Add a ball_radius() method for cones and use it to compute starting points.

diff --git a/dunshire/cones.py b/dunshire/cones.py
index 7fc8bf94993e4df86b2f57c2e6b6d6aefa65d8c4..af2415bb5f1ee18c188cfd3a1e740c655dd4b32d 100644 (file)
--- a/dunshire/cones.py
+++ b/dunshire/cones.py
@@ -77,6 +77,44 @@ class SymmetricCone:
         raise NotImplementedError
 
 
         raise NotImplementedError
 
 

+    def ball_radius(self, point):
+        """
+        Return the radius of a ball around ``point`` in this cone.
+
+        Since a radius cannot be negative, the ``point`` must be
+        contained in this cone; otherwise, an error is raised.
+
+        Parameters
+        ----------
+
+        point : matrix
+            A point contained in this cone.
+
+        Returns
+        -------
+
+        float
+            A radius ``r`` such that the ball of radius ``r`` centered at
+            ``point`` is contained entirely within this cone.
+
+        Raises
+        ------
+
+        NotImplementedError
+            Always, this method must be implemented in subclasses.
+
+        Examples
+        --------
+
+            >>> K = SymmetricCone(5)
+            >>> K.ball_radius(matrix([1,1,1,1,1]))
+            Traceback (most recent call last):
+            ...
+            NotImplementedError
+
+        """
+        raise NotImplementedError
+

 
     def dimension(self):
         """
 
     def dimension(self):
         """


@@ -201,6 +239,64 @@ class NonnegativeOrthant(SymmetricCone):
         return all([x > -options.ABS_TOL for x in point])
 
 
         return all([x > -options.ABS_TOL for x in point])
 
 

+    def ball_radius(self, point):
+        """
+        Return the radius of a ball around ``point`` in this cone.
+
+        Since a radius cannot be negative, the ``point`` must be
+        contained in this cone; otherwise, an error is raised.
+
+        The minimum distance from ``point`` to the complement of this
+        cone will always occur at its projection onto that set. It is
+        not hard to see that the projection is directly along one of the
+        coordinates, and so the minimum distance from ``point`` to the
+        boundary of this cone is the smallest coordinate of
+        ``point``. Therefore any radius less than that will work; we
+        divide it in half somewhat arbitrarily.
+
+        Parameters
+        ----------
+
+        point : matrix
+            A point contained in this cone.
+
+        Returns
+        -------
+
+        float
+            A radius ``r`` such that the ball of radius ``r`` centered at
+            ``point`` is contained entirely within this cone.
+
+        Raises
+        ------
+
+        TypeError
+            If ``point`` is not a :class:`cvxopt.base.matrix`.
+
+        TypeError
+            If ``point`` has the wrong dimensions.
+
+        ValueError
+            if ``point`` is not contained in this cone.
+
+        Examples
+        --------
+
+            >>> K = NonnegativeOrthant(5)
+            >>> K.ball_radius(matrix([1,2,3,4,5]))
+            0.5
+
+        """
+        if not isinstance(point, matrix):
+            raise TypeError('the given point is not a cvxopt.base.matrix')
+        if not point.size == (self.dimension(), 1):
+            raise TypeError('the given point has the wrong dimensions')
+        if not point in self:
+            raise ValueError('the given point does not lie in the cone')
+
+        return min(list(point)) / 2.0
+
+

 
 class IceCream(SymmetricCone):
     """
 
 class IceCream(SymmetricCone):
     """


@@ -305,6 +401,80 @@ class IceCream(SymmetricCone):
             return norm(radius) < (height + options.ABS_TOL)
 
 
             return norm(radius) < (height + options.ABS_TOL)
 
 

+    def ball_radius(self, point):
+        """
+        Return the radius of a ball around ``point`` in this cone.
+
+        Since a radius cannot be negative, the ``point`` must be
+        contained in this cone; otherwise, an error is raised.
+
+        The minimum distance from ``point`` to the complement of this
+        cone will always occur at its projection onto that set. It is
+        not hard to see that the projection is at a "down and out" angle
+        of ``pi/4`` towards the outside of the cone. If one draws a
+        right triangle involving the ``point`` and that projection, it
+        becomes clear that the distance between ``point`` and its
+        projection is a factor of ``1/sqrt(2)`` times the "horizontal"
+        distance from ``point`` to boundary of this cone. For simplicity
+        we take ``1/2`` instead.
+
+        Parameters
+        ----------
+
+        point : matrix
+            A point contained in this cone.
+
+        Returns
+        -------
+
+        float
+            A radius ``r`` such that the ball of radius ``r`` centered at
+            ``point`` is contained entirely within this cone.
+
+        Raises
+        ------
+
+        TypeError
+            If ``point`` is not a :class:`cvxopt.base.matrix`.
+
+        TypeError
+            If ``point`` has the wrong dimensions.
+
+        ValueError
+            if ``point`` is not contained in this cone.
+
+        Examples
+        --------
+
+        The height of ``x`` is one (its first coordinate), and so the
+        radius of the circle obtained from a height-one cross section is
+        also one. Note that the last two coordinates of ``x`` are half
+        of the way to the boundary of the cone, and in the direction of
+        a 30-60-90 triangle. If one follows those coordinates, they hit
+        at ``(1, sqrt(3)/2, 1/2)`` having unit norm. Thus the
+        "horizontal" distance to the boundary of the cone is ``(1 -
+        norm(x)``, which simplifies to ``1/2``. And rather than involve
+        a square root, we divide by two for a final safe radius of
+        ``1/4``.
+
+            >>> from math import sqrt
+            >>> K = IceCream(3)
+            >>> x = matrix([1, sqrt(3)/4.0, 1/4.0])
+            >>> K.ball_radius(x)
+            0.25
+
+        """
+        if not isinstance(point, matrix):
+            raise TypeError('the given point is not a cvxopt.base.matrix')
+        if not point.size == (self.dimension(), 1):
+            raise TypeError('the given point has the wrong dimensions')
+        if not point in self:
+            raise ValueError('the given point does not lie in the cone')
+
+        height = point[0]
+        radius = norm(point[1:])
+        return (height - radius) / 2.0
+

 
 class SymmetricPSD(SymmetricCone):
     r"""
 
 class SymmetricPSD(SymmetricCone):
     r"""

diff --git a/dunshire/games.py b/dunshire/games.py
index f9877b334cfa3bcf2d60c6269ecbf4340de24eb1..bb808cb592d8e013d390e69ee1b6527fc64db583 100644 (file)
--- a/dunshire/games.py
+++ b/dunshire/games.py
@@ -5,7 +5,7 @@ This module contains the main :class:`SymmetricLinearGame` class that
 knows how to solve a linear game.
 """
 from cvxopt import matrix, printing, solvers
 knows how to solve a linear game.
 """
 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)
 from .errors import GameUnsolvableException, PoorScalingException
 from .matrices import (append_col, append_row, condition_number, identity,
                        inner_product, norm, specnorm)


@@ -820,26 +820,9 @@ class SymmetricLinearGame:
         :meth:`L` is satisfied.
         """
         p = self.e2() / (norm(self.e2()) ** 2)
         :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?
-            # We use "2" because it's better numerically than sqrt(2).
-            height = self.e1()[0]
-            radius = norm(self.e1()[1:])
-            dist = (height - radius) / 2
-        else:
-            raise NotImplementedError
-
+        dist = self.K().ball_radius(self.e1())

         nu = - specnorm(self.L())/(dist*norm(self.e2()))
         nu = - specnorm(self.L())/(dist*norm(self.e2()))

-        x = matrix([nu,p], (self.dimension() + 1, 1))
+        x = matrix([nu, p], (self.dimension() + 1, 1))

         s = - self._G()*x
 
         return {'x': x, 's': s}
         s = - self._G()*x
 
         return {'x': x, 's': s}


@@ -850,29 +833,12 @@ class SymmetricLinearGame:
         Return a feasible starting point for player two.
         """
         q = self.e1() / (norm(self.e1()) ** 2)
         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?
-            # We use "2" because it's better numerically than sqrt(2).
-            height = self.e2()[0]
-            radius = norm(self.e2()[1:])
-            dist = (height - radius) / 2
-        else:
-            raise NotImplementedError
-
+        dist = self.K().ball_radius(self.e2())

         omega = specnorm(self.L())/(dist*norm(self.e1()))
         y = matrix([omega])
         z2 = q
         z1 = y*self.e2() - self.L().trans()*z2
         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))
+        z = matrix([z1, z2], (self.dimension()*2, 1))

 
         return {'y': y, 'z': z}
 
 
         return {'y': y, 'z': z}