Pass ABS_TOL to CVXOPT when solving games.

[dunshire.git] / dunshire / games.py

diff --git a/dunshire/games.py b/dunshire/games.py
index 80f5f8ae2a46d09f15e34546c511edeca286f4fa..c2ca68ea7faee4f751410a1dbf27c31517400d9f 100644 (file)
--- a/dunshire/games.py
+++ b/dunshire/games.py
@@ -7,13 +7,11 @@ knows how to solve a linear game.
 
 from cvxopt import matrix, printing, solvers
 from .cones import CartesianProduct
 
 from cvxopt import matrix, printing, solvers
 from .cones import CartesianProduct

-from .errors import GameUnsolvableException
-from .matrices import append_col, append_row, identity
+from .errors import GameUnsolvableException, PoorScalingException
+from .matrices import append_col, append_row, condition_number, identity

 from . import options
 
 printing.options['dformat'] = options.FLOAT_FORMAT
 from . import options
 
 printing.options['dformat'] = options.FLOAT_FORMAT

-solvers.options['show_progress'] = options.VERBOSE
-

 
 class Solution:
     """
 
 class Solution:
     """


@@ -221,7 +219,8 @@ class SymmetricLinearGame:
                [ 1],
           e2 = [ 1]
                [ 2]
                [ 1],
           e2 = [ 1]
                [ 2]

-               [ 3].
+               [ 3],
+          Condition((L, K, e1, e2)) = 31.834...

 
     Lists can (and probably should) be used for every argument::
 
 
     Lists can (and probably should) be used for every argument::
 


@@ -239,7 +238,8 @@ class SymmetricLinearGame:
           e1 = [ 1]
                [ 1],
           e2 = [ 1]
           e1 = [ 1]
                [ 1],
           e2 = [ 1]

-               [ 1].
+               [ 1],
+          Condition((L, K, e1, e2)) = 1.707...

 
     The points ``e1`` and ``e2`` can also be passed as some other
     enumerable type (of the correct length) without much harm, since
 
     The points ``e1`` and ``e2`` can also be passed as some other
     enumerable type (of the correct length) without much harm, since


@@ -261,7 +261,8 @@ class SymmetricLinearGame:
           e1 = [ 1]
                [ 1],
           e2 = [ 1]
           e1 = [ 1]
                [ 1],
           e2 = [ 1]

-               [ 1].
+               [ 1],
+          Condition((L, K, e1, e2)) = 1.707...

 
     However, ``L`` will always be intepreted as a list of rows, even
     if it is passed as a :class:`cvxopt.base.matrix` which is
 
     However, ``L`` will always be intepreted as a list of rows, even
     if it is passed as a :class:`cvxopt.base.matrix` which is


@@ -282,7 +283,8 @@ class SymmetricLinearGame:
           e1 = [ 1]
                [ 1],
           e2 = [ 1]
           e1 = [ 1]
                [ 1],
           e2 = [ 1]

-               [ 1].
+               [ 1],
+          Condition((L, K, e1, e2)) = 6.073...

         >>> L = cvxopt.matrix(L)
         >>> print(L)
         [ 1  3]
         >>> L = cvxopt.matrix(L)
         >>> print(L)
         [ 1  3]


@@ -297,7 +299,8 @@ class SymmetricLinearGame:
           e1 = [ 1]
                [ 1],
           e2 = [ 1]
           e1 = [ 1]
                [ 1],
           e2 = [ 1]

-               [ 1].
+               [ 1],
+          Condition((L, K, e1, e2)) = 6.073...

 
     """
     def __init__(self, L, K, e1, e2):
 
     """
     def __init__(self, L, K, e1, e2):


@@ -319,6 +322,10 @@ class SymmetricLinearGame:
         if not self._e2 in K:
             raise ValueError('the point e2 must lie in the interior of K')
 
         if not self._e2 in K:
             raise ValueError('the point e2 must lie in the interior of K')
 

+        # Cached result of the self._zero() method.
+        self._zero_col = None
+
+

     def __str__(self):
         """
         Return a string representation of this game.
     def __str__(self):
         """
         Return a string representation of this game.


@@ -327,11 +334,51 @@ class SymmetricLinearGame:
               '  L = {:s},\n' \
               '  K = {!s},\n' \
               '  e1 = {:s},\n' \
               '  L = {:s},\n' \
               '  K = {!s},\n' \
               '  e1 = {:s},\n' \

-              '  e2 = {:s}.'
+              '  e2 = {:s},\n' \
+              '  Condition((L, K, e1, e2)) = {:f}.'

         indented_L = '\n      '.join(str(self._L).splitlines())
         indented_e1 = '\n       '.join(str(self._e1).splitlines())
         indented_e2 = '\n       '.join(str(self._e2).splitlines())
         indented_L = '\n      '.join(str(self._L).splitlines())
         indented_e1 = '\n       '.join(str(self._e1).splitlines())
         indented_e2 = '\n       '.join(str(self._e2).splitlines())

-        return tpl.format(indented_L, str(self._K), indented_e1, indented_e2)
+
+        return tpl.format(indented_L,
+                          str(self._K),
+                          indented_e1,
+                          indented_e2,
+                          self.condition())
+
+
+    def _zero(self):
+        """
+        Return a column of zeros that fits ``K``.
+
+        This is used in our CVXOPT construction.
+        """
+        if self._zero_col is None:
+            # Cache it, it's constant.
+            self._zero_col = matrix(0, (self._K.dimension(), 1), tc='d')
+        return self._zero_col
+
+
+    def _A(self):
+        """
+        Return the matrix ``A`` used in our CVXOPT construction.
+
+        This matrix ``A``  appears on the right-hand side of ``Ax = b``
+        in the statement of the CVXOPT conelp program.
+        """
+        return matrix([0, self._e2], (1, self._K.dimension() + 1), 'd')
+
+
+    def _G(self):
+        r"""
+        Return the matrix ``G`` used in our CVXOPT construction.
+
+        Thus matrix ``G``that appears on the left-hand side of ``Gx + s = h``
+        in the statement of the CVXOPT conelp program.
+        """
+        I = identity(self._K.dimension())
+        return append_row(append_col(self._zero(), -I),
+                          append_col(self._e1, -self._L))

 
 
     def solution(self):
 
 
     def solution(self):


@@ -351,6 +398,10 @@ class SymmetricLinearGame:
             If the game could not be solved (if an optimal solution to its
             associated cone program was not found).
 
             If the game could not be solved (if an optimal solution to its
             associated cone program was not found).
 

+        PoorScalingException
+            If the game could not be solved because CVXOPT crashed while
+            trying to take the square root of a negative number.
+

         Examples
         --------
 
         Examples
         --------
 


@@ -366,13 +417,13 @@ class SymmetricLinearGame:
             >>> print(SLG.solution())
             Game value: -6.1724138
             Player 1 optimal:
             >>> print(SLG.solution())
             Game value: -6.1724138
             Player 1 optimal:

-              [ 0.5517241]
-              [-0.0000000]
-              [ 0.4482759]
+              [ 0.551...]
+              [-0.000...]
+              [ 0.448...]

             Player 2 optimal:
             Player 2 optimal:

-              [0.4482759]
-              [0.0000000]
-              [0.5517241]
+              [0.448...]
+              [0.000...]
+              [0.551...]

 
         The value of the following game can be computed using the fact
         that the identity is invertible::
 
         The value of the following game can be computed using the fact
         that the identity is invertible::


@@ -386,13 +437,13 @@ class SymmetricLinearGame:
             >>> print(SLG.solution())
             Game value: 0.0312500
             Player 1 optimal:
             >>> print(SLG.solution())
             Game value: 0.0312500
             Player 1 optimal:

-              [0.0312500]
-              [0.0625000]
-              [0.0937500]
+              [0.031...]
+              [0.062...]
+              [0.093...]

             Player 2 optimal:
             Player 2 optimal:

-              [0.1250000]
-              [0.1562500]
-              [0.1875000]
+              [0.125...]
+              [0.156...]
+              [0.187...]

 
         """
         # The cone "C" that appears in the statement of the CVXOPT
 
         """
         # The cone "C" that appears in the statement of the CVXOPT


@@ -403,29 +454,29 @@ class SymmetricLinearGame:
         # Ax = b in the statement of the CVXOPT conelp program.
         b = matrix([1], tc='d')
 
         # Ax = b in the statement of the CVXOPT conelp program.
         b = matrix([1], tc='d')
 

-        # A column of zeros that fits K.
-        zero = matrix(0, (self._K.dimension(), 1), tc='d')
-

         # The column vector "h" that appears on the right-hand side of
         # Gx + s = h in the statement of the CVXOPT conelp program.
         # The column vector "h" that appears on the right-hand side of
         # Gx + s = h in the statement of the CVXOPT conelp program.

-        h = matrix([zero, zero])
+        h = matrix([self._zero(), self._zero()])

 
         # The column vector "c" that appears in the objective function
         # value <c,x> in the statement of the CVXOPT conelp program.
 
         # The column vector "c" that appears in the objective function
         # value <c,x> in the statement of the CVXOPT conelp program.

-        c = matrix([-1, zero])
-
-        # The matrix "G" that appears on the left-hand side of Gx + s = h
-        # in the statement of the CVXOPT conelp program.
-        G = append_row(append_col(zero, -identity(self._K.dimension())),
-                       append_col(self._e1, -self._L))
-
-        # The matrix "A" that appears on the right-hand side of Ax = b
-        # in the statement of the CVXOPT conelp program.
-        A = matrix([0, self._e2], (1, self._K.dimension() + 1), 'd')
+        c = matrix([-1, self._zero()])

 
         # Actually solve the thing and obtain a dictionary describing
         # what happened.
 
         # Actually solve the thing and obtain a dictionary describing
         # what happened.

-        soln_dict = solvers.conelp(c, G, h, C.cvxopt_dims(), A, b)
+        try:
+            solvers.options['show_progress'] = options.VERBOSE
+            solvers.options['abs_tol'] = options.ABS_TOL
+            soln_dict = solvers.conelp(c, self._G(), h,
+                                       C.cvxopt_dims(), self._A(), b)
+        except ValueError as e:
+            if str(e) == 'math domain error':
+                # Oops, CVXOPT tried to take the square root of a
+                # negative number. Report some details about the game
+                # rather than just the underlying CVXOPT crash.
+                raise PoorScalingException(self)
+            else:
+                raise e

 
         # The optimal strategies are named ``p`` and ``q`` in the
         # background documentation, and we need to extract them from
 
         # The optimal strategies are named ``p`` and ``q`` in the
         # background documentation, and we need to extract them from


@@ -454,6 +505,12 @@ class SymmetricLinearGame:
             # objectives match (within a tolerance) and that the
             # primal/dual optimal solutions are within the cone (to a
             # tolerance as well).
             # objectives match (within a tolerance) and that the
             # primal/dual optimal solutions are within the cone (to a
             # tolerance as well).

+            #
+            # The fudge factor of two is basically unjustified, but
+            # makes intuitive sense when you imagine that the primal
+            # value could be under the true optimal by ``ABS_TOL``
+            # and the dual value could be over by the same amount.
+            #

             if abs(p1_value - p2_value) > options.ABS_TOL:
                 raise GameUnsolvableException(self, soln_dict)
             if (p1_optimal not in self._K) or (p2_optimal not in self._K):
             if abs(p1_value - p2_value) > options.ABS_TOL:
                 raise GameUnsolvableException(self, soln_dict)
             if (p1_optimal not in self._K) or (p2_optimal not in self._K):


@@ -462,6 +519,43 @@ class SymmetricLinearGame:
         return Solution(p1_value, p1_optimal, p2_optimal)
 
 
         return Solution(p1_value, p1_optimal, p2_optimal)
 
 

+    def condition(self):
+        r"""
+        Return the condition number of this game.
+
+        In the CVXOPT construction of this game, two matrices ``G`` and
+        ``A`` appear. When those matrices are nasty, numerical problems
+        can show up. We define the condition number of this game to be
+        the average of the condition numbers of ``G`` and ``A`` in the
+        CVXOPT construction. If the condition number of this game is
+        high, then you can expect numerical difficulty (such as
+        :class:`PoorScalingException`).
+
+        Returns
+        -------
+
+        float
+            A real number greater than or equal to one that measures how
+            bad this game is numerically.
+
+        Examples
+        --------
+
+        >>> from dunshire import *
+        >>> K = NonnegativeOrthant(1)
+        >>> L = [[1]]
+        >>> e1 = [1]
+        >>> e2 = e1
+        >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+        >>> actual = SLG.condition()
+        >>> expected = 1.8090169943749477
+        >>> abs(actual - expected) < options.ABS_TOL
+        True
+
+        """
+        return (condition_number(self._G()) + condition_number(self._A()))/2
+
+

     def dual(self):
         r"""
         Return the dual game to this game.
     def dual(self):
         r"""
         Return the dual game to this game.


@@ -491,7 +585,8 @@ class SymmetricLinearGame:
                    [ 3],
               e2 = [ 1]
                    [ 1]
                    [ 3],
               e2 = [ 1]
                    [ 1]

-                   [ 1].
+                   [ 1],
+              Condition((L, K, e1, e2)) = 44.476...

 
         """
         # We pass ``self._L`` right back into the constructor, because
 
         """
         # We pass ``self._L`` right back into the constructor, because