Add the player2_start() method and some tests for it.

[dunshire.git] / dunshire / games.py

diff --git a/dunshire/games.py b/dunshire/games.py
index 672810de8094df7c37005cd5106fe5b8175888c4..0a473915716de7f4be4ce7d99cbc64d87c960795 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
@@ -23,7 +24,7 @@ class Solution:
     --------
 
         >>> print(Solution(10, matrix([1,2]), matrix([3,4])))
-        Game value: 10.0000000
+        Game value: 10.000...
         Player 1 optimal:
           [ 1]
           [ 2]
@@ -809,6 +810,72 @@ 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 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?
+            height = self.e2()[0]
+            radius = norm(self.e2()[1:])
+            dist = (height - radius) / sqrt(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}
+
 
     def solution(self):
         """
@@ -844,11 +911,11 @@ 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...]
@@ -864,7 +931,7 @@ 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...]
@@ -900,8 +967,8 @@ 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...]
@@ -918,8 +985,8 @@ 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...]
@@ -933,6 +1000,7 @@ 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':