# Source code for cvxpy.atoms.lambda_max

"""

you may not use this file except in compliance with the License.
You may obtain a copy of the License at

Unless required by applicable law or agreed to in writing, software
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
"""

from cvxpy.atoms.atom import Atom
from scipy import linalg as LA
import numpy as np
import scipy.sparse as sp

[docs]class lambda_max(Atom):
""" Maximum eigenvalue; :math:\\lambda_{\\max}(A).
"""

def __init__(self, A):
super(lambda_max, self).__init__(A)

def numeric(self, values):
"""Returns the largest eigenvalue of A.

Requires that A be symmetric.
"""
lo = hi = self.args[0].shape[0]-1
return LA.eigvalsh(values[0], eigvals=(lo, hi))[0]

def _domain(self):
"""Returns constraints describing the domain of the node.
"""
return [self.args[0].H == self.args[0]]

"""Gives the (sub/super)gradient of the atom w.r.t. each argument.

Matrix expressions are vectorized, so the gradient is a matrix.

Args:
values: A list of numeric values for the arguments.

Returns:
A list of SciPy CSC sparse matrices or None.
"""
w, v = LA.eigh(values[0])
d = np.zeros(w.shape)
d[-1] = 1
d = np.diag(d)
D = v.dot(d).dot(v.T)
return [sp.csc_matrix(D.ravel(order='F')).T]

def validate_arguments(self):
"""Verify that the argument A is square.
"""
if not self.args[0].ndim == 2 or self.args[0].shape[0] != self.args[0].shape[1]:
raise ValueError("The argument '%s' to lambda_max must resolve to a square matrix."
% self.args[0].name())

def shape_from_args(self):
"""Returns the (row, col) shape of the expression.
"""
return tuple()

def sign_from_args(self):
"""Returns sign (is positive, is negative) of the expression.
"""
return (False, False)

def is_atom_convex(self):
"""Is the atom convex?
"""
return True

def is_atom_concave(self):
"""Is the atom concave?
"""
return False

def is_incr(self, idx):
"""Is the composition non-decreasing in argument idx?
"""
return False

def is_decr(self, idx):
"""Is the composition non-increasing in argument idx?
"""
return False