- def _G(self):
+ def G(self):
r"""
Return the matrix ``G`` used in our CVXOPT construction.
>>> e1 = [1,2,3]
>>> e2 = [1,1,1]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> print(SLG._G())
+ >>> 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]
append_col(self.e1(), -self.L()))
- def _c(self):
+ def c(self):
"""
Return the vector ``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())
[-1.0000000]
[ 0.0000000]
[ 0.0000000]
"""
return CartesianProduct(self._K, self._K)
- def _h(self):
+ def h(self):
r"""
Return the ``h`` vector used in our CVXOPT construction.
>>> e1 = [1,2,3]
>>> e2 = [1,1,1]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> print(SLG._h())
+ >>> print(SLG.h())
[0.0000000]
[0.0000000]
[0.0000000]
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.
+ 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
+ s = - self.G()*x
return {'x': x, 's': s}
def player2_start(self):
"""
Return a feasible starting point for player two.
+
+ This starting point is for the CVXOPT formulation and not for
+ the original game. The basic premise is that if you scale
+ :meth:`e1` by the reciprocal of its squared norm, then you get a
+ point in :meth:`K` that makes a unit inner product with
+ :meth:`e1`. We then get to choose the dual objective function
+ value such that the constraint involving :meth:`L` is satisfied.
+
+ Returns
+ -------
+
+ dict
+ A dictionary with two keys, 'y' and 'z', which contain the
+ vectors of the same name in the CVXOPT dual problem
+ formulation.
+
+ The ``1``-by-``1`` vector ``y`` consists of the dual
+ objective function value. The last :meth:`dimension` entries
+ of the vector ``z`` contain the strategy (for player two)
+ that achieves it. The remaining entries of ``z`` are
+ essentially dummy variables, computed from the equality
+ constraint in the CVXOPT dual problem.
+
"""
q = self.e1() / (norm(self.e1()) ** 2)
dist = self.K().ball_radius(self.e2())
"""
try:
opts = {'show_progress': False}
- soln_dict = solvers.conelp(self._c(),
- self._G(),
- self._h(),
+ soln_dict = solvers.conelp(self.c(),
+ self.G(),
+ self.h(),
self.C().cvxopt_dims(),
self.A(),
self.b(),
1.809...
"""
- return (condition_number(self._G()) + condition_number(self.A()))/2
+ return (condition_number(self.G()) + condition_number(self.A()))/2
def dual(self):