From e41ad668f4f16d8948181ae307cb98430b37ed1d Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Thu, 10 Nov 2016 21:29:42 -0500
Subject: [PATCH] Add the player1_start() method and two tests for it.

---
 dunshire/games.py                  | 40 ++++++++++++++++++++++++++++--
 test/symmetric_linear_game_test.py | 25 +++++++++++++++++++
 2 files changed, 63 insertions(+), 2 deletions(-)

diff --git a/dunshire/games.py b/dunshire/games.py
index 672810d..3ed89bb 100644
--- 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
@@ -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):
         """
diff --git a/test/symmetric_linear_game_test.py b/test/symmetric_linear_game_test.py
index 1e7194b..b6bd9b8 100644
--- a/test/symmetric_linear_game_test.py
+++ b/test/symmetric_linear_game_test.py
@@ -68,6 +68,31 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         self.assertTrue(p1_close and p2_close and gv_close)
 
 
+    def assert_player1_start_valid(self, G):
+        x = G.player1_start()['x']
+        s = G.player1_start()['s']
+        s1 = s[0:G.dimension()]
+        s2 = s[G.dimension():]
+        self.assert_within_tol(norm(G.A()*x - G.b()), 0)
+        self.assertTrue((s1,s2) in G.C())
+
+
+    def test_player1_start_valid_orthant(self):
+        """
+        Ensure that player one's starting point is in the orthant.
+        """
+        G = random_orthant_game()
+        self.assert_player1_start_valid(G)
+
+
+    def test_player1_start_valid_icecream(self):
+        """
+        Ensure that player one's starting point is in the ice-cream cone.
+        """
+        G = random_icecream_game()
+        self.assert_player1_start_valid(G)
+
+
     def test_condition_lower_bound(self):
         """
         Ensure that the condition number of a game is greater than or
-- 
2.44.2