"""
from cvxopt import matrix
-from matrices import norm
+from matrices import eigenvalues, norm
class SymmetricCone:
"""
There are three types of symmetric cones supported by CVXOPT:
- * The nonnegative orthant in the real n-space.
-
- * The Lorentz "ice cream" cone, or the second-order cone.
-
- * The cone of symmetric positive-semidefinite matrices.
+ 1. The nonnegative orthant in the real n-space.
+ 2. The Lorentz "ice cream" cone, or the second-order cone.
+ 3. The cone of symmetric positive-semidefinite matrices.
This class is intended to encompass them all.
"""
product of them), the only information that we need is its
dimension. We take that dimension as a parameter, and store it
for later.
+
+ INPUT:
+
+ - ``dimension`` -- the dimension of this cone.
+
"""
if dimension <= 0:
raise ValueError('cones must have dimension greater than zero')
self._dimension = dimension
+
def __contains__(self, point):
"""
Return whether or not ``point`` belongs to this cone.
+
+ EXAMPLES:
+
+ >>> K = SymmetricCone(5)
+ >>> matrix([1,2]) in K
+ Traceback (most recent call last):
+ ...
+ NotImplementedError
+
"""
raise NotImplementedError
"""
Return whether or not ``point`` belongs to the interior
of this cone.
+
+ EXAMPLES:
+
+ >>> K = SymmetricCone(5)
+ >>> K.contains_strict(matrix([1,2]))
+ Traceback (most recent call last):
+ ...
+ NotImplementedError
"""
raise NotImplementedError
should not need to be overridden in subclasses. We prefer to do
any special computation in ``__init__()`` and record the result
in ``self._dimension``.
+
+ EXAMPLES:
+
+ >>> K = SymmetricCone(5)
+ >>> K.dimension()
+ 5
+
"""
return self._dimension
class SymmetricPSD(SymmetricCone):
"""
- The nonnegative orthant in ``n`` dimensions.
+ The cone of real symmetric positive-semidefinite matrices.
+
+ This cone has a dimension ``n`` associated with it, but we let ``n``
+ refer to the dimension of the domain of our matrices and not the
+ dimension of the (much larger) space in which the matrices
+ themselves live. In other words, our ``n`` is the ``n`` that appears
+ in the usual notation `S^{n}` for symmetric matrices.
+
+ As a result, the cone ``SymmetricPSD(n)`` lives in a space of dimension
+ ``(n**2 + n)/2)``.
EXAMPLES:
>>> K = SymmetricPSD(3)
>>> print(K)
Cone of symmetric positive-semidefinite matrices on the real 3-space
+ >>> K.dimension()
+ 3
"""
def __str__(self):
return tpl.format(self.dimension())
+ def __contains__(self, point):
+ """
+ Return whether or not ``point`` belongs to this cone.
+
+ INPUT:
+
+ An instance of the ``cvxopt.base.matrix`` class having
+ dimensions ``(n,n)`` where ``n`` is the dimension of this cone.
+
+ EXAMPLES:
+
+ >>> K = SymmetricPSD(2)
+ >>> matrix([[1,0],[0,1]]) in K
+ True
+
+ >>> K = SymmetricPSD(2)
+ >>> matrix([[0,0],[0,0]]) in K
+ True
+
+ >>> K = SymmetricPSD(3)
+ >>> matrix([[2,-1,0],[-1,2,-1],[0,-1,2]]) in K
+ True
+
+ >>> K = SymmetricPSD(5)
+ >>> A = matrix([[5,4,3,2,1],
+ ... [4,5,4,3,2],
+ ... [3,4,5,4,3],
+ ... [2,3,4,5,4],
+ ... [1,2,3,4,5]])
+ >>> A in K
+ True
+
+ >>> K = SymmetricPSD(5)
+ >>> A = matrix([[1,0,0,0,0],
+ ... [0,1,0,0,0],
+ ... [0,0,0,0,0],
+ ... [0,0,0,1,0],
+ ... [0,0,0,0,1]])
+ >>> A in K
+ True
+
+ >>> K = SymmetricPSD(2)
+ >>> [[1,2],[2,3]] in K
+ Traceback (most recent call last):
+ ...
+ TypeError: the given point is not a cvxopt.base.matrix
+
+ >>> K = SymmetricPSD(3)
+ >>> matrix([[1,2],[3,4]]) in K
+ Traceback (most recent call last):
+ ...
+ TypeError: the given point has the wrong dimensions
+
+ """
+ if not isinstance(point, matrix):
+ raise TypeError('the given point is not a cvxopt.base.matrix')
+ if not point.size == (self.dimension(), self.dimension()):
+ raise TypeError('the given point has the wrong dimensions')
+ if not point.typecode == 'd':
+ point = matrix(point, (self.dimension(), self.dimension()), 'd')
+ return all([e >= 0 for e in eigenvalues(point)])
+
+
+ def contains_strict(self, point):
+ """
+ Return whether or not ``point`` belongs to the interior
+ of this cone.
+
+ INPUT:
+
+ An instance of the ``cvxopt.base.matrix`` class having
+ dimensions ``(n,n)`` where ``n`` is the dimension of this cone.
+ Its type code must be 'd'.
+
+ EXAMPLES:
+
+ >>> K = SymmetricPSD(2)
+ >>> K.contains_strict(matrix([[1,0],[0,1]]))
+ True
+
+ >>> K = SymmetricPSD(2)
+ >>> K.contains_strict(matrix([[0,0],[0,0]]))
+ False
+
+ >>> K = SymmetricPSD(3)
+ >>> matrix([[2,-1,0],[-1,2,-1],[0,-1,2]]) in K
+ True
+
+ >>> K = SymmetricPSD(5)
+ >>> A = matrix([[5,4,3,2,1],
+ ... [4,5,4,3,2],
+ ... [3,4,5,4,3],
+ ... [2,3,4,5,4],
+ ... [1,2,3,4,5]])
+ >>> A in K
+ True
+
+ >>> K = SymmetricPSD(5)
+ >>> A = matrix([[1,0,0,0,0],
+ ... [0,1,0,0,0],
+ ... [0,0,0,0,0],
+ ... [0,0,0,1,0],
+ ... [0,0,0,0,1]])
+ >>> K.contains_strict(A)
+ False
+
+ >>> K = SymmetricPSD(2)
+ >>> K.contains_strict([[1,2],[2,3]])
+ Traceback (most recent call last):
+ ...
+ TypeError: the given point is not a cvxopt.base.matrix
+
+ >>> K = SymmetricPSD(3)
+ >>> K.contains_strict(matrix([[1,2],[3,4]]))
+ Traceback (most recent call last):
+ ...
+ TypeError: the given point has the wrong dimensions
+
+ """
+ if not isinstance(point, matrix):
+ raise TypeError('the given point is not a cvxopt.base.matrix')
+ if not point.size == (self.dimension(), self.dimension()):
+ raise TypeError('the given point has the wrong dimensions')
+ if not point.typecode == 'd':
+ point = matrix(point, (self.dimension(), self.dimension()), 'd')
+ return all([e > 0 for e in eigenvalues(point)])
+
+
class CartesianProduct(SymmetricCone):
"""
A cartesian product of symmetric cones, which is itself a symmetric
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