"""
Copyright 2013 Steven Diamond
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
"""
from typing import List, Optional
import numpy as np
from cvxpy.constraints.cones import Cone
from cvxpy.expressions import cvxtypes
from cvxpy.utilities import scopes
[docs]class SOC(Cone):
"""A second-order cone constraint for each row/column.
Assumes ``t`` is a vector the same length as ``X``'s columns (rows) for
``axis == 0`` (``1``).
Attributes:
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.
"""
def __init__(self, t, X, axis: int = 0, constr_id=None) -> None:
t = cvxtypes.expression().cast_to_const(t)
if len(t.shape) >= 2 or not t.is_real():
raise ValueError("Invalid first argument.")
# Check t has one entry per cone.
if (len(X.shape) <= 1 and t.size > 1) or \
(len(X.shape) == 2 and t.size != X.shape[1-axis]) or \
(len(X.shape) == 1 and axis == 1):
raise ValueError(
"Argument dimensions %s and %s, with axis=%i, are incompatible."
% (t.shape, X.shape, axis)
)
self.axis = axis
if len(t.shape) == 0:
t = t.flatten()
super(SOC, self).__init__([t, X], constr_id)
def __str__(self) -> str:
return "SOC(%s, %s)" % (self.args[0], self.args[1])
@property
def residual(self) -> Optional[np.ndarray]:
"""
For each cone, returns:
||(t,X) - proj(t,X)||
with
proj(t,X) = (t,X) if t >= ||x||
0.5*(t/||x|| + 1)(||x||,x) if -||x|| < t < ||x||
0 if t <= -||x||
References:
https://docs.mosek.com/modeling-cookbook/practical.html#distance-to-a-cone
https://math.stackexchange.com/questions/2509986/projection-onto-the-second-order-cone
"""
t = self.args[0].value
X = self.args[1].value
if t is None or X is None:
return None
# Reduce axis = 0 to axis = 1.
if self.axis == 0:
X = X.T
promoted = X.ndim == 1
X = np.atleast_2d(X)
# Initializing with zeros makes "0 if t <= -||x||" the default case for the projection
t_proj = np.zeros(t.shape)
X_proj = np.zeros(X.shape)
norms = np.linalg.norm(X, ord=2, axis=1)
# 1. proj(t,X) = (t,X) if t >= ||x||
t_geq_x_norm = t >= norms
t_proj[t_geq_x_norm] = t[t_geq_x_norm]
X_proj[t_geq_x_norm] = X[t_geq_x_norm]
# 2. proj(t,X) = 0.5*(t/||x|| + 1)(||x||,x) if -||x|| < t < ||x||
abs_t_less_x_norm = np.abs(t) < norms
avg_coeff = 0.5 * (1 + t/norms)
X_proj[abs_t_less_x_norm] = avg_coeff[abs_t_less_x_norm, None] * X[abs_t_less_x_norm]
t_proj[abs_t_less_x_norm] = avg_coeff[abs_t_less_x_norm] * norms[abs_t_less_x_norm]
Xt = np.concatenate([X, t[:, None]], axis=1)
Xt_proj = np.concatenate([X_proj, t_proj[:, None]], axis=1)
resid = np.linalg.norm(Xt - Xt_proj, ord=2, axis=1)
# Demote back to 1D.
if promoted:
return resid[0]
else:
return resid
def get_data(self):
"""Returns info needed to reconstruct the object besides the args.
Returns
-------
list
"""
return [self.axis, self.id]
def num_cones(self):
"""The number of elementwise cones.
"""
return self.args[0].size
@property
def size(self) -> int:
"""The number of entries in the combined cones.
"""
cone_size = 1 + self.args[1].shape[self.axis]
return cone_size * self.num_cones()
def cone_sizes(self) -> List[int]:
"""The dimensions of the second-order cones.
Returns
-------
list
A list of the sizes of the elementwise cones.
"""
cone_size = 1 + self.args[1].shape[self.axis]
return [cone_size] * self.num_cones()
[docs] def is_dcp(self, dpp: bool = False) -> bool:
"""An SOC constraint is DCP if each of its arguments is affine.
"""
if dpp:
with scopes.dpp_scope():
return all(arg.is_affine() for arg in self.args)
return all(arg.is_affine() for arg in self.args)
def is_dgp(self, dpp: bool = False) -> bool:
return False
def is_dqcp(self) -> bool:
return self.is_dcp()
def save_dual_value(self, value) -> None:
cone_size = 1 + self.args[1].shape[self.axis]
value = np.reshape(value, newshape=(-1, cone_size))
t = value[:, 0]
X = value[:, 1:]
if self.axis == 0:
X = X.T
self.dual_variables[0].save_value(t)
self.dual_variables[1].save_value(X)
def _dual_cone(self, *args):
"""Implements the dual cone of the second-order cone
See Pg 85 of the MOSEK modelling cookbook for more information"""
if args is None:
return SOC(self.dual_variables[0], self.dual_variables[1], self.axis)
else:
# some assertions for verifying `args`
args_shapes = [arg.shape for arg in args]
instance_args_shapes = [arg.shape for arg in self.args]
assert len(args) == len(self.args)
assert args_shapes == instance_args_shapes
return SOC(args[0], args[1], self.axis)