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
+from .cones import CartesianProduct, IceCream, NonnegativeOrthant
from .errors import GameUnsolvableException, PoorScalingException
from .matrices import (append_col, append_row, condition_number, identity,
- inner_product)
+ inner_product, norm, specnorm)
from . import options
printing.options['dformat'] = options.FLOAT_FORMAT
--------
>>> print(Solution(10, matrix([1,2]), matrix([3,4])))
- Game value: 10.0000000
+ Game value: 10.000...
Player 1 optimal:
[ 1]
[ 2]
' e1 = {:s},\n' \
' e2 = {:s},\n' \
' Condition((L, K, e1, e2)) = {:f}.'
- indented_L = '\n '.join(str(self._L).splitlines())
- indented_e1 = '\n '.join(str(self._e1).splitlines())
- indented_e2 = '\n '.join(str(self._e2).splitlines())
+ indented_L = '\n '.join(str(self.L()).splitlines())
+ indented_e1 = '\n '.join(str(self.e1()).splitlines())
+ indented_e2 = '\n '.join(str(self.e2()).splitlines())
return tpl.format(indented_L,
- str(self._K),
+ str(self.K()),
indented_e1,
indented_e2,
self.condition())
return matrix(0, (self.dimension(), 1), tc='d')
- def _A(self):
+ def A(self):
"""
Return the matrix ``A`` used in our CVXOPT construction.
>>> e1 = [1,1,1]
>>> e2 = [1,2,3]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> print(SLG._A())
+ >>> print(SLG.A())
[0.0000000 1.0000000 2.0000000 3.0000000]
<BLANKLINE>
"""
- return matrix([0, self._e2], (1, self.dimension() + 1), 'd')
+ return matrix([0, self.e2()], (1, self.dimension() + 1), 'd')
"""
identity_matrix = identity(self.dimension())
return append_row(append_col(self._zero(), -identity_matrix),
- append_col(self._e1, -self._L))
+ append_col(self.e1(), -self.L()))
def _c(self):
return matrix([-1, self._zero()])
- def _C(self):
+ def C(self):
"""
Return the cone ``C`` used in our CVXOPT construction.
>>> e1 = [1,2,3]
>>> e2 = [1,1,1]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> print(SLG._C())
+ >>> print(SLG.C())
Cartesian product of dimension 6 with 2 factors:
* Nonnegative orthant in the real 3-space
* Nonnegative orthant in the real 3-space
@staticmethod
- def _b():
+ def b():
"""
Return the ``b`` vector used in our CVXOPT construction.
>>> e1 = [1,2,3]
>>> e2 = [1,1,1]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> print(SLG._b())
+ >>> print(SLG.b())
[1.0000000]
<BLANKLINE>
return matrix([1], tc='d')
+ def player1_start(self):
+ """
+ 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.
+ """
+ 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))
+ 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)
+
+ # 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()))
+ 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 solution(self):
"""
>>> 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...]
soln_dict = solvers.conelp(self._c(),
self._G(),
self._h(),
- self._C().cvxopt_dims(),
- self._A(),
- self._b(),
+ self.C().cvxopt_dims(),
+ self.A(),
+ self.b(),
+ primalstart=self.player1_start(),
options=opts)
except ValueError as error:
if str(error) == 'math domain error':
True
"""
- return (condition_number(self._G()) + condition_number(self._A()))/2
+ return (condition_number(self._G()) + condition_number(self.A()))/2
def dual(self):
Condition((L, K, e1, e2)) = 44.476...
"""
- # We pass ``self._L`` right back into the constructor, because
+ # We pass ``self.L()`` right back into the constructor, because
# it will be transposed there. And keep in mind that ``self._K``
# is its own dual.
- return SymmetricLinearGame(self._L,
- self._K,
- self._e2,
- self._e1)
+ return SymmetricLinearGame(self.L(),
+ self.K(),
+ self.e2(),
+ self.e1())
projects / dunshire.git / blobdiff