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.
Tests for the SymmetricLinearGame and Solution classes.
"""
+ def random_square_matrix(self, dims):
+ """
+ Generate a random square (``dims``-by-``dims``) matrix,
+ represented as a list of rows.
+ """
+ return [[uniform(-10, 10) for i in range(dims)] for j in range(dims)]
+
+
+ def random_orthant_params(self):
+ """
+ Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a
+ random game over the nonnegative orthant.
+ """
+ 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 = self.random_square_matrix(K.dimension())
+ return (L, K, e1, e2)
+
+
+ def random_icecream_params(self):
+ """
+ Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a
+ random game 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 = self.random_square_matrix(K.dimension())
+
+ return (L, K, e1, e2)
+
+
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):
"""
Every linear game has a solution, so we should be able to solve
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) = self.random_orthant_params()
self.assert_solution_exists(L, K, e1, e2)
+
def test_solution_exists_ice_cream(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) = self.random_icecream_params()
self.assert_solution_exists(L, K, e1, e2)
+
+ 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]
+ e2 = e1
+ 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):
+ """
+ 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.
+ """
+ (L, K, e1, e2) = self.random_orthant_params()
+ # Make ``L`` a matrix so that we can scale it by alpha. Its
+ # random, so who cares if it gets transposed.
+ L = matrix(L)
+ G1 = SymmetricLinearGame(L, K, e1, e2)
+ value1 = G1.solution().game_value()
+
+ alpha = uniform(0.1, 10)
+ G2 = SymmetricLinearGame(alpha*L, K, e1, e2)
+ value2 = G2.solution().game_value()
+ self.assert_within_tol(alpha*value1, value2)
+
+
+ def test_nonnegative_scaling_icecream(self):
+ """
+ The same test as :meth:`test_nonnegative_scaling_orthant`,
+ except over the ice cream cone.
+ """
+ (L, K, e1, e2) = self.random_icecream_params()
+ # Make ``L`` a matrix so that we can scale it by alpha. Its
+ # random, so who cares if it gets transposed.
+ L = matrix(L)
+ G1 = SymmetricLinearGame(L, K, e1, e2)
+ value1 = G1.solution().game_value()
+
+ alpha = uniform(0.1, 10)
+ 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))
+ G = SymmetricLinearGame(L, K, e1, e2)
+ G_soln = G.solution()
+ value_G = G_soln.game_value()
+ x_bar = G_soln.player1_optimal()
+ y_bar = G_soln.player2_optimal()
+
+ alpha = uniform(-10, 10)
+ # 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()
+ E = e1*e2.trans()
+ # Likewise, this is the "correct" representation of ``M``, but
+ # COLUMN indexed...
+ M = L + alpha*E
+
+ # so we have to transpose it when we feed it to the constructor.
+ H = SymmetricLinearGame(M.trans(), K, e1, e2)
+ value_H = H.solution().game_value()
+
+ # Make sure the same optimal pair works.
+ H_payoff = inner_product(M*x_bar, y_bar)
+
+ self.assert_within_tol(value_G + alpha, value_H)
+ self.assert_within_tol(value_H, H_payoff)
+
+
+ def test_translation_orthant(self):
+ """
+ Test that translation works over the nonnegative orthant.
+ """
+ (L, K, e1, e2) = self.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) = self.random_icecream_params()
+ self.assert_translation_works(L, K, e1, e2)
+
+
+ def assert_opposite_game_works(self, L, K, e1, e2):
+ e1 = matrix(e1, (K.dimension(), 1))
+ e2 = matrix(e2, (K.dimension(), 1))
+ G = 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.
+ H = SymmetricLinearGame(M.trans(), K, e2, e1)
+
+ G_soln = G.solution()
+ x_bar = G_soln.player1_optimal()
+ y_bar = G_soln.player2_optimal()
+ H_soln = H.solution()
+
+ # Make sure the switched optimal pair works.
+ H_payoff = inner_product(M*y_bar, x_bar)
+
+ self.assert_within_tol(-G_soln.game_value(), H_soln.game_value())
+ self.assert_within_tol(H_soln.game_value(), H_payoff)
+
+
+ def test_opposite_game_orthant(self):
+ """
+ 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.
+ """
+ (L, K, e1, e2) = self.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) = self.random_icecream_params()
+ self.assert_opposite_game_works(L, K, e1, e2)
projects / dunshire.git / blobdiff