def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
axis = data[0]
t = lu.create_var(size)
# sum(exp(x - t)) <= 1
if axis is None:
prom_t = lu.promote(t, x.size)
expr = lu.sub_expr(x, prom_t)
obj, constraints = exp.graph_implementation([expr], x.size)
obj = lu.sum_entries(obj)
elif axis == 0:
prom_size = (x.size[0], 1)
ones = lu.create_const(np.ones(prom_size), prom_size)
prom_t = lu.mul_expr(ones, t, x.size)
expr = lu.sub_expr(x, prom_t)
obj, constraints = exp.graph_implementation([expr], x.size)
const_size = (1, x.size[0])
ones = lu.create_const(np.ones(const_size), const_size)
obj = lu.mul_expr(ones, obj, size)
else: # axis == 1
prom_size = (1, x.size[1])
ones = lu.create_const(np.ones(prom_size), prom_size)
prom_t = lu.rmul_expr(t, ones, x.size)
expr = lu.sub_expr(x, prom_t)
obj, constraints = exp.graph_implementation([expr], x.size)
const_size = (x.size[1], 1)
ones = lu.create_const(np.ones(const_size), const_size)
obj = lu.rmul_expr(obj, ones, size)
ones = lu.create_const(np.ones(size), size)
constraints += [lu.create_leq(obj, ones)]
return (t, constraints)
def constr_func(aff_obj):
G_aff = G.canonical_form[0]
h_aff = h.canonical_form[0]
Gx = lu.mul_expr(G_aff, aff_obj, h_aff.size)
constraints = [lu.create_leq(Gx, h_aff)]
if A is not None:
A_const, b_const = map(self.cast_to_const, [A, b])
A_aff = A_const.canonical_form[0]
b_aff = b_const.canonical_form[0]
Ax = lu.mul_expr(A_aff, aff_obj, b_aff.size)
constraints += [lu.create_eq(Ax, b_aff)]
return constraints
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# min sum_entries(t) + kq
# s.t. x <= t + q
# 0 <= t
x = arg_objs[0]
k = lu.create_const(data[0], (1, 1))
q = lu.create_var((1, 1))
t = lu.create_var(x.size)
sum_t, constr = sum_entries.graph_implementation([t], (1, 1))
obj = lu.sum_expr([sum_t, lu.mul_expr(k, q, (1, 1))])
prom_q = lu.promote(q, x.size)
constr.append( lu.create_leq(x, lu.sum_expr([t, prom_q])) )
constr.append( lu.create_geq(t) )
return (obj, constr)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
axis = data[0]
if axis is None:
t = lu.create_var((1, 1))
promoted_t = lu.promote(t, arg_objs[0].size)
elif axis == 0:
t = lu.create_var((1, arg_objs[0].size[1]))
const_size = (arg_objs[0].size[0], 1)
ones = lu.create_const(np.ones(const_size), const_size)
promoted_t = lu.mul_expr(ones, t, arg_objs[0].size)
else: # axis == 1
t = lu.create_var((arg_objs[0].size[0], 1))
const_size = (1, arg_objs[0].size[1])
ones = lu.create_const(np.ones(const_size), const_size)
promoted_t = lu.rmul_expr(t, ones, arg_objs[0].size)
constraints = [lu.create_leq(arg_objs[0], promoted_t)]
return (t, constraints)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# TODO use log for n != 2.
v = lu.create_var((1, 1))
x = arg_objs[0]
y = arg_objs[1]
two = lu.create_const(2, (1, 1))
# SOC(x + y, [y - x, 2*v])
constraints = [
SOC(lu.sum_expr([x, y]),
[lu.sub_expr(y, x),
lu.mul_expr(two, v, (1, 1))])
]
# 0 <= x, 0 <= y
constraints += [lu.create_geq(x), lu.create_geq(y)]
return (v, constraints)
def graph_implementation(arg_objs, shape, data=None):
"""Multiply the linear expressions.
Parameters
----------
arg_objs : list
LinExpr for each argument.
shape : tuple
The shape of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# Promote shapes for compatibility with CVXCanon
lhs = arg_objs[0]
rhs = arg_objs[1]
if lu.is_const(lhs):
return (lu.mul_expr(lhs, rhs, shape), [])
elif lu.is_const(rhs):
return (lu.rmul_expr(lhs, rhs, shape), [])
else:
raise DCPError("Product of two non-constant expressions is not "
"DCP.")
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# Promote scalars.
for idx, arg in enumerate(arg_objs):
if arg.size != size:
arg_objs[idx] = lu.promote(arg, size)
x = arg_objs[0]
y = arg_objs[1]
v = lu.create_var(x.size)
two = lu.create_const(2, (1, 1))
# SOC(x + y, [y - x, 2*v])
constraints = [
SOC_Elemwise(lu.sum_expr([x, y]),
[lu.sub_expr(y, x),
lu.mul_expr(two, v, v.size)])
]
# 0 <= x, 0 <= y
constraints += [lu.create_geq(x), lu.create_geq(y)]
return (v, constraints)
def graph_implementation(arg_objs, size, data=None):
"""Sum the linear expression's entries.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
axis = data[0]
if axis is None:
obj = lu.sum_entries(arg_objs[0])
elif axis == 1:
const_size = (arg_objs[0].size[1], 1)
ones = lu.create_const(np.ones(const_size), const_size)
obj = lu.rmul_expr(arg_objs[0], ones, size)
else: # axis == 0
const_size = (1, arg_objs[0].size[0])
ones = lu.create_const(np.ones(const_size), const_size)
obj = lu.mul_expr(ones, arg_objs[0], size)
return (obj, [])
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0]
rows, cols = A.size
# Create the equivalent problem:
# minimize (trace(U) + trace(V))/2
# subject to:
# [U A; A.T V] is positive semidefinite
X = lu.create_var((rows + cols, rows + cols))
constraints = []
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:rows,rows:rows+cols] == A
index.block_eq(X, A, constraints, 0, rows, rows, rows + cols)
half = lu.create_const(0.5, (1, 1))
trace = lu.mul_expr(half, lu.trace(X), (1, 1))
# Add SDP constraint.
return (trace, [SDP(X)] + constraints)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0]
n, _ = A.size
# Requires that A is symmetric.
# A == A.T
obj, constraints = transpose.graph_implementation([A], (n, n))
constraints.append(lu.create_eq(A, obj))
# SDP constraint.
t = lu.create_var((1, 1))
I = lu.create_const(sp.eye(n, n), (n, n))
# I*t - A
expr = lu.sub_expr(lu.mul_expr(I, t, (n, n)), A)
return (t, [SDP(expr)] + constraints)
def graph_implementation(arg_objs, shape, data=None):
"""Cumulative sum via difference matrix.
Parameters
----------
arg_objs : list
LinExpr for each argument.
shape : tuple
The shape of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# Implicit O(n) definition:
# X = Y[1:,:] - Y[:-1, :]
Y = lu.create_var(shape)
axis = data[0]
dim = shape[axis]
diff_mat = get_diff_mat(dim, axis)
diff_mat = lu.create_const(diff_mat, (dim, dim), sparse=True)
if axis == 0:
diff = lu.mul_expr(diff_mat, Y)
else:
diff = lu.rmul_expr(Y, diff_mat)
return (Y, [lu.create_eq(arg_objs[0], diff)])
def graph_implementation(arg_objs, size, data=None):
"""Cumulative sum via difference matrix.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# Implicit O(n) definition:
# X = Y[:1,:] - Y[1:,:]
Y = lu.create_var(size)
axis = data[0]
dim = size[axis]
diff_mat = get_diff_mat(dim, axis)
diff_mat = lu.create_const(diff_mat, (dim, dim), sparse=True)
if axis == 0:
diff = lu.mul_expr(diff_mat, Y, size)
else:
diff = lu.rmul_expr(Y, diff_mat, size)
return (Y, [lu.create_eq(arg_objs[0], diff)])
def graph_implementation(self, arg_objs, shape, data=None):
"""Index/slice into the expression.
Parameters
----------
arg_objs : list
LinExpr for each argument.
shape : tuple
The shape of the resulting expression.
data : tuple
A tuple of slices.
Returns
-------
tuple
(LinOp, [constraints])
"""
select_mat = self._select_mat
final_shape = self._select_mat.shape
select_vec = np.reshape(select_mat, select_mat.size, order='F')
# Select the chosen entries from expr.
arg = arg_objs[0]
identity = sp.eye(self.args[0].size).tocsc()
vec_arg = lu.reshape(arg, (self.args[0].size, ))
mul_mat = identity[select_vec]
mul_const = lu.create_const(mul_mat, mul_mat.shape, sparse=True)
mul_expr = lu.mul_expr(mul_const, vec_arg, (mul_mat.shape[0], ))
obj = lu.reshape(mul_expr, final_shape)
return (obj, [])
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0]
n, _ = A.size
# Requires that A is symmetric.
# A == A.T
obj, constraints = transpose.graph_implementation([A], (n, n))
constraints.append(lu.create_eq(A, obj))
# SDP constraint.
t = lu.create_var((1, 1))
I = lu.create_const(sp.eye(n, n), (n, n))
# I*t - A
expr = lu.sub_expr(lu.mul_expr(I, t, (n, n)), A)
return (t, [SDP(expr)] + constraints)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
y = arg_objs[1] # Known to be a scalar.
v = lu.create_var((1, 1))
two = lu.create_const(2, (1, 1))
constraints = [SOC(lu.sum_expr([y, v]),
[lu.sub_expr(y, v),
lu.mul_expr(two, x, x.size)]),
lu.create_geq(y)]
return (v, constraints)
def qol_elemwise(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
y = arg_objs[1]
t = lu.create_var(x.size)
two = lu.create_const(2, (1, 1))
constraints = [SOC_Elemwise(lu.sum_expr([y, t]),
[lu.sub_expr(y, t),
lu.mul_expr(two, x, x.size)]),
lu.create_geq(y)]
return (t, constraints)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
y = arg_objs[1] # Known to be a scalar.
v = lu.create_var((1, 1))
two = lu.create_const(2, (1, 1))
constraints = [SOC(lu.sum_expr([y, v]),
[lu.sub_expr(y, v),
lu.mul_expr(two, x, x.size)]),
lu.create_geq(y)]
return (v, constraints)
def _scaled_lower_tri(self):
"""Returns a LinOp representing the lower triangular entries.
Scales the strictly lower triangular entries by
sqrt(2) as required by SCS.
"""
rows = cols = self.size[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
size = (entries, rows*cols)
coeff = sp.coo_matrix((val_arr, (row_arr, col_arr)), size).tocsc()
coeff = lu.create_const(coeff, size, sparse=True)
vect = lu.reshape(self.A, (rows*cols, 1))
return lu.mul_expr(coeff, vect, (entries, 1))
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# Promote scalars.
for idx, arg in enumerate(arg_objs):
if arg.size != size:
arg_objs[idx] = lu.promote(arg, size)
x = arg_objs[0]
y = arg_objs[1]
v = lu.create_var(x.size)
two = lu.create_const(2, (1, 1))
# SOC(x + y, [y - x, 2*v])
constraints = [
SOC_Elemwise(lu.sum_expr([x, y]),
[lu.sub_expr(y, x),
lu.mul_expr(two, v, v.size)])
]
# 0 <= x, 0 <= y
constraints += [lu.create_geq(x), lu.create_geq(y)]
return (v, constraints)
def graph_implementation(arg_objs, size, data=None):
"""Sum the linear expression's entries.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
axis = data[0]
if axis is None:
obj = lu.sum_entries(arg_objs[0])
elif axis == 1:
const_size = (arg_objs[0].size[1], 1)
ones = lu.create_const(np.ones(const_size), const_size)
obj = lu.rmul_expr(arg_objs[0], ones, size)
else: # axis == 0
const_size = (1, arg_objs[0].size[0])
ones = lu.create_const(np.ones(const_size), const_size)
obj = lu.mul_expr(ones, arg_objs[0], size)
return (obj, [])
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0]
rows, cols = A.size
# Create the equivalent problem:
# minimize (trace(U) + trace(V))/2
# subject to:
# [U A; A.T V] is positive semidefinite
X = lu.create_var((rows+cols, rows+cols))
constraints = []
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:rows,rows:rows+cols] == A
index.block_eq(X, A, constraints,
0, rows, rows, rows+cols)
half = lu.create_const(0.5, (1, 1))
trace = lu.mul_expr(half, lu.trace(X), (1, 1))
# Add SDP constraint.
return (trace, [SDP(X)] + constraints)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
w = lu.create_var(size)
v = lu.create_var(size)
two = lu.create_const(2, (1, 1))
# w**2 + 2*v
obj, constraints = square.graph_implementation([w], size)
obj = lu.sum_expr([obj, lu.mul_expr(two, v, size)])
# x <= w + v
constraints.append(lu.create_leq(x, lu.sum_expr([w, v])))
# v >= 0
constraints.append(lu.create_geq(v))
return (obj, constraints)
def qol_elemwise(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
y = arg_objs[1]
t = lu.create_var(x.size)
two = lu.create_const(2, (1, 1))
constraints = [SOC_Elemwise(lu.sum_expr([y, t]),
[lu.sub_expr(y, t),
lu.mul_expr(two, x, x.size)]),
lu.create_geq(y)]
return (t, constraints)
def gm(t, x, y):
length = t.size[0]*t.size[1]
return SOC_Axis(lu.reshape(lu.sum_expr([x, y]), (length, 1)),
lu.vstack([
lu.reshape(lu.sub_expr(x, y), (1, length)),
lu.reshape(lu.mul_expr(two, t, t.size), (1, length))
], (2, length)),
0)
def gm(t, x, y):
length = t.size[0] * t.size[1]
return SOC_Axis(
lu.reshape(lu.sum_expr([x, y]), (length, 1)),
lu.vstack([
lu.reshape(lu.sub_expr(x, y), (1, length)),
lu.reshape(lu.mul_expr(two, t, t.size), (1, length))
], (2, length)), 0)
def format_axis(t, X, axis):
"""Formats all the row/column cones for the solver.
Parameters
----------
t: The scalar part of the second-order constraint.
X: A matrix whose rows/columns are each a cone.
axis: Slice by column 0 or row 1.
Returns
-------
list
A list of LinLeqConstr that represent all the elementwise cones.
"""
# Reduce to norms of columns.
if axis == 1:
X = lu.transpose(X)
# Create matrices Tmat, Xmat such that Tmat*t + Xmat*X
# gives the format for the elementwise cone constraints.
cone_size = 1 + X.shape[0]
terms = []
# Make t_mat
mat_shape = (cone_size, 1)
t_mat = sp.coo_matrix(([1.0], ([0], [0])), mat_shape).tocsc()
t_mat = lu.create_const(t_mat, mat_shape, sparse=True)
t_vec = t
if not t.shape:
# t is scalar
t_vec = lu.reshape(t, (1, 1))
else:
# t is 1D
t_vec = lu.reshape(t, (1, t.shape[0]))
mul_shape = (cone_size, t_vec.shape[1])
terms += [lu.mul_expr(t_mat, t_vec, mul_shape)]
# Make X_mat
if len(X.shape) == 1:
X = lu.reshape(X, (X.shape[0], 1))
mat_shape = (cone_size, X.shape[0])
val_arr = (cone_size - 1) * [1.0]
row_arr = list(range(1, cone_size))
col_arr = list(range(cone_size - 1))
X_mat = sp.coo_matrix((val_arr, (row_arr, col_arr)), mat_shape).tocsc()
X_mat = lu.create_const(X_mat, mat_shape, sparse=True)
mul_shape = (cone_size, X.shape[1])
terms += [lu.mul_expr(X_mat, X, mul_shape)]
return [lu.create_geq(lu.sum_expr(terms))]
def canonicalize(self):
"""Variable must be semidefinite and symmetric.
"""
upper_tri = lu.create_var((self.size[0], 1), self.id)
fill_coeff = upper_tri_to_full(self.n)
fill_coeff = lu.create_const(fill_coeff, (self.n*self.n, self.size[0]),
sparse=True)
full_mat = lu.mul_expr(fill_coeff, upper_tri, (self.n*self.n, 1))
full_mat = lu.reshape(full_mat, (self.n, self.n))
return (upper_tri, [SDP(full_mat, enforce_sym=False)])
def graph_implementation(arg_objs, shape, data=None):
"""Reduces the atom to an affine expression and list of constraints.
minimize n^2 + 2M|s|
subject to s + n = x
Parameters
----------
arg_objs : list
LinExpr for each argument.
shape : tuple
The shape of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
M = data[0]
x = arg_objs[0]
n = lu.create_var(shape)
s = lu.create_var(shape)
two = lu.create_const(2, (1, 1))
if isinstance(M, Parameter):
M = lu.create_param(M, (1, 1))
else: # M is constant.
M = lu.create_const(M.value, (1, 1))
# n**2 + 2*M*|s|
n2, constr_sq = power.graph_implementation(
[n],
shape, (2, (Fraction(1, 2), Fraction(1, 2)))
)
abs_s, constr_abs = abs.graph_implementation([s], shape)
M_abs_s = lu.mul_expr(M, abs_s)
obj = lu.sum_expr([n2, lu.mul_expr(two, M_abs_s)])
# x == s + n
constraints = constr_sq + constr_abs
constraints.append(lu.create_eq(x, lu.sum_expr([n, s])))
return (obj, constraints)
def graph_implementation(arg_objs,size,data=None):
x = arg_objs[0]
beta,x0 = data[0],data[1]
beta_val,x0_val = beta.value,x0.value
if isinstance(beta,Parameter):
beta = lu.create_param(beta,(1,1))
else:
beta = lu.create_const(beta.value,(1,1))
if isinstance(x0,Parameter):
x0 = lu.create_param(x0,(1,1))
else:
x0 = lu.create_const(x0.value,(1,1))
xi,psi = lu.create_var(size),lu.create_var(size)
one = lu.create_const(1,(1,1))
one_over_beta = lu.create_const(1/beta_val,(1,1))
k = np.exp(-beta_val*x0_val)
k = lu.create_const(k,(1,1))
# 1/beta * (1 - exp(-beta*(xi+x0)))
xi_plus_x0 = lu.sum_expr([xi,x0])
minus_beta_times_xi_plus_x0 = lu.neg_expr(lu.mul_expr(beta,xi_plus_x0,size))
exp_xi,constr_exp = exp.graph_implementation([minus_beta_times_xi_plus_x0],size)
minus_exp_minus_etc = lu.neg_expr(exp_xi)
left_branch = lu.mul_expr(one_over_beta, lu.sum_expr([one,minus_exp_minus_etc]),size)
# psi*exp(-beta*r0)
right_branch = lu.mul_expr(k,psi,size)
obj = lu.sum_expr([left_branch,right_branch])
#x-x0 == xi + psi, xi >= 0, psi <= 0
zero = lu.create_const(0,size)
constraints = constr_exp
prom_x0 = lu.promote(x0, size)
constraints.append(lu.create_eq(x,lu.sum_expr([prom_x0,xi,psi])))
constraints.append(lu.create_geq(xi,zero))
constraints.append(lu.create_leq(psi,zero))
return (obj, constraints)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0] # m by n matrix.
n, m = A.size
# Create a matrix with Schur complement I*t - (1/t)*A.T*A.
X = lu.create_var((n+m, n+m))
t = lu.create_var((1, 1))
I_n = lu.create_const(sp.eye(n), (n, n))
I_m = lu.create_const(sp.eye(m), (m, m))
# Expand A.T.
obj, constraints = transpose.graph_implementation([A], (m, n))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n, 0:n] == I_n*t
index.block_eq(X, lu.mul_expr(I_n, t, (n, n)), constraints,
0, n, 0, n)
# X[0:n, n:n+m] == A
index.block_eq(X, A, constraints,
0, n, n, n+m)
# X[n:n+m, 0:n] == obj
index.block_eq(X, obj, constraints,
n, n+m, 0, n)
# X[n:n+m, n:n+m] == I_m*t
index.block_eq(X, lu.mul_expr(I_m, t, (m, m)), constraints,
n, n+m, n, n+m)
# Add SDP constraint.
return (t, constraints + [SDP(X)])
def canonicalize(self):
obj, constraints = super(Assign, self).canonicalize()
shape = (self.size[1], 1)
one_row_vec = lu.create_const(np.ones(shape), shape)
shape = (1, self.size[0])
one_col_vec = lu.create_const(np.ones(shape), shape)
# Row sum <= 1
row_sum = lu.rmul_expr(obj, one_row_vec, (self.size[0], 1))
constraints += [lu.create_leq(row_sum, lu.transpose(one_col_vec))]
# Col sum == 1.
col_sum = lu.mul_expr(one_col_vec, obj, (1, self.size[1]))
constraints += [lu.create_eq(col_sum, lu.transpose(one_row_vec))]
return (obj, constraints)
def canonicalize(self):
obj, constraints = super(Assign, self).canonicalize()
shape = (self.size[1], 1)
one_row_vec = lu.create_const(np.ones(shape), shape)
shape = (1, self.size[0])
one_col_vec = lu.create_const(np.ones(shape), shape)
# Row sum <= 1
row_sum = lu.rmul_expr(obj, one_row_vec, (self.size[0], 1))
constraints += [lu.create_leq(row_sum, lu.transpose(one_col_vec))]
# Col sum == 1.
col_sum = lu.mul_expr(one_col_vec, obj, (1, self.size[1]))
constraints += [lu.create_eq(col_sum, lu.transpose(one_row_vec))]
return (obj, constraints)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
minimize n^2 + 2M|s|
subject to s + n = x
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
M = data
x = arg_objs[0]
n = lu.create_var(size)
s = lu.create_var(size)
two = lu.create_const(2, (1, 1))
if isinstance(M, Parameter):
M = lu.create_param(M, (1, 1))
else: # M is constant.
M = lu.create_const(M.value, (1, 1))
# n**2 + 2*M*|s|
n2, constr_sq = square.graph_implementation([n], size)
abs_s, constr_abs = abs.graph_implementation([s], size)
M_abs_s = lu.mul_expr(M, abs_s, size)
obj = lu.sum_expr([n2, lu.mul_expr(two, M_abs_s, size)])
# x == s + n
constraints = constr_sq + constr_abs
constraints.append(lu.create_eq(x, lu.sum_expr([n, s])))
return (obj, constraints)
def format_axis(t, X, axis):
"""Formats all the row/column cones for the solver.
Parameters
----------
t: The scalar part of the second-order constraint.
X: A matrix whose rows/columns are each a cone.
axis: Slice by column 0 or row 1.
Returns
-------
list
A list of LinLeqConstr that represent all the elementwise cones.
"""
# Reduce to norms of columns.
if axis == 1:
X = lu.transpose(X)
# Create matrices Tmat, Xmat such that Tmat*t + Xmat*X
# gives the format for the elementwise cone constraints.
num_cones = t.size[0]
cone_size = 1 + X.size[0]
terms = []
# Make t_mat
mat_size = (cone_size, 1)
prod_size = (cone_size, t.size[0])
t_mat = sp.coo_matrix(([1.0], ([0], [0])), mat_size).tocsc()
t_mat = lu.create_const(t_mat, mat_size, sparse=True)
terms += [lu.mul_expr(t_mat, lu.transpose(t), prod_size)]
# Make X_mat
mat_size = (cone_size, X.size[0])
prod_size = (cone_size, X.size[1])
val_arr = (cone_size - 1)*[1.0]
row_arr = range(1, cone_size)
col_arr = range(cone_size-1)
X_mat = sp.coo_matrix((val_arr, (row_arr, col_arr)), mat_size).tocsc()
X_mat = lu.create_const(X_mat, mat_size, sparse=True)
terms += [lu.mul_expr(X_mat, X, prod_size)]
return [lu.create_geq(lu.sum_expr(terms))]
def canonicalize(self):
"""Returns the graph implementation of the object.
Marks the top level constraint as the dual_holder,
so the dual value will be saved to the EqConstraint.
Returns:
A tuple of (affine expression, [constraints]).
"""
obj, constraints = self._expr.canonical_form
half = lu.create_const(0.5, (1,1))
symm = lu.mul_expr(half, lu.sum_expr([obj, lu.transpose(obj)]),
obj.size)
dual_holder = SDP(symm, enforce_sym=False, constr_id=self.id)
return (None, constraints + [dual_holder])
def canonicalize(self):
"""Returns the graph implementation of the object.
Marks the top level constraint as the dual_holder,
so the dual value will be saved to the EqConstraint.
Returns:
A tuple of (affine expression, [constraints]).
"""
obj, constraints = self._expr.canonical_form
half = lu.create_const(0.5, (1, 1))
symm = lu.mul_expr(half, lu.sum_expr([obj, lu.transpose(obj)]),
obj.size)
dual_holder = SDP(symm, enforce_sym=False, constr_id=self.id)
return (None, constraints + [dual_holder])
def constr_func(aff_obj):
theta = [lu.create_var((1, 1)) for i in range(len(values))]
convex_objs = []
for val, theta_var in zip(values, theta):
val_aff = val.canonical_form[0]
convex_objs.append(
lu.mul_expr(val_aff, theta_var, val_aff.size))
convex_combo = lu.sum_expr(convex_objs)
one = lu.create_const(1, (1, 1))
constraints = [
lu.create_eq(aff_obj, convex_combo),
lu.create_eq(lu.sum_expr(theta), one)
]
for theta_var in theta:
constraints.append(lu.create_geq(theta_var))
return constraints
def constr_func(aff_obj):
theta = [lu.create_var((1, 1)) for i in xrange(len(values))]
convex_objs = []
for val, theta_var in zip(values, theta):
val_aff = val.canonical_form[0]
convex_objs.append(
lu.mul_expr(val_aff,
theta_var,
val_aff.size)
)
convex_combo = lu.sum_expr(convex_objs)
one = lu.create_const(1, (1, 1))
constraints = [lu.create_eq(aff_obj, convex_combo),
lu.create_eq(lu.sum_expr(theta), one)]
for theta_var in theta:
constraints.append(lu.create_geq(theta_var))
return constraints
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
t = lu.create_var(size)
# log(1 + exp(x)) <= t <=> exp(-t) + exp(x - t) <= 1
'''
obj0, constr0 = exp.graph_implementation([lu.neg_expr(t)], size)
obj1, constr1 = exp.graph_implementation([lu.sub_expr(x, t)], size)
lhs = lu.sum_expr([obj0, obj1])
ones = lu.create_const(np.mat(np.ones(size)), size)
constr = constr0 + constr1 + [lu.create_leq(lhs, ones)]
'''
s = data[0]
if isinstance(s, Parameter):
s = lu.create_param(s, (1, 1))
else: # M is constant.
s = lu.create_const(s, (1, 1))
#Wrong sign?
obj0, constr0 = exp.graph_implementation([lu.neg_expr(t)], size)
obj1, constr1 = exp.graph_implementation([lu.sub_expr(s, lu.sum_expr([t, x]))], size)
obj2, constr2 = exp.graph_implementation([lu.sub_expr(lu.neg_expr(s), lu.sum_expr([t, x]))], size)
obj3, constr3 = exp.graph_implementation([lu.sub_expr(lu.neg_expr(t), lu.mul_expr(2, x, size))], size)
lhs = lu.sum_expr([obj0, obj1, obj2, obj3])
ones = lu.create_const(np.mat(np.ones(size)), size)
constr = constr0 + constr1 + constr2 + constr3 + [lu.create_leq(lhs, ones)]
return (t, constr)
def graph_implementation(arg_objs, size, data=None):
"""Multiply the linear expressions.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
return (lu.mul_expr(arg_objs[0], arg_objs[1], size), [])
def graph_implementation(arg_objs, size, data=None):
"""Multiply the linear expressions.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
return (lu.mul_expr(arg_objs[0], arg_objs[1], size), [])
def graph_implementation(self,
arg_objs,
shape: Tuple[int, ...],
data=None) -> Tuple[lo.LinOp, List[Constraint]]:
"""Sum the linear expression's entries.
Parameters
----------
arg_objs : list
LinExpr for each argument.
shape : tuple
The shape of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
axis = data[0]
keepdims = data[1]
if axis is None:
obj = lu.sum_entries(arg_objs[0], shape=shape)
elif axis == 1:
if keepdims:
const_shape = (arg_objs[0].shape[1], 1)
else:
const_shape = (arg_objs[0].shape[1], )
ones = lu.create_const(np.ones(const_shape), const_shape)
obj = lu.rmul_expr(arg_objs[0], ones, shape)
else: # axis == 0
if keepdims:
const_shape = (1, arg_objs[0].shape[0])
else:
const_shape = (arg_objs[0].shape[0], )
ones = lu.create_const(np.ones(const_shape), const_shape)
obj = lu.mul_expr(ones, arg_objs[0], shape)
return (obj, [])
def graph_implementation(arg_objs, size, data=None):
"""Multiply the linear expressions.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# Promote the right hand side to a diagonal matrix if necessary.
if size[1] != 1 and arg_objs[1].size == (1, 1):
arg = lu.promote(arg_objs[1], (size[1], 1))
arg_objs[1] = lu.diag_vec(arg)
return (lu.mul_expr(arg_objs[0], arg_objs[1], size), [])
def graph_implementation(arg_objs, size, data=None):
"""Multiply the expressions elementwise.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# One of the arguments is a scalar, so we can use normal multiplication.
if arg_objs[0].size != arg_objs[1].size:
return (lu.mul_expr(arg_objs[0], arg_objs[1], size), [])
else:
return (lu.mul_elemwise(arg_objs[0], arg_objs[1]), [])
def graph_implementation(arg_objs, size, data=None):
"""Multiply the linear expressions.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# Promote the right hand side to a diagonal matrix if necessary.
if size[1] != 1 and arg_objs[1].size == (1, 1):
arg = lu.promote(arg_objs[1], (size[1], 1))
arg_objs[1] = lu.diag_vec(arg)
return (lu.mul_expr(arg_objs[0], arg_objs[1], size), [])
def graph_implementation(arg_objs, size, data=None):
"""Multiply the expressions elementwise.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
# One of the arguments is a scalar, so we can use normal multiplication.
if arg_objs[0].size != arg_objs[1].size:
return (lu.mul_expr(arg_objs[0], arg_objs[1], size), [])
else:
return (lu.mul_elemwise(arg_objs[0], arg_objs[1]), [])
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0] # m by n matrix.
n, m = A.size
# Create the equivalent problem:
# minimize (trace(U) + trace(V))/2
# subject to:
# [U A; A.T V] is positive semidefinite
X = lu.create_var((n+m, n+m))
# Expand A.T.
obj, constraints = transpose.graph_implementation([A], (m, n))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n,n:n+m] == A
index.block_eq(X, A, constraints,
0, n, n, n+m)
# X[n:n+m,0:n] == obj
index.block_eq(X, obj, constraints,
n, n+m, 0, n)
diag = [index.get_index(X, constraints, i, i) for i in range(n+m)]
half = lu.create_const(0.5, (1, 1))
trace = lu.mul_expr(half, lu.sum_expr(diag), (1, 1))
# Add SDP constraint.
return (trace, [SDP(X)] + constraints)
def format_elemwise(vars_):
"""Formats all the elementwise cones for the solver.
Parameters
----------
vars_ : list
A list of the LinOp expressions in the elementwise cones.
Returns
-------
list
A list of LinLeqConstr that represent all the elementwise cones.
"""
# Create matrices Ai such that 0 <= A0*x0 + ... + An*xn
# gives the format for the elementwise cone constraints.
spacing = len(vars_)
# Matrix spaces out columns of the LinOp expressions.
mat_shape = (spacing * vars_[0].shape[0], vars_[0].shape[0])
terms = []
for i, var in enumerate(vars_):
mat = get_spacing_matrix(mat_shape, spacing, i)
terms.append(lu.mul_expr(mat, var))
return [lu.create_geq(lu.sum_expr(terms))]
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0] # m by n matrix.
n, m = A.size
# Create the equivalent problem:
# minimize (trace(U) + trace(V))/2
# subject to:
# [U A; A.T V] is positive semidefinite
X = lu.create_var((n + m, n + m))
# Expand A.T.
obj, constraints = transpose.graph_implementation([A], (m, n))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n,n:n+m] == A
index.block_eq(X, A, constraints, 0, n, n, n + m)
# X[n:n+m,0:n] == obj
index.block_eq(X, obj, constraints, n, n + m, 0, n)
diag = [index.get_index(X, constraints, i, i) for i in range(n + m)]
half = lu.create_const(0.5, (1, 1))
trace = lu.mul_expr(half, lu.sum_expr(diag), (1, 1))
# Add SDP constraint.
return (trace, [SDP(X)] + constraints)
def format_elemwise(vars_):
"""Formats all the elementwise cones for the solver.
Parameters
----------
vars_ : list
A list of the LinOp expressions in the elementwise cones.
Returns
-------
list
A list of LinLeqConstr that represent all the elementwise cones.
"""
# Create matrices Ai such that 0 <= A0*x0 + ... + An*xn
# gives the format for the elementwise cone constraints.
spacing = len(vars_)
prod_size = (spacing*vars_[0].size[0], vars_[0].size[1])
# Matrix spaces out columns of the LinOp expressions.
mat_size = (spacing*vars_[0].size[0], vars_[0].size[0])
terms = []
for i, var in enumerate(vars_):
mat = get_spacing_matrix(mat_size, spacing, i)
terms.append(lu.mul_expr(mat, var, prod_size))
return [lu.create_geq(lu.sum_expr(terms))]
def gm(t, x, y):
return SOC_Elemwise(lu.sum_expr([x, y]),
[lu.sub_expr(x, y),
lu.mul_expr(two, t, t.size)])
def constr_func(aff_obj):
G_aff = G.canonical_form[0]
h_aff = h.canonical_form[0]
Gx = lu.mul_expr(G_aff, aff_obj, h_aff.size)
constraints = [lu.create_leq(Gx, h_aff)]
return constraints
