from cvxopt import matrix, printing, solvers
from cones import CartesianProduct, IceCream, NonnegativeOrthant
from errors import GameUnsolvableException
-from matrices import append_col, append_row, identity, inner_product
+from matrices import append_col, append_row, identity, inner_product, norm
import options
printing.options['dformat'] = options.FLOAT_FORMAT
# feeding it to CVXOPT.
self._L = matrix(L, (K.dimension(), K.dimension())).trans()
- if not K.contains_strict(self._e1):
+ if not self._e1 in K:
raise ValueError('the point e1 must lie in the interior of K')
- if not K.contains_strict(self._e2):
+ if not self._e2 in K:
raise ValueError('the point e2 must lie in the interior of K')
def __str__(self):
# what happened.
soln_dict = solvers.conelp(c, G, h, C.cvxopt_dims(), A, b)
+ p1_value = -soln_dict['primal objective']
+ p2_value = -soln_dict['dual objective']
+ p1_optimal = soln_dict['x'][1:]
+ p2_optimal = soln_dict['z'][self._K.dimension():]
+
# The "status" field contains "optimal" if everything went
# according to plan. Other possible values are "primal
- # infeasible", "dual infeasible", "unknown", all of which
- # mean we didn't get a solution. That should never happen,
- # because by construction our game has a solution, and thus
- # the cone program should too.
- if soln_dict['status'] != 'optimal':
+ # infeasible", "dual infeasible", "unknown", all of which mean
+ # we didn't get a solution. The "infeasible" ones are the
+ # worst, since they indicate that CVXOPT is convinced the
+ # problem is infeasible (and that cannot happen).
+ if soln_dict['status'] in ['primal infeasible', 'dual infeasible']:
raise GameUnsolvableException(soln_dict)
-
- p1_value = soln_dict['x'][0]
- p1_optimal = soln_dict['x'][1:]
- p2_optimal = soln_dict['z'][self._K.dimension():]
+ elif soln_dict['status'] == 'unknown':
+ # When we get a status of "unknown", we may still be able
+ # to salvage a solution out of the returned
+ # dictionary. Often this is the result of numerical
+ # difficulty and we can simply check that the primal/dual
+ # objectives match (within a tolerance) and that the
+ # primal/dual optimal solutions are within the cone (to a
+ # tolerance as well).
+ if abs(p1_value - p2_value) > options.ABS_TOL:
+ raise GameUnsolvableException(soln_dict)
+ if (p1_optimal not in self._K) or (p2_optimal not in self._K):
+ raise GameUnsolvableException(soln_dict)
return Solution(p1_value, p1_optimal, p2_optimal)
+
def dual(self):
r"""
Return the dual game to this game.
self._e1)
+
+def _random_matrix(dims):
+ """
+ Generate a random square (``dims``-by-``dims``) matrix,
+ represented as a list of rows. This is used only by the
+ :class:`SymmetricLinearGameTest` class.
+ """
+ return [[uniform(-10, 10) for i in range(dims)] for j in range(dims)]
+
+def _random_nonnegative_matrix(dims):
+ """
+ Generate a random square (``dims``-by-``dims``) matrix with
+ nonnegative entries, represented as a list of rows. This is used
+ only by the :class:`SymmetricLinearGameTest` class.
+ """
+ L = _random_matrix(dims)
+ return [[abs(entry) for entry in row] for row in L]
+
+def _random_diagonal_matrix(dims):
+ """
+ Generate a random square (``dims``-by-``dims``) matrix with nonzero
+ entries only on the diagonal, represented as a list of rows. This is
+ used only by the :class:`SymmetricLinearGameTest` class.
+ """
+ return [[uniform(-10, 10)*int(i == j) for i in range(dims)]
+ for j in range(dims)]
+
+def _random_orthant_params():
+ """
+ Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a
+ random game over the nonnegative orthant. This is only used by
+ the :class:`SymmetricLinearGameTest` class.
+ """
+ ambient_dim = randint(1, 10)
+ K = NonnegativeOrthant(ambient_dim)
+ e1 = [uniform(0.5, 10) for idx in range(K.dimension())]
+ e2 = [uniform(0.5, 10) for idx in range(K.dimension())]
+ L = _random_matrix(K.dimension())
+ return (L, K, e1, e2)
+
+
+def _random_icecream_params():
+ """
+ Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a
+ random game over the ice cream cone. This is only used by
+ the :class:`SymmetricLinearGameTest` class.
+ """
+ # Use a minimum dimension of two to avoid divide-by-zero in
+ # the fudge factor we make up later.
+ ambient_dim = randint(2, 10)
+ K = IceCream(ambient_dim)
+ e1 = [1] # Set the "height" of e1 to one
+ e2 = [1] # And the same for e2
+
+ # If we choose the rest of the components of e1,e2 randomly
+ # between 0 and 1, then the largest the squared norm of the
+ # non-height part of e1,e2 could be is the 1*(dim(K) - 1). We
+ # need to make it less than one (the height of the cone) so
+ # that the whole thing is in the cone. The norm of the
+ # non-height part is sqrt(dim(K) - 1), and we can divide by
+ # twice that.
+ fudge_factor = 1.0 / (2.0*sqrt(K.dimension() - 1.0))
+ e1 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)]
+ e2 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)]
+ L = _random_matrix(K.dimension())
+
+ return (L, K, e1, e2)
+
+
class SymmetricLinearGameTest(TestCase):
"""
Tests for the SymmetricLinearGame and Solution classes.
"""
-
def assert_within_tol(self, first, second):
"""
Test that ``first`` and ``second`` are equal within our default
self.assertTrue(abs(first - second) < options.ABS_TOL)
+ def assert_norm_within_tol(self, first, second):
+ """
+ Test that ``first`` and ``second`` vectors are equal in the
+ sense that the norm of their difference is within our default
+ tolerance.
+ """
+ self.assert_within_tol(norm(first - second), 0)
+
+
def assert_solution_exists(self, L, K, e1, e2):
"""
Given the parameters needed to construct a SymmetricLinearGame,
"""
G = SymmetricLinearGame(L, K, e1, e2)
soln = G.solution()
+
+ # The matrix() constructor assumes that ``L`` is a list of
+ # columns, so we transpose it to agree with what
+ # SymmetricLinearGame() thinks.
L_matrix = matrix(L).trans()
expected = inner_product(L_matrix*soln.player1_optimal(),
soln.player2_optimal())
self.assert_within_tol(soln.game_value(), expected)
- def test_solution_exists_nonnegative_orthant(self):
+
+ def test_solution_exists_orthant(self):
"""
Every linear game has a solution, so we should be able to solve
every symmetric linear game over the NonnegativeOrthant. Pick
optimal solutions should give us the optimal game value when we
apply the payoff operator to them.
"""
- ambient_dim = randint(1, 10)
- K = NonnegativeOrthant(ambient_dim)
- e1 = [uniform(0.1, 10) for idx in range(K.dimension())]
- e2 = [uniform(0.1, 10) for idx in range(K.dimension())]
- L = [[uniform(-10, 10) for i in range(K.dimension())]
- for j in range(K.dimension())]
+ (L, K, e1, e2) = _random_orthant_params()
self.assert_solution_exists(L, K, e1, e2)
- def test_solution_exists_ice_cream(self):
+
+ def test_solution_exists_icecream(self):
"""
Like :meth:`test_solution_exists_nonnegative_orthant`, except
over the ice cream cone.
"""
- # Use a minimum dimension of two to avoid divide-by-zero in
- # the fudge factor we make up later.
- ambient_dim = randint(2, 10)
- K = IceCream(ambient_dim)
- e1 = [1] # Set the "height" of e1 to one
- e2 = [1] # And the same for e2
-
- # If we choose the rest of the components of e1,e2 randomly
- # between 0 and 1, then the largest the squared norm of the
- # non-height part of e1,e2 could be is the 1*(dim(K) - 1). We
- # need to make it less than one (the height of the cone) so
- # that the whole thing is in the cone. The norm of the
- # non-height part is sqrt(dim(K) - 1), and we can divide by
- # twice that.
- fudge_factor = 1.0 / (2.0*sqrt(K.dimension() - 1.0))
- e1 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)]
- e2 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)]
- L = [[uniform(-10, 10) for i in range(K.dimension())]
- for j in range(K.dimension())]
+ (L, K, e1, e2) = _random_icecream_params()
self.assert_solution_exists(L, K, e1, e2)
- def test_negative_value_Z_operator(self):
+ def test_negative_value_z_operator(self):
"""
Test the example given in Gowda/Ravindran of a Z-matrix with
negative game value on the nonnegative orthant.
"""
K = NonnegativeOrthant(2)
- e1 = [1,1]
+ e1 = [1, 1]
e2 = e1
- L = [[1,-2],[-2,1]]
+ L = [[1, -2], [-2, 1]]
G = SymmetricLinearGame(L, K, e1, e2)
self.assertTrue(G.solution().game_value() < -options.ABS_TOL)
- def test_nonnegative_scaling_orthant(self):
+ def assert_scaling_works(self, L, K, e1, e2):
"""
Test that scaling ``L`` by a nonnegative number scales the value
- of the game by the same number. Use the nonnegative orthant as
- our cone.
- """
- ambient_dim = randint(1, 10)
- K = NonnegativeOrthant(ambient_dim)
- e1 = [uniform(0.1, 10) for idx in range(K.dimension())]
- e2 = [uniform(0.1, 10) for idx in range(K.dimension())]
- L = matrix([[uniform(-10, 10) for i in range(K.dimension())]
- for j in range(K.dimension())])
- G1 = SymmetricLinearGame(L, K, e1, e2)
- value1 = G1.solution().game_value()
- alpha = uniform(0.1, 10)
+ of the game by the same number.
+ """
+ # Make ``L`` a matrix so that we can scale it by alpha. Its
+ # random, so who cares if it gets transposed.
+ L = matrix(L)
+ game1 = SymmetricLinearGame(L, K, e1, e2)
+ value1 = game1.solution().game_value()
- G2 = SymmetricLinearGame(alpha*L, K, e1, e2)
- value2 = G2.solution().game_value()
+ alpha = uniform(0.1, 10)
+ game2 = SymmetricLinearGame(alpha*L, K, e1, e2)
+ value2 = game2.solution().game_value()
self.assert_within_tol(alpha*value1, value2)
- def test_nonnegative_scaling_icecream(self):
+ def test_scaling_orthant(self):
+ """
+ Test that scaling ``L`` by a nonnegative number scales the value
+ of the game by the same number over the nonnegative orthant.
+ """
+ (L, K, e1, e2) = _random_orthant_params()
+ self.assert_scaling_works(L, K, e1, e2)
+
+
+ def test_scaling_icecream(self):
"""
The same test as :meth:`test_nonnegative_scaling_orthant`,
except over the ice cream cone.
"""
- # Use a minimum dimension of two to avoid divide-by-zero in
- # the fudge factor we make up later.
- ambient_dim = randint(2, 10)
- K = IceCream(ambient_dim)
- e1 = [1] # Set the "height" of e1 to one
- e2 = [1] # And the same for e2
-
- # If we choose the rest of the components of e1,e2 randomly
- # between 0 and 1, then the largest the squared norm of the
- # non-height part of e1,e2 could be is the 1*(dim(K) - 1). We
- # need to make it less than one (the height of the cone) so
- # that the whole thing is in the cone. The norm of the
- # non-height part is sqrt(dim(K) - 1), and we can divide by
- # twice that.
- fudge_factor = 1.0 / (2.0*sqrt(K.dimension() - 1.0))
- e1 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)]
- e2 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)]
- L = matrix([[uniform(-10, 10) for i in range(K.dimension())]
- for j in range(K.dimension())])
-
- G1 = SymmetricLinearGame(L, K, e1, e2)
- value1 = G1.solution().game_value()
- alpha = uniform(0.1, 10)
+ (L, K, e1, e2) = _random_icecream_params()
+ self.assert_scaling_works(L, K, e1, e2)
- G2 = SymmetricLinearGame(alpha*L, K, e1, e2)
- value2 = G2.solution().game_value()
- self.assert_within_tol(alpha*value1, value2)
+ def assert_translation_works(self, L, K, e1, e2):
+ """
+ Check that translating ``L`` by alpha*(e1*e2.trans()) increases
+ the value of the associated game by alpha.
+ """
+ e1 = matrix(e1, (K.dimension(), 1))
+ e2 = matrix(e2, (K.dimension(), 1))
+ game1 = SymmetricLinearGame(L, K, e1, e2)
+ soln1 = game1.solution()
+ value1 = soln1.game_value()
+ x_bar = soln1.player1_optimal()
+ y_bar = soln1.player2_optimal()
+
+ # Make ``L`` a CVXOPT matrix so that we can do math with
+ # it. Note that this gives us the "correct" representation of
+ # ``L`` (in agreement with what G has), but COLUMN indexed.
+ alpha = uniform(-10, 10)
+ L = matrix(L).trans()
+ tensor_prod = e1*e2.trans()
+
+ # Likewise, this is the "correct" representation of ``M``, but
+ # COLUMN indexed...
+ M = L + alpha*tensor_prod
+
+ # so we have to transpose it when we feed it to the constructor.
+ game2 = SymmetricLinearGame(M.trans(), K, e1, e2)
+ value2 = game2.solution().game_value()
+
+ self.assert_within_tol(value1 + alpha, value2)
+
+ # Make sure the same optimal pair works.
+ self.assert_within_tol(value2, inner_product(M*x_bar, y_bar))
+
+
+ def test_translation_orthant(self):
+ """
+ Test that translation works over the nonnegative orthant.
+ """
+ (L, K, e1, e2) = _random_orthant_params()
+ self.assert_translation_works(L, K, e1, e2)
+
+
+ def test_translation_icecream(self):
+ """
+ The same as :meth:`test_translation_orthant`, except over the
+ ice cream cone.
+ """
+ (L, K, e1, e2) = _random_icecream_params()
+ self.assert_translation_works(L, K, e1, e2)
+
+
+ def assert_opposite_game_works(self, L, K, e1, e2):
+ """
+ Check the value of the "opposite" game that gives rise to a
+ value that is the negation of the original game. Comes from
+ some corollary.
+ """
+ e1 = matrix(e1, (K.dimension(), 1))
+ e2 = matrix(e2, (K.dimension(), 1))
+ game1 = SymmetricLinearGame(L, K, e1, e2)
+
+ # Make ``L`` a CVXOPT matrix so that we can do math with
+ # it. Note that this gives us the "correct" representation of
+ # ``L`` (in agreement with what G has), but COLUMN indexed.
+ L = matrix(L).trans()
+
+ # Likewise, this is the "correct" representation of ``M``, but
+ # COLUMN indexed...
+ M = -L.trans()
+
+ # so we have to transpose it when we feed it to the constructor.
+ game2 = SymmetricLinearGame(M.trans(), K, e2, e1)
+
+ soln1 = game1.solution()
+ x_bar = soln1.player1_optimal()
+ y_bar = soln1.player2_optimal()
+ soln2 = game2.solution()
+
+ self.assert_within_tol(-soln1.game_value(), soln2.game_value())
+
+ # Make sure the switched optimal pair works.
+ self.assert_within_tol(soln2.game_value(),
+ inner_product(M*y_bar, x_bar))
+
+
+ def test_opposite_game_orthant(self):
+ """
+ Test the value of the "opposite" game over the nonnegative
+ orthant.
+ """
+ (L, K, e1, e2) = _random_orthant_params()
+ self.assert_opposite_game_works(L, K, e1, e2)
+
+
+ def test_opposite_game_icecream(self):
+ """
+ Like :meth:`test_opposite_game_orthant`, except over the
+ ice-cream cone.
+ """
+ (L, K, e1, e2) = _random_icecream_params()
+ self.assert_opposite_game_works(L, K, e1, e2)
+
+
+ def assert_orthogonality(self, L, K, e1, e2):
+ """
+ Two orthogonality relations hold at an optimal solution, and we
+ check them here.
+ """
+ game = SymmetricLinearGame(L, K, e1, e2)
+ soln = game.solution()
+ x_bar = soln.player1_optimal()
+ y_bar = soln.player2_optimal()
+ value = soln.game_value()
+
+ # Make these matrices so that we can compute with them.
+ L = matrix(L).trans()
+ e1 = matrix(e1, (K.dimension(), 1))
+ e2 = matrix(e2, (K.dimension(), 1))
+
+ ip1 = inner_product(y_bar, L*x_bar - value*e1)
+ self.assert_within_tol(ip1, 0)
+
+ ip2 = inner_product(value*e2 - L.trans()*y_bar, x_bar)
+ self.assert_within_tol(ip2, 0)
+
+
+ def test_orthogonality_orthant(self):
+ """
+ Check the orthgonality relationships that hold for a solution
+ over the nonnegative orthant.
+ """
+ (L, K, e1, e2) = _random_orthant_params()
+ self.assert_orthogonality(L, K, e1, e2)
+
+
+ def test_orthogonality_icecream(self):
+ """
+ Check the orthgonality relationships that hold for a solution
+ over the ice-cream cone.
+ """
+ (L, K, e1, e2) = _random_icecream_params()
+ self.assert_orthogonality(L, K, e1, e2)
+
+
+ def test_positive_operator_value(self):
+ """
+ Test that a positive operator on the nonnegative orthant gives
+ rise to a a game with a nonnegative value.
+
+ This test theoretically applies to the ice-cream cone as well,
+ but we don't know how to make positive operators on that cone.
+ """
+ (_, K, e1, e2) = _random_orthant_params()
+
+ # Ignore that L, we need a nonnegative one.
+ L = _random_nonnegative_matrix(K.dimension())
+
+ game = SymmetricLinearGame(L, K, e1, e2)
+ self.assertTrue(game.solution().game_value() >= -options.ABS_TOL)
projects / dunshire.git / blobdiff