Add the player1_start() method and two tests for it.

[dunshire.git] / dunshire / games.py

diff --git a/dunshire/games.py b/dunshire/games.py
index 71da5edbc561de3608e324c309c8ea3914213ce1..3ed89bb3f2f70b30d0313cbe5a578e4f53e47421 100644 (file)
--- a/dunshire/games.py
+++ b/dunshire/games.py
@@ -4,12 +4,13 @@ Symmetric linear games and their solutions.
 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
@@ -581,7 +582,7 @@ class SymmetricLinearGame:
         return matrix(0, (self.dimension(), 1), tc='d')
 
 
-    def _A(self):
+    def A(self):
         """
         Return the matrix ``A`` used in our CVXOPT construction.
 
@@ -609,7 +610,7 @@ class SymmetricLinearGame:
             >>> 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>
 
@@ -698,7 +699,7 @@ class SymmetricLinearGame:
         return matrix([-1, self._zero()])
 
 
-    def _C(self):
+    def C(self):
         """
         Return the cone ``C`` used in our CVXOPT construction.
 
@@ -720,7 +721,7 @@ class SymmetricLinearGame:
             >>> 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
@@ -770,7 +771,7 @@ class SymmetricLinearGame:
 
 
     @staticmethod
-    def _b():
+    def b():
         """
         Return the ``b`` vector used in our CVXOPT construction.
 
@@ -801,7 +802,7 @@ class SymmetricLinearGame:
             >>> e1 = [1,2,3]
             >>> e2 = [1,1,1]
             >>> SLG = SymmetricLinearGame(L, K, e1, e2)
-            >>> print(SLG._b())
+            >>> print(SLG.b())
             [1.0000000]
             <BLANKLINE>
 
@@ -809,6 +810,41 @@ class SymmetricLinearGame:
         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):
         """
@@ -930,9 +966,9 @@ class SymmetricLinearGame:
             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(),
                                        options=opts)
         except ValueError as error:
             if str(error) == 'math domain error':
@@ -1036,7 +1072,7 @@ class SymmetricLinearGame:
         True
 
         """
-        return (condition_number(self._G()) + condition_number(self._A()))/2
+        return (condition_number(self._G()) + condition_number(self.A()))/2
 
 
     def dual(self):