def unique_equalityset(C, D, b, af, bf, abs_tol=1e-7, verbose=0):
"""Return equality set E with the following property:
P_E = {x | af x = bf} intersection P
where P is the polytope C x + D y < b
The inequalities have to be satisfied with equality everywhere on
the face defined by af and bf.
"""
if D is not None:
A = np.hstack([C, D])
a = np.hstack([af, np.zeros(D.shape[1])])
else:
A = C
a = af
E = []
for i in range(A.shape[0]):
A_i = np.array(A[i, :])
b_i = b[i]
sol = solvers._solve_lp_using_cvxopt(c=A_i, G=A, h=b, A=a.T, b=bf)
if sol['status'] != "optimal":
raise Exception("unique_equalityset: LP returned status " +
str(sol['status']))
if np.abs(sol['primal objective'] - b_i) < abs_tol:
# Constraint is active everywhere
E.append(i)
if len(E) == 0:
raise Exception("unique_equalityset: empty E")
return np.array(E)
def unique_equalityset(C, D, b, af, bf, abs_tol=1e-7, verbose=0):
"""Return equality set E with the following property:
P_E = {x | af x = bf} intersection P
where P is the polytope C x + D y < b
The inequalities have to be satisfied with equality everywhere on
the face defined by af and bf.
"""
if D is not None:
A = np.hstack([C, D])
a = np.hstack([af, np.zeros(D.shape[1])])
else:
A = C
a = af
E = []
for i in range(A.shape[0]):
A_i = np.array(A[i, :])
b_i = b[i]
sol = solvers._solve_lp_using_cvxopt(
c=A_i, G=A, h=b,
A=a.T, b=bf)
if sol['status'] != "optimal":
raise Exception(
"unique_equalityset: LP returned status " +
str(sol['status']))
if np.abs(sol['primal objective'] - b_i) < abs_tol:
# Constraint is active everywhere
E.append(i)
if len(E) == 0:
raise Exception("unique_equalityset: empty E")
return np.array(E)
def is_dual_degenerate(c, G, h, A, b, x_opt, z_opt, abs_tol=1e-7):
"""Return `True` if pair of dual problems is dual degenerate.
Checks if the pair of dual problems::
(P): min c'x (D): max h'z + b'y
s.t Gx <= h s.t G'z + A'y = c
Ax = b z <= 0
is dual degenerate, i.e. if (P) has several optimal solutions.
Optimal solutions x* and z* are required.
Input:
`G,h,A,b`: Parameters of (P)
`x_opt`: One optimal solution to (P)
`z_opt`: The optimal solution to (D) corresponding to
_inequality constraints_ in (P)
Output:
`dual`: Boolean indicating whether (P) has many optimal solutions.
"""
D = -G
d = -h.flatten()
mu = -z_opt.flatten() # mu >= 0
# Active constraints
I = np.nonzero(np.abs(np.dot(D, x_opt).flatten() - d) < abs_tol)[0]
# Positive elements in dual opt
J = np.nonzero(mu > abs_tol)[0]
# i, j
i = mu < abs_tol # Zero elements in dual opt
i = i.astype(int)
j = np.zeros(len(mu), dtype=int)
j[I] = 1 # 1 if active
# Indices where active constraints have 0 dual opt
L = np.nonzero(i + j == 2)[0]
# sizes
nI = len(I)
nJ = len(J)
nL = len(L)
# constraints
DI = D[I, :] # Active constraints
DJ = D[J, :] # Constraints with positive lagrange mult
DL = D[L, :] # Active constraints with zero dual opt
dual = 0
if A is None:
test = DI
else:
test = np.vstack([DI, A])
if rank(test) < np.amin(DI.shape):
return True
else:
if len(L) > 0:
if A is None:
Ae = DJ
else:
Ae = np.vstack([DJ, A])
be = np.zeros(Ae.shape[0])
Ai = -DL
bi = np.zeros(nL)
sol = solvers._solve_lp_using_cvxopt(c=-np.sum(DL, axis=0),
G=Ai,
h=bi,
A=Ae,
b=be)
if sol['status'] == "dual infeasible":
# Dual infeasible -> primal unbounded -> value>epsilon
return True
if sol['primal objective'] > abs_tol:
return True
return False
def adjacent(C, D, b, rid_fac, abs_tol=1e-7):
"""Compute the (unique) adjacent facet.
@param rid_fac: A Ridge_Facet object containing the parameters for
a facet and one of its ridges.
@return: (E_adj,a_adj,b_adj): The equality set and parameters for
the adjacent facet such that::
P_{E_adj} = P intersection {x | a_adj x = b_adj}
"""
E = rid_fac.E_0
af = rid_fac.af
bf = rid_fac.bf
#
E_r = rid_fac.E_r
ar = rid_fac.ar
br = rid_fac.br
# shape
d = C.shape[1]
k = D.shape[1]
# E_r slices
C_er = C[E_r, :]
D_er = D[E_r, :]
b_er = b[E_r]
# stack
c = -np.hstack([ar, np.zeros(k)])
G = np.hstack([C_er, D_er])
h = b_er
A = np.hstack([af, np.zeros(k)])
sol = solvers._solve_lp_using_cvxopt(c, G, h, A=A.T, b=bf * (1 - 0.01))
if sol['status'] != "optimal":
print(G)
print(h)
print(af)
print(bf)
print(ar)
print(br)
print(np.dot(af, ar))
data = {}
data["C"] = C
data["D"] = D
data["b"] = b
sio.savemat("matlabdata", data)
pickle.dump(data, open("polytope.p", "wb"))
raise Exception("adjacent: Lp returned status " + str(sol['status']))
opt_sol = np.array(sol['x']).flatten()
dual_opt_sol = np.array(sol['z']).flatten()
x_opt = opt_sol[range(0, d)]
y_opt = opt_sol[range(d, d + k)]
if is_dual_degenerate(c.flatten(),
G,
h,
A,
bf * (1 - 0.01),
opt_sol,
dual_opt_sol,
abs_tol=abs_tol):
# If degenerate, compute affine hull and take preimage
E_temp = np.nonzero(np.abs(np.dot(G, opt_sol) - h) < abs_tol)[0]
a_temp, b_temp = proj_aff(C_er[E_temp, :],
D_er[E_temp, :],
b_er[E_temp],
expected_dim=1,
abs_tol=abs_tol)
E_adj = unique_equalityset(C, D, b, a_temp, b_temp, abs_tol=abs_tol)
if len(E_adj) == 0:
data = {}
data["C"] = C
data["D"] = D
data["b"] = b
data["Er"] = E_r + 1
data["ar"] = ar
data["br"] = br
data["Ef"] = E + 1
data["af"] = af
data["bf"] = bf
sio.savemat("matlabdata", data)
raise Exception(
"adjacent: equality set computation returned empty set")
else:
r = np.abs(np.dot(C, x_opt) + np.dot(D, y_opt) - b) < abs_tol
E_adj = np.nonzero(r)[0]
C_eadj = C[E_adj, :]
D_eadj = D[E_adj, :]
b_eadj = b[E_adj]
af_adj, bf_adj = proj_aff(C_eadj, D_eadj, b_eadj, abs_tol=abs_tol)
return E_adj, af_adj, bf_adj
def ridge(C, D, b, E, af, bf, abs_tol=1e-7, verbose=0):
"""Compute all ridges of a facet in the projection.
Input:
`C,D,b`: Original polytope data
`E,af,bf`: Equality set and affine hull of a facet in the projection
Output:
`ridge_list`: A list containing all the ridges of
the facet as Ridge objects
"""
d = C.shape[1]
k = D.shape[1]
Er_list = []
q = C.shape[0]
E_c = np.setdiff1d(range(q), E)
# E slices
C_E = C[E, :]
D_E = D[E, :]
b_E = b[E, :]
# E_c slices
C_Ec = C[E_c, :]
D_Ec = D[E_c, :]
b_Ec = b[E_c]
# dots
S = C_Ec - np.dot(np.dot(D_Ec, linalg.pinv(D_E)), C_E)
L = np.dot(D_Ec, null_space(D_E))
t = b_Ec - np.dot(D_Ec, np.dot(linalg.pinv(D_E), b_E))
if rank(np.hstack([C_E, D_E])) < k + 1:
if verbose > 1:
print("Doing recursive ESP call")
u, s, v = linalg.svd(np.array([af]), full_matrices=1)
sigma = s[0]
v = v.T * u[0, 0] # Correct sign
V_hat = v[:, [0]]
V_tilde = v[:, range(1, v.shape[1])]
Cnew = np.dot(S, V_tilde)
Dnew = L
bnew = t - np.dot(S, V_hat).flatten() * bf / sigma
Anew = np.hstack([Cnew, Dnew])
xc2, yc2, cen2 = cheby_center(Cnew, Dnew, bnew)
bnew = bnew - np.dot(Cnew, xc2).flatten() - np.dot(Dnew, yc2).flatten()
Gt, gt, E_t = esp(Cnew,
Dnew,
bnew,
centered=True,
abs_tol=abs_tol,
verbose=0)
if (len(E_t[0]) == 0) or (len(E_t[1]) == 0):
raise Exception(
"ridge: recursive call did not return any equality sets")
for i in range(len(E_t)):
E_f = E_t[i]
er = np.sort(np.hstack([E, E_c[E_f]]))
ar = np.dot(Gt[i, :], V_tilde.T).flatten()
br0 = gt[i].flatten()
# Make orthogonal to facet
ar = ar - af * np.dot(af.flatten(), ar.flatten())
br = br0 - bf * np.dot(af.flatten(), ar.flatten())
# Normalize and make ridge equation point outwards
norm = np.sqrt(np.sum(ar * ar))
ar = ar * np.sign(br) / norm
br = br * np.sign(br) / norm
# Restore center
br = br + np.dot(Gt[i, :], xc2) / norm
if len(ar) > d:
raise Exception("ridge: wrong length of new ridge!")
Er_list.append(Ridge(er, ar, br))
else:
if verbose > 0:
print("Doing direct calculation of ridges")
X = np.arange(S.shape[0])
while len(X) > 0:
i = X[0]
X = np.setdiff1d(X, i)
if np.linalg.norm(S[i, :]) < abs_tol:
continue
Si = S[i, :]
Si = Si / np.linalg.norm(Si)
if np.linalg.norm(af - np.dot(Si, af) * Si) > abs_tol:
test1 = null_space(np.vstack(
[np.hstack([af, bf]),
np.hstack([S[i, :], t[i]])]),
nonempty=True)
test2 = np.hstack([S, np.array([t]).T])
test = np.dot(test1.T, test2.T)
test = np.sum(np.abs(test), 0)
Q_i = np.nonzero(test > abs_tol)[0]
Q = np.nonzero(test < abs_tol)[0]
X = np.setdiff1d(X, Q)
# Have Q_i
Sq = S[Q_i, :]
tq = t[Q_i]
c = np.zeros(d + 1)
c[0] = 1
Gup = np.hstack([-np.ones([Sq.shape[0], 1]), Sq])
Gdo = np.hstack([-1, np.zeros(Sq.shape[1])])
G = np.vstack([Gup, Gdo])
h = np.hstack([tq, 1])
Al = np.zeros([2, 1])
Ar = np.vstack([af, S[i, :]])
A = np.hstack([Al, Ar])
bb = np.hstack([bf, t[i]])
sol = solvers._solve_lp_using_cvxopt(c, G, h, A=A, b=bb)
if sol['status'] == 'optimal':
tau = sol['x'][0]
if tau < -abs_tol:
ar = np.array([S[i, :]]).flatten()
br = t[i].flatten()
# Make orthogonal to facet
ar = ar - af * np.dot(af.flatten(), ar.flatten())
br = br - bf * np.dot(af.flatten(), ar.flatten())
# Normalize and make ridge equation point outwards
norm = np.sqrt(np.sum(ar * ar))
ar = ar / norm
br = br / norm
# accumulate
Er_list.append(
Ridge(np.sort(np.hstack([E, E_c[Q]])), ar, br))
return Er_list
def is_dual_degenerate(c, G, h, A, b, x_opt, z_opt, abs_tol=1e-7):
"""Return `True` if pair of dual problems is dual degenerate.
Checks if the pair of dual problems::
(P): min c'x (D): max h'z + b'y
s.t Gx <= h s.t G'z + A'y = c
Ax = b z <= 0
is dual degenerate, i.e. if (P) has several optimal solutions.
Optimal solutions x* and z* are required.
Input:
`G,h,A,b`: Parameters of (P)
`x_opt`: One optimal solution to (P)
`z_opt`: The optimal solution to (D) corresponding to
_inequality constraints_ in (P)
Output:
`dual`: Boolean indicating whether (P) has many optimal solutions.
"""
D = - G
d = - h.flatten()
mu = - z_opt.flatten() # mu >= 0
# Active constraints
I = np.nonzero(np.abs(np.dot(D, x_opt).flatten() - d) < abs_tol)[0]
# Positive elements in dual opt
J = np.nonzero(mu > abs_tol)[0]
# i, j
i = mu < abs_tol # Zero elements in dual opt
i = i.astype(int)
j = np.zeros(len(mu), dtype=int)
j[I] = 1 # 1 if active
# Indices where active constraints have 0 dual opt
L = np.nonzero(i + j == 2)[0]
# sizes
nI = len(I)
nJ = len(J)
nL = len(L)
# constraints
DI = D[I, :] # Active constraints
DJ = D[J, :] # Constraints with positive lagrange mult
DL = D[L, :] # Active constraints with zero dual opt
dual = 0
if A is None:
test = DI
else:
test = np.vstack([DI, A])
if rank(test) < np.amin(DI.shape):
return True
else:
if len(L) > 0:
if A is None:
Ae = DJ
else:
Ae = np.vstack([DJ, A])
be = np.zeros(Ae.shape[0])
Ai = - DL
bi = np.zeros(nL)
sol = solvers._solve_lp_using_cvxopt(
c= - np.sum(DL, axis=0), G=Ai,
h=bi, A=Ae, b=be)
if sol['status'] == "dual infeasible":
# Dual infeasible -> primal unbounded -> value>epsilon
return True
if sol['primal objective'] > abs_tol:
return True
return False
def adjacent(C, D, b, rid_fac, abs_tol=1e-7):
"""Compute the (unique) adjacent facet.
@param rid_fac: A Ridge_Facet object containing the parameters for
a facet and one of its ridges.
@return: (E_adj,a_adj,b_adj): The equality set and parameters for
the adjacent facet such that::
P_{E_adj} = P intersection {x | a_adj x = b_adj}
"""
E = rid_fac.E_0
af = rid_fac.af
bf = rid_fac.bf
#
E_r = rid_fac.E_r
ar = rid_fac.ar
br = rid_fac.br
# shape
d = C.shape[1]
k = D.shape[1]
# E_r slices
C_er = C[E_r, :]
D_er = D[E_r, :]
b_er = b[E_r]
# stack
c = -np.hstack([ar, np.zeros(k)])
G = np.hstack([C_er, D_er])
h = b_er
A = np.hstack([af, np.zeros(k)])
sol = solvers._solve_lp_using_cvxopt(
c, G, h, A=A.T, b=bf * (1 - 0.01))
if sol['status'] != "optimal":
print(G)
print(h)
print(af)
print(bf)
print(ar)
print(br)
print(np.dot(af, ar))
data = {}
data["C"] = C
data["D"] = D
data["b"] = b
sio.savemat("matlabdata", data)
pickle.dump(data, open("polytope.p", "wb"))
raise Exception(
"adjacent: Lp returned status " + str(sol['status']))
opt_sol = np.array(sol['x']).flatten()
dual_opt_sol = np.array(sol['z']).flatten()
x_opt = opt_sol[range(0, d)]
y_opt = opt_sol[range(d, d + k)]
if is_dual_degenerate(
c.flatten(), G, h, A, bf * (1 - 0.01),
opt_sol, dual_opt_sol, abs_tol=abs_tol):
# If degenerate, compute affine hull and take preimage
E_temp = np.nonzero(np.abs(np.dot(G, opt_sol) - h) < abs_tol)[0]
a_temp, b_temp = proj_aff(
C_er[E_temp, :], D_er[E_temp, :], b_er[E_temp],
expected_dim=1, abs_tol=abs_tol)
E_adj = unique_equalityset(C, D, b, a_temp, b_temp, abs_tol=abs_tol)
if len(E_adj) == 0:
data = {}
data["C"] = C
data["D"] = D
data["b"] = b
data["Er"] = E_r + 1
data["ar"] = ar
data["br"] = br
data["Ef"] = E + 1
data["af"] = af
data["bf"] = bf
sio.savemat("matlabdata", data)
raise Exception(
"adjacent: equality set computation returned empty set")
else:
r = np.abs(np.dot(C, x_opt) + np.dot(D, y_opt) - b) < abs_tol
E_adj = np.nonzero(r)[0]
C_eadj = C[E_adj, :]
D_eadj = D[E_adj, :]
b_eadj = b[E_adj]
af_adj, bf_adj = proj_aff(C_eadj, D_eadj, b_eadj, abs_tol=abs_tol)
return E_adj, af_adj, bf_adj
def ridge(C, D, b, E, af, bf, abs_tol=1e-7, verbose=0):
"""Compute all ridges of a facet in the projection.
Input:
`C,D,b`: Original polytope data
`E,af,bf`: Equality set and affine hull of a facet in the projection
Output:
`ridge_list`: A list containing all the ridges of
the facet as Ridge objects
"""
d = C.shape[1]
k = D.shape[1]
Er_list = []
q = C.shape[0]
E_c = np.setdiff1d(range(q), E)
# E slices
C_E = C[E, :]
D_E = D[E, :]
b_E = b[E, :]
# E_c slices
C_Ec = C[E_c, :]
D_Ec = D[E_c, :]
b_Ec = b[E_c]
# dots
S = C_Ec - np.dot(np.dot(D_Ec, linalg.pinv(D_E)), C_E)
L = np.dot(D_Ec, null_space(D_E))
t = b_Ec - np.dot(D_Ec, np.dot(linalg.pinv(D_E), b_E))
if rank(np.hstack([C_E, D_E])) < k + 1:
if verbose > 1:
print("Doing recursive ESP call")
u, s, v = linalg.svd(np.array([af]), full_matrices=1)
sigma = s[0]
v = v.T * u[0, 0] # Correct sign
V_hat = v[:, [0]]
V_tilde = v[:, range(1, v.shape[1])]
Cnew = np.dot(S, V_tilde)
Dnew = L
bnew = t - np.dot(S, V_hat).flatten() * bf / sigma
Anew = np.hstack([Cnew, Dnew])
xc2, yc2, cen2 = cheby_center(Cnew, Dnew, bnew)
bnew = bnew - np.dot(Cnew, xc2).flatten() - np.dot(Dnew, yc2).flatten()
Gt, gt, E_t = esp(
Cnew, Dnew, bnew,
centered=True, abs_tol=abs_tol, verbose=0)
if (len(E_t[0]) == 0) or (len(E_t[1]) == 0):
raise Exception(
"ridge: recursive call did not return any equality sets")
for i in range(len(E_t)):
E_f = E_t[i]
er = np.sort(np.hstack([E, E_c[E_f]]))
ar = np.dot(Gt[i, :], V_tilde.T).flatten()
br0 = gt[i].flatten()
# Make orthogonal to facet
ar = ar - af * np.dot(af.flatten(), ar.flatten())
br = br0 - bf * np.dot(af.flatten(), ar.flatten())
# Normalize and make ridge equation point outwards
norm = np.sqrt(np.sum(ar * ar))
ar = ar * np.sign(br) / norm
br = br * np.sign(br) / norm
# Restore center
br = br + np.dot(Gt[i, :], xc2) / norm
if len(ar) > d:
raise Exception("ridge: wrong length of new ridge!")
Er_list.append(Ridge(er, ar, br))
else:
if verbose > 0:
print("Doing direct calculation of ridges")
X = np.arange(S.shape[0])
while len(X) > 0:
i = X[0]
X = np.setdiff1d(X, i)
if np.linalg.norm(S[i, :]) < abs_tol:
continue
Si = S[i, :]
Si = Si / np.linalg.norm(Si)
if np.linalg.norm(af - np.dot(Si, af) * Si) > abs_tol:
test1 = null_space(
np.vstack([
np.hstack([af, bf]),
np.hstack([S[i, :], t[i]])]),
nonempty=True)
test2 = np.hstack([S, np.array([t]).T])
test = np.dot(test1.T, test2.T)
test = np.sum(np.abs(test), 0)
Q_i = np.nonzero(test > abs_tol)[0]
Q = np.nonzero(test < abs_tol)[0]
X = np.setdiff1d(X, Q)
# Have Q_i
Sq = S[Q_i, :]
tq = t[Q_i]
c = np.zeros(d + 1)
c[0] = 1
Gup = np.hstack([-np.ones([Sq.shape[0], 1]), Sq])
Gdo = np.hstack([-1, np.zeros(Sq.shape[1])])
G = np.vstack([Gup, Gdo])
h = np.hstack([tq, 1])
Al = np.zeros([2, 1])
Ar = np.vstack([af, S[i, :]])
A = np.hstack([Al, Ar])
bb = np.hstack([bf, t[i]])
sol = solvers._solve_lp_using_cvxopt(
c, G, h, A=A, b=bb)
if sol['status'] == 'optimal':
tau = sol['x'][0]
if tau < -abs_tol:
ar = np.array([S[i, :]]).flatten()
br = t[i].flatten()
# Make orthogonal to facet
ar = ar - af * np.dot(af.flatten(), ar.flatten())
br = br - bf * np.dot(af.flatten(), ar.flatten())
# Normalize and make ridge equation point outwards
norm = np.sqrt(np.sum(ar * ar))
ar = ar / norm
br = br / norm
# accumulate
Er_list.append(
Ridge(np.sort(np.hstack([E, E_c[Q]])), ar, br))
return Er_list