from .errors import GameUnsolvableException, PoorScalingException
from .matrices import (append_col, append_row, condition_number, identity,
inner_product, norm, specnorm)
-from . import options
+from .options import ABS_TOL, FLOAT_FORMAT, DEBUG_FLOAT_FORMAT
-printing.options['dformat'] = options.FLOAT_FORMAT
+printing.options['dformat'] = FLOAT_FORMAT
class Solution:
return CartesianProduct(self._K, self._K)
def _h(self):
- """
+ r"""
Return the ``h`` vector used in our CVXOPT construction.
The ``h`` vector appears on the right-hand side of :math:`Gx + s
@staticmethod
def b():
- """
+ r"""
Return the ``b`` vector used in our CVXOPT construction.
The vector ``b`` appears on the right-hand side of :math:`Ax =
def _L_specnorm(self):
"""
- Compute the spectral norm of ``L`` and cache it.
+ Compute the spectral norm of :meth:`L` and cache it.
+
+ The spectral norm of the matrix :meth:`L` is used in a few
+ places. Since it can be expensive to compute, we want to cache
+ its value. That is not possible in :func:`specnorm`, which lies
+ outside of a class, so this is the place to do it.
+
+ Returns
+ -------
+
+ float
+ A nonnegative real number; the largest singular value of
+ the matrix :meth:`L`.
+
"""
if self._L_specnorm_value is None:
self._L_specnorm_value = specnorm(self.L())
return self._L_specnorm_value
- def epsilon_scale(self, solution):
- # Don't return anything smaller than 1... we can't go below
- # out "minimum tolerance."
+
+ def tolerance_scale(self, solution):
+ r"""
+ Return a scaling factor that should be applied to ``ABS_TOL``
+ for this game.
+
+ When performing certain comparisons, the default tolernace
+ ``ABS_TOL`` may not be appropriate. For example, if we expect
+ ``x`` and ``y`` to be within ``ABS_TOL`` of each other, than the
+ inner product of ``L*x`` and ``y`` can be as far apart as the
+ spectral norm of ``L`` times the sum of the norms of ``x`` and
+ ``y``. Such a comparison is made in :meth:`solution`, and in
+ many of our unit tests.
+
+ The returned scaling factor found from the inner product mentioned
+ above is
+
+ .. math::
+
+ \left\lVert L \right\rVert_{2}
+ \left( \left\lVert \bar{x} \right\rVert
+ + \left\lVert \bar{y} \right\rVert
+ \right),
+
+ where :math:`\bar{x}` and :math:`\bar{y}` are optimal solutions
+ for players one and two respectively. This scaling factor is not
+ formally justified, but attempting anything smaller leads to
+ test failures.
+
+ .. warning::
+
+ Optimal solutions are not unique, so the scaling factor
+ obtained from ``solution`` may not work when comparing other
+ solutions.
+
+ Parameters
+ ----------
+
+ solution : Solution
+ A solution of this game, used to obtain the norms of the
+ optimal strategies.
+
+ Returns
+ -------
+
+ float
+ A scaling factor to be multiplied by ``ABS_TOL`` when
+ making comparisons involving solutions of this game.
+
+ """
norm_p1_opt = norm(solution.player1_optimal())
norm_p2_opt = norm(solution.player2_optimal())
scale = self._L_specnorm()*(norm_p1_opt + norm_p2_opt)
+
+ # Don't return anything smaller than 1... we can't go below
+ # out "minimum tolerance."
return max(1, scale)
[2.506...]
[0.000...]
+ This is another one that was difficult numerically, and caused
+ trouble even after we fixed the first two::
+
+ >>> from dunshire import *
+ >>> L = [[57.22233908627052301199, 41.70631373437460354126],
+ ... [83.04512571985074487202, 57.82581810406928468637]]
+ >>> K = NonnegativeOrthant(2)
+ >>> e1 = [7.31887017043399268346, 0.89744171905822367474]
+ >>> e2 = [0.11099824781179848388, 6.12564670639315345113]
+ >>> SLG = SymmetricLinearGame(L,K,e1,e2)
+ >>> print(SLG.solution())
+ Game value: 70.437...
+ Player 1 optimal:
+ [9.009...]
+ [0.000...]
+ Player 2 optimal:
+ [0.136...]
+ [0.000...]
+
+ And finally, here's one that returns an "optimal" solution, but
+ whose primal/dual objective function values are far apart::
+
+ >>> from dunshire import *
+ >>> L = [[ 6.49260076597376212248, -0.60528030227678542019],
+ ... [ 2.59896077096751731972, -0.97685530240286766457]]
+ >>> K = IceCream(2)
+ >>> e1 = [1, 0.43749513972645248661]
+ >>> e2 = [1, 0.46008379832200291260]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.solution())
+ Game value: 11.596...
+ Player 1 optimal:
+ [ 1.852...]
+ [-1.852...]
+ Player 2 optimal:
+ [ 1.777...]
+ [-1.777...]
+
"""
try:
opts = {'show_progress': False}
# Oops, CVXOPT tried to take the square root of a
# negative number. Report some details about the game
# rather than just the underlying CVXOPT crash.
- printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT
+ printing.options['dformat'] = DEBUG_FLOAT_FORMAT
raise PoorScalingException(self)
else:
raise error
# that CVXOPT is convinced the problem is infeasible (and that
# cannot happen).
if soln_dict['status'] in ['primal infeasible', 'dual infeasible']:
- printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT
+ printing.options['dformat'] = DEBUG_FLOAT_FORMAT
raise GameUnsolvableException(self, soln_dict)
# For the game value, we could use any of:
# it) because otherwise CVXOPT might return "unknown" and give
# us two points in the cone that are nowhere near optimal.
#
- if abs(p1_value - p2_value) > self.epsilon_scale(soln)*options.ABS_TOL:
- printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT
+ if abs(p1_value - p2_value) > self.tolerance_scale(soln)*ABS_TOL:
+ printing.options['dformat'] = DEBUG_FLOAT_FORMAT
raise GameUnsolvableException(self, soln_dict)
# And we also check that the points it gave us belong to the
# cone, just in case...
if (p1_optimal not in self._K) or (p2_optimal not in self._K):
- printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT
+ printing.options['dformat'] = DEBUG_FLOAT_FORMAT
raise GameUnsolvableException(self, soln_dict)
return soln
>>> e1 = [1]
>>> e2 = e1
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> actual = SLG.condition()
- >>> expected = 1.8090169943749477
- >>> abs(actual - expected) < options.ABS_TOL
- True
+ >>> SLG.condition()
+ 1.809...
"""
return (condition_number(self._G()) + condition_number(self.A()))/2
projects / dunshire.git / blobdiff