+ Thus matrix :math:`G` appears on the left-hand side of :math:`Gx
+ + s = h` in the `statement of the CVXOPT conelp program
+ <http://cvxopt.org/userguide/coneprog.html#linear-cone-programs>`_.
+
+ .. 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 + self.dimension())`` matrix.
+
+ 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.G())
+ [ 0.0000000 -1.0000000 0.0000000 0.0000000]
+ [ 0.0000000 0.0000000 -1.0000000 0.0000000]
+ [ 0.0000000 0.0000000 0.0000000 -1.0000000]
+ [ 1.0000000 -4.0000000 -5.0000000 -6.0000000]
+ [ 2.0000000 -7.0000000 -8.0000000 -9.0000000]
+ [ 3.0000000 -10.0000000 -11.0000000 -12.0000000]
+ <BLANKLINE>
+
+ """
+ identity_matrix = identity(self.dimension())
+ return append_row(append_col(self._zero(), -identity_matrix),
+ append_col(self.e1(), -self.L()))
+
+
+ def c(self):
+ r"""
+ Return the vector ``c`` used in our CVXOPT construction.
+
+ The column vector :math:`c` appears in the objective function
+ value :math:`\left\langle c,x \right\rangle` in the `statement
+ of the CVXOPT conelp program
+ <http://cvxopt.org/userguide/coneprog.html#linear-cone-programs>`_.
+
+ .. warning::
+
+ It is not safe to cache any of the matrices passed to
+ CVXOPT, because it can clobber them.
+
+ Returns
+ -------
+
+ matrix
+ A :meth:`dimension`-by-``1`` column vector.
+
+ 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.c())
+ [-1.0000000]
+ [ 0.0000000]
+ [ 0.0000000]
+ [ 0.0000000]
+ <BLANKLINE>
+
+ """
+ return matrix([-1, self._zero()])
+
+
+ def C(self):
+ """
+ Return the cone ``C`` used in our CVXOPT construction.
+
+ This is the cone over which the `CVXOPT conelp program
+ <http://cvxopt.org/userguide/coneprog.html#linear-cone-programs>`_
+ takes place.
+
+ Returns
+ -------
+
+ CartesianProduct
+ The cartesian product of ``K`` with itself.
+
+ 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.C())
+ Cartesian product of dimension 6 with 2 factors:
+ * Nonnegative orthant in the real 3-space
+ * Nonnegative orthant in the real 3-space
+
+ """
+ return CartesianProduct(self._K, self._K)
+
+ def h(self):
+ r"""
+ Return the ``h`` vector used in our CVXOPT construction.
+
+ The :math:`h` vector appears on the right-hand side of :math:`Gx
+ + s = h` in the `statement of the CVXOPT conelp program
+ <http://cvxopt.org/userguide/coneprog.html#linear-cone-programs>`_.
+
+ .. 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():
+ r"""
+ Return the ``b`` vector used in our CVXOPT construction.
+
+ The vector :math:`b` appears on the right-hand side of :math:`Ax
+ = b` in the `statement of the CVXOPT conelp program
+ <http://cvxopt.org/userguide/coneprog.html#linear-cone-programs>`_.
+
+ 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 scale
+ :meth:`e2` by the reciprocal of its squared norm, 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.
+
+ Returns
+ -------
+
+ dict
+ A dictionary with two keys, ``'x'`` and ``'s'``, which
+ contain the vectors of the same name in the CVXOPT primal
+ problem formulation.
+
+ The vector ``x`` consists of the primal objective function
+ value concatenated with the strategy (for player one) that
+ achieves it. The vector ``s`` is essentially a dummy
+ variable, and is computed from the equality constraing in
+ the CVXOPT primal problem.
+
+ """
+ p = self.e2() / (norm(self.e2()) ** 2)
+ 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):