This module contains the main :class:`SymmetricLinearGame` class that
knows how to solve a linear game.
"""
-from math import sqrt
-
from cvxopt import matrix, printing, solvers
-from .cones import CartesianProduct, IceCream, NonnegativeOrthant
+from .cones import CartesianProduct
from .errors import GameUnsolvableException, PoorScalingException
from .matrices import (append_col, append_row, condition_number, identity,
inner_product, norm, specnorm)
if not self._e2 in K:
raise ValueError('the point e2 must lie in the interior of K')
+ # Initial value of cached method.
+ self._L_specnorm_value = None
def __str__(self):
:meth:`L` is satisfied.
"""
p = self.e2() / (norm(self.e2()) ** 2)
-
- # Compute the distance from p to the outside of K.
- if isinstance(self.K(), NonnegativeOrthant):
- # How far is it to a wall?
- dist = min(list(self.e1()))
- elif isinstance(self.K(), IceCream):
- # How far is it to the boundary of the ball that defines
- # the ice-cream cone at a given height? Now draw a
- # 45-45-90 triangle and the shortest distance to the
- # outside of the cone should be 1/sqrt(2) of that.
- # It works in R^2, so it works everywhere, right?
- height = self.e1()[0]
- radius = norm(self.e1()[1:])
- dist = (height - radius) / sqrt(2)
- else:
- raise NotImplementedError
-
- nu = - specnorm(self.L())/(dist*norm(self.e2()))
- x = matrix([nu,p], (self.dimension() + 1, 1))
+ dist = self.K().ball_radius(self.e1())
+ nu = - self._L_specnorm()/(dist*norm(self.e2()))
+ x = matrix([nu, p], (self.dimension() + 1, 1))
s = - self._G()*x
return {'x': x, 's': s}
Return a feasible starting point for player two.
"""
q = self.e1() / (norm(self.e1()) ** 2)
-
- # Compute the distance from p to the outside of K.
- if isinstance(self.K(), NonnegativeOrthant):
- # How far is it to a wall?
- dist = min(list(self.e2()))
- elif isinstance(self.K(), IceCream):
- # How far is it to the boundary of the ball that defines
- # the ice-cream cone at a given height? Now draw a
- # 45-45-90 triangle and the shortest distance to the
- # outside of the cone should be 1/sqrt(2) of that.
- # It works in R^2, so it works everywhere, right?
- height = self.e2()[0]
- radius = norm(self.e2()[1:])
- dist = (height - radius) / sqrt(2)
- else:
- raise NotImplementedError
-
- omega = specnorm(self.L())/(dist*norm(self.e1()))
+ dist = self.K().ball_radius(self.e2())
+ omega = self._L_specnorm()/(dist*norm(self.e1()))
y = matrix([omega])
z2 = q
z1 = y*self.e2() - self.L().trans()*z2
- z = matrix([z1,z2], (self.dimension()*2, 1))
+ z = matrix([z1, z2], (self.dimension()*2, 1))
return {'y': y, 'z': z}
+ def _L_specnorm(self):
+ """
+ Compute the spectral norm of ``L`` and cache it.
+ """
+ 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."
+ norm_p1_opt = norm(solution.player1_optimal())
+ norm_p2_opt = norm(solution.player2_optimal())
+ scale = self._L_specnorm()*(norm_p1_opt + norm_p2_opt)
+ return max(1, scale)
+
+
def solution(self):
"""
Solve this linear game and return a :class:`Solution`.