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 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 quad_over_lin_canon(expr, args):
# quad_over_lin := sum_{ij} X^2_{ij} / y
x = args[0]
y = args[1].flatten()
# precondition: shape == ()
t = Variable(1,)
# (y+t, y-t, 2*x) must lie in the second-order cone,
# where y+t is the scalar part of the second-order
# cone constraint.
constraints = [SOC(
t=y+t,
X=hstack([y-t, 2*x.flatten()]), axis=0
), y >= 0]
return t, constraints
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 bmat(block_lists):
"""Constructs a block matrix.
Takes a list of lists. Each internal list is stacked horizontally.
The internal lists are stacked vertically.
Parameters
----------
block_lists : list of lists
The blocks of the block matrix.
Return
------
CVXPY expression
The CVXPY expression representing the block matrix.
"""
row_blocks = [hstack(*blocks) for blocks in block_lists]
return vstack(*row_blocks)
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 deep_flatten(x):
# base cases
if isinstance(x, Expression):
if len(x.shape) == 1:
return x
else:
return x.flatten()
elif isinstance(x, np.ndarray) or isinstance(x, (int, float)):
x = Expression.cast_to_const(x)
return x.flatten()
# recursion
if isinstance(x, list):
y = []
for x0 in x:
x1 = deep_flatten(x0)
y.append(x1)
y = hstack(y)
return y
msg = 'The input to deep_flatten must be an Expression, a NumPy array, an int'\
+ ' or float, or a nested list thereof. Received input of type %s' % type(x)
raise ValueError(msg)
def prod(expr, axis=None, keepdims: bool = False) -> Prod:
"""Multiply the entries of an expression.
The semantics of this atom are the same as np.prod.
This atom is log-log affine, but it is neither convex nor concave.
Parameters
----------
expr : Expression or list[Expression, Numeric]
The expression to multiply the entries of, or a list of Expressions
and numeric types.
axis : int
The axis along which to take the product; ignored if `expr` is a list.
keepdims : bool
Whether to drop dimensions after taking the product; ignored if `expr`
is a list.
"""
if isinstance(expr, list):
return Prod(hstack(expr))
else:
return Prod(expr, axis, keepdims)
def mixed_norm(X, p=2, q=1):
"""Lp,q norm; :math:` (\sum_k (\sum_l \lvert x_{k,l} \rvert )^q/p)^{1/q}`.
Parameters
----------
X : Expression or numeric constant
The matrix to take the l_{p,q} norm of.
p : int or str, optional
The type of inner norm.
q : int or str, optional
The type of outer norm.
Returns
-------
Expression
An Expression representing the mixed norm.
"""
X = Expression.cast_to_const(X)
# inner norms
vecnorms = [norm(X[i, :], p) for i in range(X.size[0])]
# outer norm
return norm(hstack(*vecnorms), q)
def mixed_norm(X, p=2, q=1):
"""Lp,q norm; :math:` (\sum_k (\sum_l \lvert x_{k,l} \rvert )^q/p)^{1/q}`.
Parameters
----------
X : Expression or numeric constant
The matrix to take the l_{p,q} norm of.
p : int or str, optional
The type of inner norm.
q : int or str, optional
The type of outer norm.
Returns
-------
Expression
An Expression representing the mixed norm.
"""
X = Expression.cast_to_const(X)
# inner norms
vecnorms = [norm(X[i, :], p) for i in range(X.size[0])]
# outer norm
return norm(hstack(*vecnorms), q)
def col_norm(X, p=2):
X = Expression.cast_to_const(X)
vecnorms = [norm(X[:, i], p) for i in range(X.size[1])]
return hstack(*vecnorms)
def col_norm(X, p=2):
X = Expression.cast_to_const(X)
vecnorms = [ norm(X[:, i], p) for i in range(X.size[1]) ]
return hstack(*vecnorms)
def row_norm(X, p=2):
X = Expression.cast_to_const(X)
vecnorms = [ norm(X[i, :], p) for i in range(X.size[0]) ]
return hstack(*vecnorms).T
