+ Return the ``h`` vector used in our CVXOPT construction.
+
+ The ``h`` vector appears on the right-hand side of :math:`Gx + s
+ = h` in the statement of the CVXOPT conelp program.
+
+ .. warning::
+
+ It is not safe to cache any of the matrices passed to
+ CVXOPT, because it can clobber them.
+
+ Returns
+ -------
+
+ matrix
+ A ``2*self.dimension()``-by-``1`` column vector of zeros.
+
+ Examples
+ --------
+
+ >>> from dunshire import *
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[4,5,6],[7,8,9],[10,11,12]]
+ >>> e1 = [1,2,3]
+ >>> e2 = [1,1,1]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG._h())
+ [0.0000000]
+ [0.0000000]
+ [0.0000000]
+ [0.0000000]
+ [0.0000000]
+ [0.0000000]
+ <BLANKLINE>
+
+ """
+
+ return matrix([self._zero(), self._zero()])
+
+
+ @staticmethod
+ def b():
+ """
+ Return the ``b`` vector used in our CVXOPT construction.
+
+ The vector ``b`` appears on the right-hand side of :math:`Ax =
+ b` in the statement of the CVXOPT conelp program.
+
+ This method is static because the dimensions and entries of
+ ``b`` are known beforehand, and don't depend on any other
+ properties of the game.
+
+ .. warning::
+
+ It is not safe to cache any of the matrices passed to
+ CVXOPT, because it can clobber them.
+
+ Returns
+ -------
+
+ matrix
+ A ``1``-by-``1`` matrix containing a single entry ``1``.
+
+ Examples
+ --------
+
+ >>> from dunshire import *
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[4,5,6],[7,8,9],[10,11,12]]
+ >>> e1 = [1,2,3]
+ >>> e2 = [1,1,1]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> 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?
+ # We use "2" because it's better numerically than sqrt(2).
+ height = self.e1()[0]
+ radius = norm(self.e1()[1:])
+ dist = (height - radius) / 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?
+ # We use "2" because it's better numerically than sqrt(2).
+ height = self.e2()[0]
+ radius = norm(self.e2()[1:])
+ dist = (height - radius) / 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}