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, 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 format(self):
"""Formats SOC constraints as inequalities for the solver.
"""
constraints = [lu.create_geq(self.t)]
for elem in self.x_elems:
constraints.append(lu.create_geq(elem))
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)
"""
# 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 __SCS_format(self):
eq_constr = self._get_eq_constr()
term = self._scaled_lower_tri()
if self.constr_id is None:
leq_constr = lu.create_geq(term)
else:
leq_constr = lu.create_geq(term, constr_id=self.constr_id)
return ([eq_constr], [leq_constr])
def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
leq_constr = [lu.create_geq(self.t)]
for elem in self.x_elems:
leq_constr.append(lu.create_geq(elem))
return ([], leq_constr)
def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
leq_constr = [lu.create_geq(self.t)]
for elem in self.x_elems:
leq_constr.append(lu.create_geq(elem))
return ([], leq_constr)
def __CVXOPT_format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
eq_constr = self._get_eq_constr()
if self.constr_id is None:
leq_constr = lu.create_geq(self.A)
else:
leq_constr = lu.create_geq(self.A, constr_id=self.constr_id)
return ([eq_constr], [leq_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)
"""
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 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 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)
"""
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 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 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((1, 1))
promoted_t = lu.promote(t, x.size)
constraints = [
lu.create_geq(lu.sum_expr([x, promoted_t])),
lu.create_leq(x, promoted_t)
]
return (t, 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]
t = lu.create_var((1, 1))
constraints = [ExpCone(t, x, y),
lu.create_geq(y)] # 0 <= y
# -t - x + y
obj = lu.sub_expr(y, lu.sum_expr([x, t]))
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)
"""
# 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)
"""
x = arg_objs[0]
y = arg_objs[1]
t = lu.create_var((1, 1))
constraints = [ExpCone(t, x, y),
lu.create_geq(y)] # 0 <= y
# -t - x + y
obj = lu.sub_expr(y, lu.sum_expr([x, t]))
return (obj, constraints)
def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
leq_constr = lu.create_geq(self.expr, constr_id=self.constr_id)
return [leq_constr]
def format(self, eq_constr, leq_constr, dims, solver):
"""Formats SOC constraints as inequalities for the solver.
Parameters
----------
eq_constr : list
A list of the equality constraints in the canonical problem.
leq_constr : list
A list of the inequality constraints in the canonical problem.
dims : dict
A dict with the dimensions of the conic constraints.
solver : str
The solver being called.
"""
leq_constr.append(lu.create_geq(self.t))
for elem in self.x_elems:
leq_constr.append(lu.create_geq(elem))
# Update dims.
dims[s.SOC_DIM].append(self.size[0])
def format(self, eq_constr, leq_constr, dims, solver):
"""Formats SOC constraints as inequalities for the solver.
Parameters
----------
eq_constr : list
A list of the equality constraints in the canonical problem.
leq_constr : list
A list of the inequality constraints in the canonical problem.
dims : dict
A dict with the dimensions of the conic constraints.
solver : str
The solver being called.
"""
leq_constr.append(lu.create_geq(self.t))
for elem in self.x_elems:
leq_constr.append(lu.create_geq(elem))
# Update dims.
dims[s.SOC_DIM].append(self.size[0])
def constraints(self):
obj, constraints = super(BoolVar, self).canonicalize()
one = lu.create_const(1, (1, 1))
constraints += [lu.create_geq(obj),
lu.create_leq(obj, one)]
for i in range(self.size[0]):
row_sum = lu.sum_expr([self[i, j] for j in range(self.size[0])])
col_sum = lu.sum_expr([self[j, i] for j in range(self.size[0])])
constraints += [lu.create_eq(row_sum, one),
lu.create_eq(col_sum, one)]
return constraints
def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
eq_constr = lu.create_eq(self.A, lu.transpose(self.A))
leq_constr = lu.create_geq(self.A)
return ([eq_constr], [leq_constr])
def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
eq_constr = lu.create_eq(self.A, lu.transpose(self.A))
leq_constr = lu.create_geq(self.A)
return ([eq_constr], [leq_constr])
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 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 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 format(self, eq_constr, leq_constr, dims, solver):
"""Formats SDP constraints as inequalities for the solver.
Parameters
----------
eq_constr : list
A list of the equality constraints in the canonical problem.
leq_constr : list
A list of the inequality constraints in the canonical problem.
dims : dict
A dict with the dimensions of the conic constraints.
solver : str
The solver being called.
"""
# A == A.T
eq_constr.append(lu.create_eq(self.A, lu.transpose(self.A)))
# 0 <= A
leq_constr.append(lu.create_geq(self.A))
# Update dims.
dims[s.EQ_DIM] += self.size[0]*self.size[1]
dims[s.SDP_DIM].append(self.size[0])
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)
# x >= 0 implied by x >= t^2.
obj, constraints = square.graph_implementation([t], size)
return (t, constraints + [lu.create_leq(obj, x), lu.create_geq(t)])
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)
# x >= 0 implied by x >= t^2.
obj, constraints = square.graph_implementation([t], size)
return (t, constraints + [lu.create_leq(obj, x), lu.create_geq(t)])
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((1, 1))
promoted_t = lu.promote(t, x.size)
constraints = [lu.create_geq(lu.sum_expr([x, promoted_t])), lu.create_leq(x, promoted_t)]
return (t, 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 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 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 format(self):
"""Formats SDP constraints as inequalities for the solver.
"""
# 0 <= A
return [lu.create_geq(self.A)]
def graph_implementation(arg_objs, size, data=None):
r"""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)
Notes
-----
Implementation notes.
For general ``p``, the p-norm is equivalent to the following convex inequalities:
.. math::
x_i &\leq r_i\\
-x_i &\leq r_i\\
r_i &\leq s_i^{1/p} t^{1 - 1/p}\\
\sum_i s_i &\leq t,
where :math:`p \geq 1`.
These inequalities are also correct for :math:`p = +\infty` if we interpret :math:`1/\infty` as :math:`0`.
Although the inequalities above are correct, for a few special cases, we can represent the p-norm
more efficiently and with fewer variables and inequalities.
- For :math:`p = 1`, we use the representation
.. math::
x_i &\leq r_i\\
-x_i &\leq r_i\\
\sum_i r_i &\leq t
- For :math:`p = \infty`, we use the representation
.. math::
x_i &\leq t\\
-x_i &\leq t
Note that we don't need the :math:`s` variables or the sum inequality.
- For :math:`p = 2`, we use the natural second-order cone representation
.. math::
\|x\|_2 \leq t
Note that we could have used the set of inequalities given above if we wanted an alternate decomposition
of a large second-order cone into into several smaller inequalities.
"""
p, w = data
x = arg_objs[0]
t = None # dummy value so linter won't complain about initialization
if p != 1:
t = lu.create_var((1, 1))
if p == 2:
return t, [SOC(t, [x])]
if p == np.inf:
r = lu.promote(t, x.size)
else:
r = lu.create_var(x.size)
constraints = [lu.create_geq(lu.sum_expr([x, r])),
lu.create_leq(x, r)]
if p == 1:
return lu.sum_entries(r), constraints
if p == np.inf:
return t, constraints
# otherwise do case of general p
s = lu.create_var(x.size)
# todo: no need to run gm_constr to form the tree each time. we only need to form the tree once
constraints += gm_constrs(r, [s, lu.promote(t, x.size)], w)
constraints += [lu.create_leq(lu.sum_entries(s), t)]
return t, constraints
def canonicalize(self):
obj, constraints = super(BoolVar, self).canonicalize()
one = lu.create_const(1, (1, 1))
constraints += [lu.create_geq(obj),
lu.create_leq(obj, one)]
return (obj, constraints)
def graph_implementation(arg_objs, size, data=None):
r"""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)
Notes
-----
Implementation notes.
- For general :math:`p \geq 1`, the inequality :math:`\|x\|_p \leq t`
is equivalent to the following convex inequalities:
.. math::
|x_i| &\leq r_i^{1/p} t^{1 - 1/p}\\
\sum_i r_i &= t.
These inequalities happen to also be correct for :math:`p = +\infty`,
if we interpret :math:`1/\infty` as :math:`0`.
- For general :math:`0 < p < 1`, the inequality :math:`\|x\|_p \geq t`
is equivalent to the following convex inequalities:
.. math::
r_i &\leq x_i^{p} t^{1 - p}\\
\sum_i r_i &= t.
- For general :math:`p < 0`, the inequality :math:`\|x\|_p \geq t`
is equivalent to the following convex inequalities:
.. math::
t &\leq x_i^{-p/(1-p)} r_i^{1/(1 - p)}\\
\sum_i r_i &= t.
Although the inequalities above are correct, for a few special cases, we can represent the p-norm
more efficiently and with fewer variables and inequalities.
- For :math:`p = 1`, we use the representation
.. math::
x_i &\leq r_i\\
-x_i &\leq r_i\\
\sum_i r_i &= t
- For :math:`p = \infty`, we use the representation
.. math::
x_i &\leq t\\
-x_i &\leq t
Note that we don't need the :math:`r` variable or the sum inequality.
- For :math:`p = 2`, we use the natural second-order cone representation
.. math::
\|x\|_2 \leq t
Note that we could have used the set of inequalities given above if we wanted an alternate decomposition
of a large second-order cone into into several smaller inequalities.
"""
p = data[0]
x = arg_objs[0]
t = lu.create_var((1, 1))
constraints = []
# first, take care of the special cases of p = 2, inf, and 1
if p == 2:
return t, [SOC(t, [x])]
if p == np.inf:
t_ = lu.promote(t, x.size)
return t, [lu.create_leq(x, t_), lu.create_geq(lu.sum_expr([x, t_]))]
# we need an absolute value constraint for the symmetric convex branches (p >= 1)
# we alias |x| as x from this point forward to make the code pretty :)
if p >= 1:
absx = lu.create_var(x.size)
constraints += [lu.create_leq(x, absx), lu.create_geq(lu.sum_expr([x, absx]))]
x = absx
if p == 1:
return lu.sum_entries(x), constraints
# now, we take care of the remaining convex and concave branches
# to create the rational powers, we need a new variable, r, and
# the constraint sum(r) == t
r = lu.create_var(x.size)
t_ = lu.promote(t, x.size)
constraints += [lu.create_eq(lu.sum_entries(r), t)]
# make p a fraction so that the input weight to gm_constrs
# is a nice tuple of fractions.
p = Fraction(p)
if p < 0:
constraints += gm_constrs(t_, [x, r], (-p / (1 - p), 1 / (1 - p)))
if 0 < p < 1:
constraints += gm_constrs(r, [x, t_], (p, 1 - p))
if p > 1:
constraints += gm_constrs(x, [r, t_], (1 / p, 1 - 1 / p))
return t, constraints
def canonicalize(self):
obj, constraints = super(Boolean, self).canonicalize()
one = lu.create_const(np.ones(self.size), self.size)
constraints += [lu.create_geq(obj), lu.create_leq(obj, one)]
return (obj, constraints)
def __SCS_format(self):
eq_constr = self._get_eq_constr()
term = self._scaled_lower_tri()
leq_constr = lu.create_geq(term)
return ([eq_constr], [leq_constr])
def canonicalize(self):
"""Enforce that var >= 0.
"""
obj, constr = super(NonNegative, self).canonicalize()
return (obj, constr + [lu.create_geq(obj)])
def canonicalize(self):
obj, constraints = super(BoolVar, self).canonicalize()
one = lu.create_const(1, (1, 1))
constraints += [lu.create_geq(obj), lu.create_leq(obj, one)]
return (obj, constraints)
def canonicalize(self):
obj, constraints = super(Boolean, self).canonicalize()
one = lu.create_const(np.ones(self.size), self.size)
constraints += [lu.create_geq(obj),
lu.create_leq(obj, one)]
return (obj, constraints)
def canonicalize(self):
"""Enforce that var >= 0.
"""
obj, constr = super(NonNegative, self).canonicalize()
return (obj, constr + [lu.create_geq(obj)])
