def graph_implementation(arg_objs, size, data=None):
"""Index/slice into the expression.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data : tuple
A tuple of slices.
Returns
-------
tuple
(LinOp, [constraints])
"""
obj = lu.index(arg_objs[0], size, data[0])
return (obj, [])
def graph_implementation(arg_objs, size, data=None):
"""Index/slice into the expression.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data : tuple
A tuple of slices.
Returns
-------
tuple
(LinOp, [constraints])
"""
obj = lu.index(arg_objs[0], size, data[0])
return (obj, [])
def graph_implementation(self,
arg_objs,
shape: Tuple[int, ...],
data=None) -> Tuple[lo.LinOp, List[Constraint]]:
"""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])
"""
obj = lu.index(arg_objs[0], shape, data[0])
return (obj, [])
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]
axis = data[1]
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:
if axis is None:
return t, [SOC(t, [x])]
elif axis == 0:
size = (1, x.size[1])
t = lu.create_var(size)
return t, [SOC_Elemwise(
t, [lu.index(x, size, (slice(i, i+1), slice(0, x.size[1])))
for i in range(x.size[0])])]
else: # axis == 1
size = (x.size[0], 1)
t = lu.create_var(size)
return t, [SOC_Elemwise(
t, [lu.index(x, size, (slice(0, x.size[0]), slice(i, i+1)))
for i in range(x.size[1])])]
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
