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
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 solution(self):
"""