X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=ae1426a2c611f7e315f94fc5fea0e98f1da0905b;hb=cd77ba5250ed98ece623730c26af845366847487;hp=3ed89bb3f2f70b30d0313cbe5a578e4f53e47421;hpb=e41ad668f4f16d8948181ae307cb98430b37ed1d;p=dunshire.git

diff --git a/dunshire/games.py b/dunshire/games.py
index 3ed89bb..ae1426a 100644
--- a/dunshire/games.py
+++ b/dunshire/games.py
@@ -4,10 +4,8 @@ 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, IceCream, NonnegativeOrthant
+from .cones import CartesianProduct
 from .errors import GameUnsolvableException, PoorScalingException
 from .matrices import (append_col, append_row, condition_number, identity,
                        inner_product, norm, specnorm)
@@ -24,7 +22,7 @@ class Solution:
     --------
 
         >>> print(Solution(10, matrix([1,2]), matrix([3,4])))
-        Game value: 10.0000000
+        Game value: 10.000...
         Player 1 optimal:
           [ 1]
           [ 2]
@@ -324,6 +322,8 @@ class SymmetricLinearGame:
         if not self._e2 in K:
             raise ValueError('the point e2 must lie in the interior of K')
 
+        # Initial value of cached method.
+        self._L_specnorm_value = None
 
 
     def __str__(self):
@@ -822,30 +822,46 @@ class SymmetricLinearGame:
         :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))
+        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):
+        """
+        Return a feasible starting point for player two.
+        """
+        q = self.e1() / (norm(self.e1()) ** 2)
+        dist = self.K().ball_radius(self.e2())
+        omega = self._L_specnorm()/(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}
+
+
+    def _L_specnorm(self):
+        """
+        Compute the spectral norm of ``L`` and cache it.
+        """
+        if self._L_specnorm_value is None:
+            self._L_specnorm_value = specnorm(self.L())
+        return self._L_specnorm_value
+
+    def epsilon_scale(self, solution):
+        # Don't return anything smaller than 1... we can't go below
+        # out "minimum tolerance."
+        norm_p1_opt = norm(solution.player1_optimal())
+        norm_p2_opt = norm(solution.player2_optimal())
+        scale = self._L_specnorm()*(norm_p1_opt + norm_p2_opt)
+        return max(1, scale)
+
+
     def solution(self):
         """
         Solve this linear game and return a :class:`Solution`.
@@ -880,11 +896,11 @@ class SymmetricLinearGame:
             >>> e2 = [1,1,1]
             >>> SLG = SymmetricLinearGame(L, K, e1, e2)
             >>> print(SLG.solution())
-            Game value: -6.1724138
+            Game value: -6.172...
             Player 1 optimal:
-              [ 0.551...]
-              [-0.000...]
-              [ 0.448...]
+              [0.551...]
+              [0.000...]
+              [0.448...]
             Player 2 optimal:
               [0.448...]
               [0.000...]
@@ -900,7 +916,7 @@ class SymmetricLinearGame:
             >>> e2 = [4,5,6]
             >>> SLG = SymmetricLinearGame(L, K, e1, e2)
             >>> print(SLG.solution())
-            Game value: 0.0312500
+            Game value: 0.031...
             Player 1 optimal:
               [0.031...]
               [0.062...]
@@ -936,8 +952,8 @@ class SymmetricLinearGame:
             >>> print(SLG.solution())
             Game value: 18.767...
             Player 1 optimal:
-              [-0.000...]
-              [ 9.766...]
+              [0.000...]
+              [9.766...]
             Player 2 optimal:
               [1.047...]
               [0.000...]
@@ -954,8 +970,8 @@ class SymmetricLinearGame:
             >>> print(SLG.solution())
             Game value: 24.614...
             Player 1 optimal:
-              [ 6.371...]
-              [-0.000...]
+              [6.371...]
+              [0.000...]
             Player 2 optimal:
               [2.506...]
               [0.000...]
@@ -969,6 +985,7 @@ class SymmetricLinearGame:
                                        self.C().cvxopt_dims(),
                                        self.A(),
                                        self.b(),
+                                       primalstart=self.player1_start(),
                                        options=opts)
         except ValueError as error:
             if str(error) == 'math domain error':