def log_normcdf(x):
"""Elementwise log of the cumulative distribution function of a standard normal random variable.
Implementation is a quadratic approximation with modest accuracy over [-4, 4].
"""
A = scipy.sparse.diags(
np.sqrt(
[
0.02301291,
0.08070214,
0.16411522,
0.09003495,
0.08200854,
0.01371543,
0.04641081,
]
)
)
b = np.array([[3.0, 2.0, 1.0, 0.0, -1.0, -2.5, -3.5]]).reshape(-1, 1)
x = Expression.cast_to_const(x)
flat_x = reshape(x, (1, x.size))
y = A @ (b @ np.ones(flat_x.shape) - np.ones(b.shape) @ flat_x)
out = -sum_(maximum(y, 0) ** 2, axis=0)
return reshape(out, x.shape)
def gm(t, x, y):
length = t.size
return SOC(t=reshape(x + y, (length, )),
X=vstack(
[reshape(x - y, (1, length)),
reshape(2 * t, (1, length))]),
axis=0)
def log_normcdf(x): # noqa: E501
"""Elementwise log of the cumulative distribution function of a standard normal random variable.
The implementation is a quadratic approximation with modest accuracy over [-4, 4].
For details on the nature of the approximation, refer to
`CVXPY GitHub PR #1224 <https://github.com/cvxpy/cvxpy/pull/1224#issue-793221374>`_.
.. note::
SciPy's analog of ``log_normcdf`` is called `log_ndtr <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.log_ndtr.html>`_.
We opted not to use that name because its meaning would not be obvious to the casual user.
"""
A = scipy.sparse.diags(
np.sqrt(
[
0.02301291,
0.08070214,
0.16411522,
0.09003495,
0.08200854,
0.01371543,
0.04641081,
]
)
)
b = np.array([[3.0, 2.0, 1.0, 0.0, -1.0, -2.5, -3.5]]).reshape(-1, 1)
x = Expression.cast_to_const(x)
flat_x = reshape(x, (1, x.size))
y = A @ (b @ np.ones(flat_x.shape) - np.ones(b.shape) @ flat_x)
out = -sum_(maximum(y, 0) ** 2, axis=0)
return reshape(out, x.shape)
def test_reshape_with_lists(self) -> None:
n = 2
a = Variable([n, n])
b = Variable(n**2)
c = reshape(b, [n, n])
self.assertEqual((a + c).shape, (n, n))
d = reshape(b, (n, n))
self.assertEqual((a + d).shape, (n, n))
def get_special_slice(expr, key):
"""Indexing using logical indexing or a list of indices.
Parameters
----------
expr : Expression
The expression being indexed/sliced into.
key : tuple
ndarrays or lists.
Returns
-------
Expression
An expression representing the index/slice.
"""
expr = index.cast_to_const(expr)
# Order the entries of expr and select them using key.
idx_mat = np.arange(expr.size[0]*expr.size[1])
idx_mat = np.reshape(idx_mat, expr.size, order='F')
select_mat = idx_mat[key]
if select_mat.ndim == 2:
final_size = select_mat.shape
else: # Always cast 1d arrays as column vectors.
final_size = (select_mat.size, 1)
select_vec = np.reshape(select_mat, select_mat.size, order='F')
# Select the chosen entries from expr.
identity = sp.eye(expr.size[0]*expr.size[1]).tocsc()
return reshape(identity[select_vec]*vec(expr), *final_size)
def scaled_lower_tri(matrix):
"""Returns an expression representing the lower triangular entries
Scales the strictly lower triangular entries by sqrt(2), as required
by SCS.
Parameters
----------
matrix : Expression
A 2-dimensional CVXPY expression.
Returns
-------
Expression
An expression representing the (scaled) lower triangular part of
the supplied matrix expression.
"""
rows = cols = matrix.shape[0]
entries = rows * (cols + 1)//2
row_arr = np.arange(0, entries)
lower_diag_indices = np.tril_indices(rows)
col_arr = np.sort(np.ravel_multi_index(lower_diag_indices, (rows, cols), order='F'))
val_arr = np.zeros((rows, cols))
val_arr[lower_diag_indices] = np.sqrt(2)
np.fill_diagonal(val_arr, 1)
val_arr = np.ravel(val_arr, order='F')
val_arr = val_arr[np.nonzero(val_arr)]
shape = (entries, rows*cols)
coeff = Constant(sp.csc_matrix((val_arr, (row_arr, col_arr)), shape))
vectorized_matrix = reshape(matrix, (rows*cols, 1))
return coeff * vectorized_matrix
def vec_to_upper_tri(expr, strict=False):
expr = Expression.cast_to_const(expr)
ell = expr.shape[0]
if strict:
# n * (n-1)/2 == ell
n = ((8 * ell + 1)**0.5 + 1) // 2
else:
# n * (n+1)/2 == ell
n = ((8 * ell + 1)**0.5 - 1) // 2
n = int(n)
# form a matrix P, of shape (n**2, ell).
# the i-th block of n rows of P gives the entries of the i-th row
# of the upper-triangular matrix associated with expr.
# compute expr2 = P @ expr
# compute expr3 = reshape(expr2, shape=(n, n)).T
# expr3 is the matrix formed by reading length-n blocks of expr2,
# and letting each block form a row of expr3.
P_rows = []
P_row = 0
for mat_row in range(n):
entries_in_row = n - mat_row
if strict:
entries_in_row -= 1
P_row += n - entries_in_row # these are zeros
P_rows.extend(range(P_row, P_row + entries_in_row))
P_row += entries_in_row
P_cols = np.arange(ell)
P_vals = np.ones(P_cols.size)
P = csc_matrix((P_vals, (P_rows, P_cols)), shape=(n**2, ell))
expr2 = P @ expr
expr3 = reshape(expr2, (n, n)).T
return expr3
def diag(expr):
"""Extracts the diagonal from a matrix or makes a vector a diagonal matrix.
Parameters
----------
expr : Expression or numeric constant
A vector or square matrix.
Returns
-------
Expression
An Expression representing the diagonal vector/matrix.
"""
expr = AffAtom.cast_to_const(expr)
if expr.is_vector():
if expr.size[1] == 1:
return diag_vec(expr)
# Convert a row vector to a column vector.
else:
expr = reshape(expr, expr.size[1], 1)
return diag_vec(expr)
elif expr.size[0] == expr.size[1]:
return diag_mat(expr)
else:
raise ValueError("Argument to diag must be a vector or square matrix.")
def var_cone_canon(expr, args):
"""Expand implicit constraints on variable.
"""
# Convert attributes into constraints.
new_attr = expr.attributes.copy()
for key in ['nonneg', 'nonpos', 'symmetric', 'PSD', 'NSD']:
if new_attr[key]:
new_attr[key] = False
if expr.is_symmetric():
n = expr.shape[0]
shape = (n * (n + 1) // 2, 1)
upper_tri = Variable(shape, var_id=expr.id, **new_attr)
fill_coeff = Constant(upper_tri_to_full(n))
full_mat = fill_coeff * upper_tri
obj = reshape(full_mat, (n, n))
else:
obj = Variable(expr.shape, var_id=expr.id, **new_attr)
constr = []
if expr.is_nonneg():
constr.append(obj >= 0)
elif expr.is_nonpos():
constr.append(obj <= 0)
elif expr.attributes['PSD']:
constr.append(obj >> 0)
elif expr.attributes['NSD']:
constr.append(obj << 0)
return (obj, constr)
def sum_canon(expr, args):
X = args[0]
if expr.axis is None:
summation = explicit_sum(X)
canon, _ = add_canon(summation, summation.args)
return reshape(canon, expr.shape), []
if expr.axis == 0:
X = X.T
rows = []
for i in range(X.shape[0]):
summation = explicit_sum(X[i])
canon, _ = add_canon(summation, summation.args)
rows.append(canon)
canon = hstack(rows)
return reshape(canon, expr.shape), []
def mulexpression_canon(expr, args):
lhs = args[0]
rhs = args[1]
lhs_shape, rhs_shape, _ = mul_shapes_promote(lhs.shape, rhs.shape)
lhs = reshape(lhs, lhs_shape)
rhs = reshape(rhs, rhs_shape)
rows = []
# TODO(akshayka): Parallelize this for large matrices.
for i in range(lhs.shape[0]):
row = []
for j in range(rhs.shape[1]):
arr = hstack([lhs[i, k] + rhs[k, j] for k in range(lhs.shape[1])])
row.append(log_sum_exp(arr))
rows.append(row)
mat = bmat(rows)
if mat.shape != expr.shape:
mat = reshape(mat, expr.shape)
return mat, []
def special_index_canon(expr, args):
select_mat = expr._select_mat
final_shape = expr._select_mat.shape
select_vec = np.reshape(select_mat, select_mat.size, order='F')
# Select the chosen entries from expr.
arg = args[0]
identity = sp.eye(arg.size).tocsc()
lowered = reshape(identity[select_vec] * vec(arg), final_shape)
return lowered, []
def add_canon(expr, args):
if expr.is_scalar():
return log_sum_exp(hstack(args)), []
rows = []
summands = [promote(s, expr.shape) if s.is_scalar() else s for s in args]
if len(expr.shape) == 1:
for i in range(expr.shape[0]):
row = []
row.append(
log_sum_exp(hstack([summand[i] for summand in summands])))
rows.append(row)
return reshape(bmat(rows), expr.shape), []
else:
for i in range(expr.shape[0]):
row = []
for j in range(expr.shape[1]):
row.append(
log_sum_exp(hstack([summand[i, j]
for summand in summands])))
rows.append(row)
return reshape(bmat(rows), expr.shape), []
def vec(X):
"""Flattens the matrix X into a vector in column-major order.
Parameters
----------
X : Expression or numeric constant
The matrix to flatten.
Returns
-------
Expression
An Expression representing the flattened matrix.
"""
X = Expression.cast_to_const(X)
return reshape(X, (X.size, ))
def grad(self):
"""Gives the (sub/super)gradient of the expression w.r.t. each variable.
Matrix expressions are vectorized, so the gradient is a matrix.
None indicates variable values unknown or outside domain.
Returns:
A map of variable to SciPy CSC sparse matrix or None.
"""
select_vec = np.reshape(self._select_mat,
self._select_mat.size,
order='F')
identity = sp.eye(self.args[0].size).tocsc()
lowered = reshape(identity[select_vec] @ vec(self.args[0]),
self._shape)
return lowered.grad
def vec(X):
"""Flattens the matrix X into a vector in column-major order.
Parameters
----------
X : Expression or numeric constant
The matrix to flatten.
Returns
-------
Expression
An Expression representing the flattened matrix.
"""
X = Expression.cast_to_const(X)
return reshape(X, X.size[0]*X.size[1], 1)
def pow_nd_canon(con, args):
"""
con : PowConeND
We can extract metadata from this.
For example, con.alpha and con.axis.
args : tuple of length two
W,z = args[0], args[1]
"""
alpha, axis = con.get_data()
alpha = alpha.value
W, z = args
if axis == 1:
W = W.T
alpha = alpha.T
if W.ndim == 1:
W = reshape(W, (W.size, 1))
alpha = np.reshape(alpha, (W.size, 1))
n, k = W.shape
if n == 2:
can_con = PowCone3D(W[0, :], W[1, :], z, alpha[0, :])
else:
T = Variable(shape=(n - 2, k))
x_3d, y_3d, z_3d, alpha_3d = [], [], [], []
for j in range(k):
x_3d.append(W[:-1, j])
y_3d.append(T[:, j])
y_3d.append(W[n - 1, j])
z_3d.append(z[j])
z_3d.append(T[:, j])
r_nums = alpha[:, j]
r_dens = np.cumsum(r_nums[::-1])[::-1]
# ^ equivalent to [np.sum(alpha[i:, j]) for i in range(n)]
r = r_nums / r_dens
alpha_3d.append(r[:n - 1])
x_3d = hstack(x_3d)
y_3d = hstack(y_3d)
z_3d = hstack(z_3d)
alpha_p3d = hstack(alpha_3d)
# TODO: Ideally we should construct x,y,z,alpha_p3d by
# applying suitable sparse matrices to W,z,T, rather
# than using the hstack atom. (hstack will probably
# result in longer compile times).
can_con = PowCone3D(x_3d, y_3d, z_3d, alpha_p3d)
# Return a single PowCone3D constraint defined over all auxiliary
# variables needed for the reduction to go through.
# There are no "auxiliary constraints" beyond this one.
return can_con, []
def ILP_form(m, P, sinkresource, n, C, Ce, Cr, Tt, Tc, B, E, e, stagemat, Adj):
# -- Variables --
T = cp.Variable(P + 1) # task time
M = cp.Variable(P) # models resources
x = cp.Variable((n, m), boolean=True) # binary decisions
# -- Objective --
obj = cp.Minimize(T[P])
# -- Constraints --
constraints = []
# (1) completion time (Note that M(i) = M_{i-1})
constraints.append(T[0] == 0)
for p in range(1, P + 1):
constraints.append(T[p] == T[p - 1] + M[p - 1])
# (2) find the slowest process
for p in range(P):
for i in range(n):
for j in range(m):
constraints.append(Tt[i,j] * stagemat[p,i] * x[i,j] + \
stagemat[p,i] * (Adj[:,i].reshape(1,n) @ x @ Tc[:,j]) <= M[p])
# (3) resource constraints
for p in range(P):
for j in range(m):
constraints.append(stagemat[p] @ multiply(reshape(x[:, j], (
n, 1)), C[:, j].reshape(n, 1)) <= B[j])
# (5) assign tasks
for i in range(n):
constraints.append(sum(x[i]) == 1)
# -- Solve --
prob = cp.Problem(obj, constraints)
val = prob.solve()
print('T.value:', T.value)
print('M.value:', M.value)
print('x.value:\n', x.value)
print('T:', val)
return x.value
def tv(value, *args):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
args : Matrix constants/expressions
Additional matrices extending the third dimension of value.
Returns
-------
Expression
An Expression representing the total variation.
"""
# Accept single list as argument.
if isinstance(value, list) and len(args) == 0:
args = value[1:]
value = value[0]
value = Expression.cast_to_const(value)
rows, cols = value.size
if value.is_scalar():
raise ValueError("tv cannot take a scalar argument.")
# L1 norm for vectors.
elif value.is_vector():
return norm(value[1:] - value[0:max(rows, cols)-1], 1)
# L2 norm for matrices.
else:
args = list(map(Expression.cast_to_const, args))
values = [value] + list(args)
diffs = []
for mat in values:
diffs += [
mat[0:rows-1, 1:cols] - mat[0:rows-1, 0:cols-1],
mat[1:rows, 0:cols-1] - mat[0:rows-1, 0:cols-1],
]
length = diffs[0].size[0]*diffs[1].size[1]
stacked = vstack(*[reshape(diff, 1, length) for diff in diffs])
return sum_entries(norm(stacked, p='fro', axis=0))
def tv(value, *args):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
args : Matrix constants/expressions
Additional matrices extending the third dimension of value.
Returns
-------
Expression
An Expression representing the total variation.
"""
# Accept single list as argument.
if isinstance(value, list) and len(args) == 0:
args = value[1:]
value = value[0]
value = Expression.cast_to_const(value)
rows, cols = value.size
if value.is_scalar():
raise ValueError("tv cannot take a scalar argument.")
# L1 norm for vectors.
elif value.is_vector():
return norm(value[1:] - value[0:max(rows, cols)-1], 1)
# L2 norm for matrices.
else:
args = map(Expression.cast_to_const, args)
values = [value] + list(args)
diffs = []
for mat in values:
diffs += [
mat[0:rows-1, 1:cols] - mat[0:rows-1, 0:cols-1],
mat[1:rows, 0:cols-1] - mat[0:rows-1, 0:cols-1],
]
length = diffs[0].size[0]*diffs[1].size[1]
stacked = vstack(*[reshape(diff, 1, length) for diff in diffs])
return sum_entries(norm(stacked, p='fro', axis=0))
def tvnorm2d(value, Dx, Dy):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
Returns
-------
Expression
An Expression representing the total variation.
"""
value = Expression.cast_to_const(value)
len = value.size[0]
diffs = [Dx * value, Dy * value]
stack = vstack(*[reshape(diff, 1, len) for diff in diffs])
return sum_entries(cvxnorm(stack, p='fro', axis=0))
def scaled_lower_tri(matrix):
"""Returns an expression representing the lower triangular entries
Scales the strictly lower triangular entries by sqrt(2), as required
by SCS.
Parameters
----------
matrix : Expression
A 2-dimensional CVXPY expression.
Returns
-------
Expression
An expression representing the (scaled) lower triangular part of
the supplied matrix expression.
"""
rows = cols = matrix.shape[0]
entries = rows * (cols + 1) // 2
val_arr = []
row_arr = []
col_arr = []
count = 0
for j in range(cols):
for i in range(rows):
if j <= i:
# Index in the original matrix.
col_arr.append(j * rows + i)
# Index in the extracted vector.
row_arr.append(count)
if j == i:
val_arr.append(1.0)
else:
val_arr.append(np.sqrt(2))
count += 1
shape = (entries, rows * cols)
coeff = Constant(
sp.coo_matrix((val_arr, (row_arr, col_arr)), shape).tocsc())
vectorized_matrix = reshape(matrix, (rows * cols, 1))
return coeff * vectorized_matrix
def tv(value, *args):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
args : Matrix constants/expressions
Additional matrices extending the third dimension of value.
Returns
-------
Expression
An Expression representing the total variation.
"""
value = Expression.cast_to_const(value)
if value.ndim == 0:
raise ValueError("tv cannot take a scalar argument.")
# L1 norm for vectors.
elif value.ndim == 1:
return norm(value[1:] - value[0:value.shape[0]-1], 1)
# L2 norm for matrices.
else:
rows, cols = value.shape
args = map(Expression.cast_to_const, args)
values = [value] + list(args)
diffs = []
for mat in values:
diffs += [
mat[0:rows-1, 1:cols] - mat[0:rows-1, 0:cols-1],
mat[1:rows, 0:cols-1] - mat[0:rows-1, 0:cols-1],
]
length = diffs[0].shape[0]*diffs[1].shape[1]
stacked = vstack([reshape(diff, (1, length)) for diff in diffs])
return sum(norm(stacked, p=2, axis=0))
def tvnorm_anisotropic_2d(signal, Dx, Dy):
magnitudes = pnorm(signal, 2, axis=1)
diffs = [Dx * magnitudes, Dy * magnitudes]
stack = vstack(*[reshape(diff, 1, magnitudes.size[0]) for diff in diffs])
return sum_entries(pnorm(stack, 2, axis=0))
def LP_relax(m,
P,
sinkresource,
n,
C,
Ce,
Cr,
Tt,
Tc,
B,
E,
e,
stagemat,
Adj,
verbose=1):
# -- Variables --
T = cp.Variable(P + 1) # task time
M = cp.Variable(P) # models resources
xl = cp.Variable((n, m), boolean=False) # binary decisions
# -- Objective --
obj = cp.Minimize(T[P])
# -- Constraints --
constraints = []
# (1) completion time (Note that M(i) = M_{i-1})
constraints.append(T[0] == 0)
for p in range(1, P + 1):
constraints.append(T[p] == T[p - 1] + M[p - 1])
# (2) find the slowest process
for p in range(P):
for i in range(n):
for j in range(m):
constraints.append(Tt[i,j] * stagemat[p,i] * xl[i,j] + \
stagemat[p,i] * (Adj[:,i].reshape(1,n) @ xl @ Tc[:,j]) <= M[p])
# (3) resource constraints
for p in range(P):
for j in range(m):
constraints.append(
stagemat[p] @ multiply(reshape(xl[:, j], (
n, 1)), C[:, j].reshape(n, 1)) <= B[j])
# (5) assign tasks
for i in range(n):
constraints.append(sum(xl[i]) == 1)
# (6) bounded between 0 and 1
for i in range(n):
for j in range(m):
constraints.append(xl[i, j] >= 0)
for i in range(n):
for j in range(m):
constraints.append(xl[i, j] <= 1)
# -- Solve --
prob = cp.Problem(obj, constraints)
val = prob.solve()
if verbose == 1:
print('prob.solve:', val)
print('T.value:', np.around(T.value, 3))
print('M.value:', np.around(M.value, 3))
print('xl.value:\n', np.around(xl.value, 3))
print('T:', val)
return val, xl.value