def __obj(self, λ):
'''
defines the objective function for section between A and B during the Trade-off analysis
parameters:
λ(float): a constant greater than 0
'''
if not ((isinstance(λ, float) or isinstance(λ, int)) and λ > 0):
raise ValueError('λ must be a float and is > 0')
self.objFunc = norm(self.Σ_p_sqrt * self.a,
2) + λ * norm(self.Σ_n_sqrt * self.a, 2)
self.obj = cp.Minimize(self.objFunc)
def tv(value):
"""Total variation of a vector or matrix.
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)
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:
row_diff = value[0:rows - 1, 1:cols] - value[0:rows - 1, 0:cols - 1]
col_diff = value[1:rows, 0:cols - 1] - value[0:rows - 1, 0:cols - 1]
return sum_entries(norm2_elemwise(row_diff, col_diff))
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 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 mixed_norm(X, p=2, q=1):
"""Lp,q norm; :math:`(\\sum_k (\\sum_l \\lvert x_{k,l} \\rvert^p)^{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, p, axis=1)
# outer norm
return norm(vecnorms, q)
def sum_squares(expr):
"""The sum of the squares of the entries.
Parameters
----------
expr: Expression
The expression to take the sum of squares of.
Returns
-------
Expression
An expression representing the sum of squares.
"""
return square(norm(expr, "fro"))
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 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)
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],
]
return sum_entries(norm2_elemwise(*diffs))
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)
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],
]
return sum_entries(norm2_elemwise(*diffs))
print(f"\n\nMatrix U: {U}")
print(f"Shape of U: {U.shape}.")
Q = deepcopy(U[:, 4:len(destination)])
print(f"\n\nMatrix Q: {Q}")
print(f"Shape of Q: {Q.shape}.")
# Solve SDP problem
A = cp.Variable(shape=(len(destination), len(destination)))
A_bar = Q.transpose() * A * Q
print(f"\n\nShape of A: {A.shape}.")
print(f"Shape of A_bar: {A_bar.shape}.")
objective = cp.Maximize(lambda_min(-A_bar))
constraint = [A * N == 0, norm(A) <= 10]
problem = cp.Problem(objective, constraint)
problem.solve()
gain_A = np.array(A.value).round(5)
# gain_A = np.matrix(A.value)
print(f"Status: {problem.status}. \n")
print(f"Optimal value: {problem.value}. \n")
print(f"Optimal variable: \n{gain_A}. \n")
print(f"Shape of gain_A: {gain_A.shape}")
gain_Ai = np.zeros((number_of_agent, number_of_agent, 2,
2)) # I try putting everything nice and neat
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
