def A(self):
- """
+ r"""
Return the matrix ``A`` used in our CVXOPT construction.
- This matrix ``A`` appears on the right-hand side of ``Ax = b``
- in the statement of the CVXOPT conelp program.
+ This matrix :math`A` appears on the right-hand side of :math:`Ax
+ = b` in the statement of the CVXOPT conelp program.
.. warning::
r"""
Return the matrix ``G`` used in our CVXOPT construction.
- Thus matrix ``G`` appears on the left-hand side of ``Gx + s = h``
- in the statement of the CVXOPT conelp program.
+ Thus matrix :math:`G` appears on the left-hand side of :math:`Gx
+ + s = h` in the statement of the CVXOPT conelp program.
.. warning::
def c(self):
- """
+ r"""
Return the vector ``c`` used in our CVXOPT construction.
- The column vector ``c`` appears in the objective function
- value ``<c,x>`` in the statement of the CVXOPT conelp program.
+ The column vector :math:`c` appears in the objective function
+ value :math:`\left\langle c,x \right\rangle` in the statement of
+ the CVXOPT conelp program.
.. warning::
"""
Return the cone ``C`` used in our CVXOPT construction.
- The cone ``C`` is the cone over which the conelp program takes
- place.
+ This is the cone over which the conelp program takes place.
Returns
-------
r"""
Return the ``h`` vector used in our CVXOPT construction.
- The ``h`` vector appears on the right-hand side of :math:`Gx + s
+ The :math:`h` vector appears on the right-hand side of :math:`Gx + s
= h` in the statement of the CVXOPT conelp program.
.. warning::
Return a feasible starting point for player one.
This starting point is for the CVXOPT formulation and not for
- the original game. The basic premise is that if you normalize
- :meth:`e2`, then you get a point in :meth:`K` that makes a unit
- inner product with :meth:`e2`. We then get to choose the primal
- objective function value such that the constraint involving
- :meth:`L` is satisfied.
+ the original game. The basic premise is that if you scale
+ :meth:`e2` by the reciprocal of its squared norm, then you get a
+ point in :meth:`K` that makes a unit inner product with
+ :meth:`e2`. We then get to choose the primal objective function
+ value such that the constraint involving :meth:`L` is satisfied.
+
+ Returns
+ -------
+
+ dict
+ A dictionary with two keys, 'x' and 's', which contain the
+ vectors of the same name in the CVXOPT primal problem
+ formulation.
+
+ The vector ``x`` consists of the primal objective function
+ value concatenated with the strategy (for player one) that
+ achieves it. The vector ``s`` is essentially a dummy
+ variable, and is computed from the equality constraing in
+ the CVXOPT primal problem.
+
"""
p = self.e2() / (norm(self.e2()) ** 2)
dist = self.K().ball_radius(self.e1())
def player2_start(self):
"""
Return a feasible starting point for player two.
+
+ This starting point is for the CVXOPT formulation and not for
+ the original game. The basic premise is that if you scale
+ :meth:`e1` by the reciprocal of its squared norm, then you get a
+ point in :meth:`K` that makes a unit inner product with
+ :meth:`e1`. We then get to choose the dual objective function
+ value such that the constraint involving :meth:`L` is satisfied.
+
+ Returns
+ -------
+
+ dict
+ A dictionary with two keys, 'y' and 'z', which contain the
+ vectors of the same name in the CVXOPT dual problem
+ formulation.
+
+ The ``1``-by-``1`` vector ``y`` consists of the dual
+ objective function value. The last :meth:`dimension` entries
+ of the vector ``z`` contain the strategy (for player two)
+ that achieves it. The remaining entries of ``z`` are
+ essentially dummy variables, computed from the equality
+ constraint in the CVXOPT dual problem.
+
"""
q = self.e1() / (norm(self.e1()) ** 2)
dist = self.K().ball_radius(self.e2())
A nonnegative real number; the largest singular value of
the matrix :meth:`L`.
+ Examples
+ --------
+
+ >>> from dunshire import *
+ >>> from dunshire.matrices import specnorm
+ >>> L = [[1,2],[3,4]]
+ >>> K = NonnegativeOrthant(2)
+ >>> e1 = [1,1]
+ >>> e2 = e1
+ >>> SLG = SymmetricLinearGame(L,K,e1,e2)
+ >>> specnorm(SLG.L()) == SLG._L_specnorm()
+ True
+
"""
if self._L_specnorm_value is None:
self._L_specnorm_value = specnorm(self.L())
A scaling factor to be multiplied by ``ABS_TOL`` when
making comparisons involving solutions of this game.
+ Examples
+ --------
+
+ The spectral norm of ``L`` in this case is around ``5.464``, and
+ the optimal strategies both have norm one, so we expect the
+ tolerance scale to be somewhere around ``2 * 5.464``, or
+ ``10.929``::
+
+ >>> from dunshire import *
+ >>> L = [[1,2],[3,4]]
+ >>> K = NonnegativeOrthant(2)
+ >>> e1 = [1,1]
+ >>> e2 = e1
+ >>> SLG = SymmetricLinearGame(L,K,e1,e2)
+ >>> SLG.tolerance_scale(SLG.solution())
+ 10.929...
+
"""
norm_p1_opt = norm(solution.player1_optimal())
norm_p2_opt = norm(solution.player2_optimal())
# same. Even if CVXOPT bails out due to numerical difficulty,
# it will have some candidate points in mind. If those
# candidates are good enough, we take them. We do the same
- # check (perhaps pointlessly so) for "optimal" results.
+ # check for "optimal" results.
#
# First we check that the primal/dual objective values are
- # close enough (one could be low by ABS_TOL, the other high by
- # it) because otherwise CVXOPT might return "unknown" and give
- # us two points in the cone that are nowhere near optimal.
+ # close enough because otherwise CVXOPT might return "unknown"
+ # and give us two points in the cone that are nowhere near
+ # optimal. And in fact, we need to ensure that they're close
+ # for "optimal" results, too, because we need to know how
+ # lenient to be in our testing.
#
if abs(p1_value - p2_value) > self.tolerance_scale(soln)*ABS_TOL:
printing.options['dformat'] = DEBUG_FLOAT_FORMAT
projects / dunshire.git / blobdiff