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)
--------
>>> print(Solution(10, matrix([1,2]), matrix([3,4])))
- Game value: 10.0000000
+ Game value: 10.000...
Player 1 optimal:
[ 1]
[ 2]
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}
+ def player2_start(self):
+ """
+ Return a feasible starting point for player two.
+ """
+ q = self.e1() / (norm(self.e1()) ** 2)
+ 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))
+
+ 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 solution(self):
"""
Solve this linear game and return a :class:`Solution`.
>>> e2 = [1,1,1]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
>>> print(SLG.solution())
- Game value: -6.1724138
+ Game value: -6.172...
Player 1 optimal:
- [ 0.551...]
- [-0.000...]
- [ 0.448...]
+ [0.551...]
+ [0.000...]
+ [0.448...]
Player 2 optimal:
[0.448...]
[0.000...]
>>> e2 = [4,5,6]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
>>> print(SLG.solution())
- Game value: 0.0312500
+ Game value: 0.031...
Player 1 optimal:
[0.031...]
[0.062...]
>>> print(SLG.solution())
Game value: 18.767...
Player 1 optimal:
- [-0.000...]
- [ 9.766...]
+ [0.000...]
+ [9.766...]
Player 2 optimal:
[1.047...]
[0.000...]
>>> print(SLG.solution())
Game value: 24.614...
Player 1 optimal:
- [ 6.371...]
- [-0.000...]
+ [6.371...]
+ [0.000...]
Player 2 optimal:
[2.506...]
[0.000...]
self.C().cvxopt_dims(),
self.A(),
self.b(),
+ primalstart=self.player1_start(),
options=opts)
except ValueError as error:
if str(error) == 'math domain error':