inter_par :
:param xc :
:param R2:
:param y0:
:param K0:
:param A_simplex:
:param b_simplex:
:param lb_simplex:
:param ub_simplex:
:param y_safe:
:param L_safe:
:return:
'''
xE = inter_par.xi
n = xE.shape[0]
# Determine if the boundary corner exists in simplex, if boundary corner detected:
# e(x) = || x - x' ||^2_2
# else, e(x) is the regular uncertainty function.
exist = unevaluated_vertices_identification(xE, simplex)
# ------- ADD THE FOLLOWING LINE WHEN DEBUGGING --------
# simplex = xi[:, tri[ind, :]]
# ------- ADD THE FOLLOWING LINE WHEN DEBUGGING --------
# Find the minimizer of the search fucntion in a simplex using SNOPT package.
R2, xc = Utils.circhyp(simplex, n)
# x is the center of this simplex
x = np.dot(simplex, np.ones([n + 1, 1]) / (n + 1))
# First find minimizer xr on reduced model, then find the 2D point corresponding to xr. Constrained optm.
A_simplex, b_simplex = Utils.search_simplex_bounds(simplex)
lb_simplex = np.min(simplex, axis=1)
ub_simplex = np.max(simplex, axis=1)
inf = 1.0e+20
m = n + 1 # The number of constraints which is determined by the number of simplex boundaries.
assert m == A_simplex.shape[0], 'The No. of simplex constraints is wrong'
# TODO: multiple safe constraints in future.
# nF: The number of problem functions in F(x), including the objective function, linear and nonlinear constraints.
# ObjRow indicates the number of objective row in F(x).
ObjRow = 1
# solve for constrained minimization of safe learning within each open ball of the vertices of simplex.
# Then choose the one with the minimum continuous function value.
x_solver = np.empty(shape=[n, 0])
y_solver = []
for i in range(n + 1):
vertex = simplex[:, i].reshape(-1, 1)
# First find the y_safe[vertex]:
val, idx, x_nn = Utils.mindis(vertex, xE)
if val > 1e-10:
# This vertex is a boundary corner point. No safe-guarantee, we do not optimize around support points.
continue
else:
# TODO: multiple safe constraints in future.
safe_bounds = y_safe[idx]
if n > 1:
# The first function in F(x) is the objective function, the rest are m simplex constraints.
# The last part of functions in F(x) is the safe constraints.
# In high dimension, A_simplex make sure that linear_derivative_A won't be all zero.
nF = 1 + m + 1 # the last 1 is the safe constraint.
Flow = np.hstack((-inf, b_simplex.T[0], -safe_bounds))
Fupp = inf * np.ones(nF)
# The lower and upper bounds of variables x.
xlow = np.copy(lb_simplex)
xupp = np.copy(ub_simplex)
# For the nonlinear components, enter any nonzero value in G to indicate the location
# of the nonlinear derivatives (in this case, 2).
# A must be properly defined with the correct derivative values.
linear_derivative_A = np.vstack((np.zeros(
(1, n)), A_simplex, np.zeros((1, n))))
nonlinear_derivative_G = np.vstack((2 * np.ones(
(1, n)), np.zeros((m, n)), 2 * np.ones((1, n))))
else:
# For 1D problem, the simplex constraint is defined in x bounds.
# TODO multiple safe cons.
# 2 = 1 obj + 1 safe con. Plus one redundant constraint to make matrix A suitable.
nF = 2 + 1
Flow = np.array([-inf, -safe_bounds, -inf])
Fupp = np.array([inf, inf, inf])
xlow = np.min(simplex) * np.ones(n)
xupp = np.max(simplex) * np.ones(n)
linear_derivative_A = np.vstack((np.zeros(
(1, n)), np.zeros((1, n)), np.ones((1, n))))
nonlinear_derivative_G = np.vstack((2 * np.ones(
(2, n)), np.zeros((1, n))))
x0 = x.T[0]
# ------- ADD THE FOLLOWING LINE WHEN DEBUGGING --------
# cd dogs
# ------- ADD THE FOLLOWING LINE WHEN DEBUGGING --------
save_opt_for_snopt_ck(n, nF, inter_par, xc, R2, K, A_simplex,
L_safe, vertex, exist)
# Since adaptiveK using ( p(x) - f0 ) / e(x), the objective function is nonlinear.
# The constraints are generated by simplex bounds, all linear.
options = SNOPT_options()
options.setOption('Infinite bound', inf)
options.setOption('Verify level', 3)
options.setOption('Verbose', False)
options.setOption('Print level', -1)
options.setOption('Scale option', 2)
options.setOption('Print frequency', -1)
options.setOption('Summary', 'No')
result = snopta(dogsobj,
n,
nF,
x0=x0,
name='DeltaDOGS_snopt',
xlow=xlow,
xupp=xupp,
Flow=Flow,
Fupp=Fupp,
ObjRow=ObjRow,
A=linear_derivative_A,
G=nonlinear_derivative_G,
options=options)
x_solver = np.hstack((x_solver, result.x.reshape(-1, 1)))
y_solver.append(result.objective)
y_solver = np.array(y_solver)
y = np.min(y_solver)
x = x_solver[:, np.argmin(y_solver)].reshape(-1, 1)
return x, y
def tringulation_search_bound(inter_par, xi, y0, K0, ind_min):
inf = 1e+20
n = xi.shape[0]
xm, ym = interpolation.inter_min(xi[:, ind_min], inter_par)
sc_min = inf
# cse=1
if ym > y0:
ym = inf
# cse =2
# construct Deluanay tringulation
if n == 1:
sx = sorted(range(xi.shape[1]), key=lambda x: xi[:, x])
tri = np.zeros((xi.shape[1] - 1, 2))
tri[:, 0] = sx[:xi.shape[1] - 1]
tri[:, 1] = sx[1:]
tri = tri.astype(np.int32)
else:
options = 'Qt Qbb Qc' if n <= 3 else 'Qt Qbb Qc Qx'
tri = Delaunay(xi.T, qhull_options=options).simplices
keep = np.ones(len(tri), dtype=bool)
for i, t in enumerate(tri):
if abs(np.linalg.det(np.hstack(
(xi.T[t], np.ones([1, n + 1]).T)))) < 1E-15:
keep[i] = False # Point is coplanar, we don't want to keep it
tri = tri[keep]
Sc = np.zeros([np.shape(tri)[0]])
Scl = np.zeros([np.shape(tri)[0]])
for ii in range(np.shape(tri)[0]):
R2, xc = Utils.circhyp(xi[:, tri[ii, :]], n)
# if R2 != np.inf:
if R2 < inf:
# initialze with body center of each simplex
x = np.dot(xi[:, tri[ii, :]], np.ones([n + 1, 1]) / (n + 1))
Sc[ii] = (interpolation.interpolate_val(x, inter_par) -
y0) / (R2 - np.linalg.norm(x - xc)**2)
if np.sum(ind_min == tri[ii, :]):
Scl[ii] = np.copy(Sc[ii])
else:
Scl[ii] = inf
else:
Scl[ii] = inf
Sc[ii] = inf
# Global one
if np.min(Sc) < 0:
func = 'p'
# The minimum of Sc is negative, minimize p(x) instead.
Scp = np.zeros(tri.shape[0])
Scpl = np.zeros(tri.shape[0])
for ii in range(tri.shape[0]):
x = np.dot(xi[:, tri[ii, :]], np.ones([n + 1, 1]) / (n + 1))
Scp[ii] = interpolation.interpolate_val(x, inter_par)
if np.sum(ind_min == tri[ii, :]):
Scpl[ii] = np.copy(Scp[ii])
else:
Scpl[ii] = inf
else:
Scpl[ii] = inf
Scp[ii] = inf
# Globally minimize p(x)
ind = np.argmin(Scp)
x = np.dot(xi[:, tri[ind, :]], np.ones([n + 1, 1]) / (n + 1))
simplex_bnds = Utils.search_bounds(xi[:, tri[ind, :]])
xm, ym = AdaptiveK_Search_p(x, inter_par, simplex_bnds)
# Locally minimize p(x)
ind = np.argmin(Scpl)
x = np.dot(xi[:, tri[ind, :]], np.ones([n + 1, 1]) / (n + 1))
simplex_bnds = Utils.search_bounds(xi[:, tri[ind, :]])
xml, yml = AdaptiveK_Search_p(x, inter_par, simplex_bnds)
else:
func = 'sc'
# Minimize sc(x).
# Global one, the minimum of Sc has the minimum value of all circumcenters.
ind = np.argmin(Sc)
R2, xc = Utils.circhyp(xi[:, tri[ind, :]], n)
x = np.dot(xi[:, tri[ind, :]], np.ones([n + 1, 1]) / (n + 1))
simplex_bnds = Utils.search_bounds(xi[:, tri[ind, :]])
xm, ym = Adaptive_K_Search(x, inter_par, xc, R2, y0, K0, simplex_bnds)
# Local one
ind = np.argmin(Scl)
R2, xc = Utils.circhyp(xi[:, tri[ind, :]], n)
x = np.dot(xi[:, tri[ind, :]], np.ones([n + 1, 1]) / (n + 1))
simplex_bnds = Utils.search_bounds(xi[:, tri[ind, :]])
xml, yml = Adaptive_K_Search(x, inter_par, xc, R2, y0, K0,
simplex_bnds)
if yml < ym:
xm = np.copy(xml)
ym = np.copy(yml)
result = 'local'
else:
result = 'glob'
xm = xm.reshape(-1, 1)
ym = ym[0, 0]
return xm, ym, result, func
def triangulation_search_bound_snopt(inter_par, xi, y0, ind_min, y_safe,
L_safe):
# reddir is a vector
inf = 1e+20
n = xi.shape[0] # The dimension of the reduced model.
# 0: Build up the Delaunay triangulation based on reduced subspace.
if n == 1:
sx = sorted(range(xi.shape[1]), key=lambda x: xi[:, x])
tri = np.zeros((xi.shape[1] - 1, 2))
tri[:, 0] = sx[:xi.shape[1] - 1]
tri[:, 1] = sx[1:]
tri = tri.astype(np.int32)
else:
options = 'Qt Qbb Qc' if n <= 3 else 'Qt Qbb Qc Qx'
tri = Delaunay(xi.T, qhull_options=options).simplices
keep = np.ones(len(tri), dtype=bool)
for i, t in enumerate(tri):
if abs(np.linalg.det(np.hstack(
(xi.T[t], np.ones([1, n + 1]).T)))) < 1E-15:
keep[i] = False # Point is coplanar, we don't want to keep it
tri = tri[keep]
# Sc contains the continuous search function value of the center of each Delaunay simplex
# 1: Identify the minimizer of adaptive K continuous search function
Sc = np.zeros([np.shape(tri)[0]])
Scl = np.zeros([np.shape(tri)[0]])
for ii in range(np.shape(tri)[0]):
R2, xc = Utils.circhyp(xi[:, tri[ii, :]], n)
if R2 < inf:
# initialize with body center of each simplex
x = np.dot(xi[:, tri[ii, :]], np.ones([n + 1, 1]) / (n + 1))
Sc[ii] = (interpolation.interpolate_val(x, inter_par) -
y0) / (R2 - np.linalg.norm(x - xc)**2)
if np.sum(ind_min == tri[ii, :]):
Scl[ii] = np.copy(Sc[ii])
else:
Scl[ii] = inf
else:
Scl[ii] = inf
Sc[ii] = inf
if np.min(Sc) < 0:
func = 'p'
# The minimum of Sc is negative, minimize p(x) instead.
Scp = np.zeros(tri.shape[0])
for ii in range(tri.shape[0]):
x = np.dot(xi[:, tri[ii, :]], np.ones([n + 1, 1]) / (n + 1))
Scp[ii] = interpolation.interpolate_val(x, inter_par)
# Globally minimize p(x)
ind = np.argmin(Scp)
x = np.dot(xi[:, tri[ind, :]], np.ones([n + 1, 1]) / (n + 1))
xm, ym = adaptiveK_p_snopt_min(x, inter_par, y_safe, L_safe)
result = 'glob'
else:
func = 'sc'
# Global one, the minimum of Sc has the minimum value of all circumcenters.
ind = np.argmin(Sc)
R2, xc = Utils.circhyp(xi[:, tri[ind, :]], n)
# x is the center of this simplex
x = np.dot(xi[:, tri[ind, :]], np.ones([n + 1, 1]) / (n + 1))
# First find minimizer xr on reduced model, then find the 2D point corresponding to xr. Constrained optm.
A_simplex, b_simplex = Utils.search_simplex_bounds(xi[:, tri[ind, :]])
lb_simplex = np.min(xi[:, tri[ind, :]], axis=1)
ub_simplex = np.max(xi[:, tri[ind, :]], axis=1)
xm, ym = adaptiveK_search_snopt_min(x, inter_par, xc, R2, y0, K0,
A_simplex, b_simplex, lb_simplex,
ub_simplex, y_safe, L_safe)
# Local one
ind = np.argmin(Scl)
R2, xc = Utils.circhyp(xi[:, tri[ind, :]], n)
x = np.dot(xi[:, tri[ind, :]], np.ones([n + 1, 1]) / (n + 1))
A_simplex, b_simplex = Utils.search_simplex_bounds(xi[:, tri[ind, :]])
lb_simplex = np.min(xi[:, tri[ind, :]], axis=1)
ub_simplex = np.max(xi[:, tri[ind, :]], axis=1)
xml, yml = adaptiveK_search_snopt_min(x, inter_par, xc, R2, y0, K0,
A_simplex, b_simplex, lb_simplex,
ub_simplex, y_safe, L_safe)
if yml < ym:
xm = np.copy(xml)
ym = np.copy(yml)
result = 'local'
else:
result = 'glob'
return xm, ym, result, func