-# These few are used only for tests.
+"""
+Unit tests for the :class:`SymmetricLinearGame` class.
+"""
+
+MAX_COND = 250
+"""
+The maximum condition number of a randomly-generated game.
+"""
+
+RANDOM_MAX = 10
+"""
+When generating uniform random real numbers, this will be used as the
+largest allowed magnitude. It keeps our condition numbers down and other
+properties within reason.
+"""
+
from math import sqrt
from random import randint, uniform
from unittest import TestCase
identity, inner_product)
from dunshire import options
-def _random_matrix(dims):
+
+def random_matrix(dims):
"""
- Generate a random square (``dims``-by-``dims``) matrix. This is used
- only by the :class:`SymmetricLinearGameTest` class.
+ Generate a random square matrix.
+
+ Parameters
+ ----------
+
+ dims : int
+ The number of rows/columns you want in the returned matrix.
+
+ Returns
+ -------
+
+ matrix
+ A new matrix whose entries are random floats chosen uniformly from
+ the interval [-RANDOM_MAX, RANDOM_MAX].
+
+ Examples
+ --------
+
+ >>> A = random_matrix(3)
+ >>> A.size
+ (3, 3)
+
"""
- return matrix([[uniform(-10, 10) for i in range(dims)]
- for j in range(dims)])
+ return matrix([[uniform(-RANDOM_MAX, RANDOM_MAX) for _ in range(dims)]
+ for _ in range(dims)])
+
-def _random_nonnegative_matrix(dims):
+def random_nonnegative_matrix(dims):
"""
- Generate a random square (``dims``-by-``dims``) matrix with
- nonnegative entries. This is used only by the
- :class:`SymmetricLinearGameTest` class.
+ Generate a random square matrix with nonnegative entries.
+
+ Parameters
+ ----------
+
+ dims : int
+ The number of rows/columns you want in the returned matrix.
+
+ Returns
+ -------
+
+ matrix
+ A new matrix whose entries are random floats chosen uniformly from
+ the interval [0, RANDOM_MAX].
+
+ Examples
+ --------
+
+ >>> A = random_nonnegative_matrix(3)
+ >>> A.size
+ (3, 3)
+ >>> all([entry >= 0 for entry in A])
+ True
+
"""
- L = _random_matrix(dims)
+ L = random_matrix(dims)
return matrix([abs(entry) for entry in L], (dims, dims))
-def _random_diagonal_matrix(dims):
+
+def random_diagonal_matrix(dims):
"""
- Generate a random square (``dims``-by-``dims``) matrix with nonzero
- entries only on the diagonal. This is used only by the
- :class:`SymmetricLinearGameTest` class.
+ Generate a random square matrix with zero off-diagonal entries.
+
+ These matrices are Lyapunov-like on the nonnegative orthant, as is
+ fairly easy to see.
+
+ Parameters
+ ----------
+
+ dims : int
+ The number of rows/columns you want in the returned matrix.
+
+ Returns
+ -------
+
+ matrix
+ A new matrix whose diagonal entries are random floats chosen
+ uniformly from the interval [-RANDOM_MAX, RANDOM_MAX] and whose
+ off-diagonal entries are zero.
+
+ Examples
+ --------
+
+ >>> A = random_diagonal_matrix(3)
+ >>> A.size
+ (3, 3)
+ >>> A[0,1] == A[0,2] == A[1,0] == A[2,0] == A[1,2] == A[2,1] == 0
+ True
+
"""
- return matrix([[uniform(-10, 10)*int(i == j) for i in range(dims)]
+ return matrix([[uniform(-RANDOM_MAX, RANDOM_MAX)*int(i == j)
+ for i in range(dims)]
for j in range(dims)])
-def _random_skew_symmetric_matrix(dims):
+def random_skew_symmetric_matrix(dims):
"""
- Generate a random skew-symmetrix (``dims``-by-``dims``) matrix.
+ Generate a random skew-symmetrix matrix.
+
+ Parameters
+ ----------
+
+ dims : int
+ The number of rows/columns you want in the returned matrix.
+
+ Returns
+ -------
+
+ matrix
+ A new skew-matrix whose strictly above-diagonal entries are
+ random floats chosen uniformly from the interval
+ [-RANDOM_MAX, RANDOM_MAX].
Examples
--------
+ >>> A = random_skew_symmetric_matrix(3)
+ >>> A.size
+ (3, 3)
+
>>> from dunshire.matrices import norm
- >>> A = _random_skew_symmetric_matrix(randint(1, 10))
+ >>> A = random_skew_symmetric_matrix(randint(1, 10))
>>> norm(A + A.trans()) < options.ABS_TOL
True
return strict_ut - strict_ut.trans()
-def _random_lyapunov_like_icecream(dims):
- """
- Generate a random Lyapunov-like matrix over the ice-cream cone in
- ``dims`` dimensions.
+def random_lyapunov_like_icecream(dims):
+ r"""
+ Generate a random matrix Lyapunov-like on the ice-cream cone.
+
+ The form of these matrices is cited in Gowda and Tao
+ [GowdaTao]_. The scalar ``a`` and the vector ``b`` (using their
+ notation) are easy to generate. The submatrix ``D`` is a little
+ trickier, but it can be found noticing that :math:`C + C^{T} = 0`
+ for a skew-symmetric matrix :math:`C` implying that :math:`C + C^{T}
+ + \left(2a\right)I = \left(2a\right)I`. Thus we can stick an
+ :math:`aI` with each of :math:`C,C^{T}` and let those be our
+ :math:`D,D^{T}`.
+
+ Parameters
+ ----------
+
+ dims : int
+ The dimension of the ice-cream cone (not of the matrix you want!)
+ on which the returned matrix should be Lyapunov-like.
+
+ Returns
+ -------
+
+ matrix
+ A new matrix, Lyapunov-like on the ice-cream cone in ``dims``
+ dimensions, whose free entries are random floats chosen uniformly
+ from the interval [-10, 10].
+
+ References
+ ----------
+
+ .. [GowdaTao] M. S. Gowda and J. Tao. On the bilinearity rank of a
+ proper cone and Lyapunov-like transformations. Mathematical
+ Programming, 147:155-170, 2014.
+
+ Examples
+ --------
+
+ >>> L = random_lyapunov_like_icecream(3)
+ >>> L.size
+ (3, 3)
+ >>> x = matrix([1,1,0])
+ >>> s = matrix([1,-1,0])
+ >>> abs(inner_product(L*x, s)) < options.ABS_TOL
+ True
+
"""
a = matrix([uniform(-10, 10)], (1, 1))
- b = matrix([uniform(-10, 10) for idx in range(dims-1)], (dims-1, 1))
- D = _random_skew_symmetric_matrix(dims-1) + a*identity(dims-1)
+ b = matrix([uniform(-10, 10) for _ in range(dims-1)], (dims-1, 1))
+ D = random_skew_symmetric_matrix(dims-1) + a*identity(dims-1)
row1 = append_col(a, b.trans())
row2 = append_col(b, D)
return append_row(row1, row2)
-def _random_orthant_params():
+def random_orthant_game():
"""
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.
+ random game over the nonnegative orthant, and return the
+ corresponding :class:`SymmetricLinearGame`.
+
+ We keep going until we generate a game with a condition number under
+ 5000.
"""
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, matrix(e1), matrix(e2))
+ e1 = [uniform(0.5, 10) for _ in range(K.dimension())]
+ e2 = [uniform(0.5, 10) for _ in range(K.dimension())]
+ L = random_matrix(K.dimension())
+ G = SymmetricLinearGame(L, K, e1, e2)
+
+ if G._condition() <= MAX_COND:
+ return G
+ else:
+ return random_orthant_game()
-def _random_icecream_params():
+def random_icecream_game():
"""
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.
+ random game over the ice-cream cone, and return the corresponding
+ :class:`SymmetricLinearGame`.
"""
# Use a minimum dimension of two to avoid divide-by-zero in
# the fudge factor we make up later.
# 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())
+ e1 += [fudge_factor*uniform(0, 1) for _ in range(K.dimension() - 1)]
+ e2 += [fudge_factor*uniform(0, 1) for _ in range(K.dimension() - 1)]
+ L = random_matrix(K.dimension())
+ G = SymmetricLinearGame(L, K, e1, e2)
- return (L, K, matrix(e1), matrix(e2))
+ if G._condition() <= MAX_COND:
+ return G
+ else:
+ return random_icecream_game()
# Tell pylint to shut up about the large number of methods.
"""
def assert_within_tol(self, first, second):
"""
- Test that ``first`` and ``second`` are equal within our default
- tolerance.
- """
- self.assertTrue(abs(first - second) < options.ABS_TOL)
-
+ Test that ``first`` and ``second`` are equal within a multiple of
+ our default tolerances.
- def assert_norm_within_tol(self, first, second):
+ The factor of two is because if we compare two solutions, both
+ of which may be off by ``ABS_TOL``, then the result could be off
+ by ``2*ABS_TOL``. The factor of ``RANDOM_MAX`` allows for
+ scaling a result (by ``RANDOM_MAX``) that may be off by
+ ``ABS_TOL``. The final factor of two is to allow for the edge
+ cases where we get an "unknown" result and need to lower the
+ CVXOPT tolerance by a factor of two.
"""
- 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)
+ self.assertTrue(abs(first - second) < 2*2*RANDOM_MAX*options.ABS_TOL)
- def assert_solution_exists(self, L, K, e1, e2):
+ def assert_solution_exists(self, G):
"""
- Given the parameters needed to construct a SymmetricLinearGame,
- ensure that that game has a solution.
+ Given a SymmetricLinearGame, ensure that it has a solution.
"""
- # The matrix() constructor assumes that ``L`` is a list of
- # columns, so we transpose it to agree with what
- # SymmetricLinearGame() thinks.
- G = SymmetricLinearGame(L.trans(), K, e1, e2)
soln = G.solution()
- expected = inner_product(L*soln.player1_optimal(),
+ expected = inner_product(G._L*soln.player1_optimal(),
soln.player2_optimal())
self.assert_within_tol(soln.game_value(), expected)
+
+ def test_condition_lower_bound(self):
+ """
+ Ensure that the condition number of a game is greater than or
+ equal to one.
+
+ It should be safe to compare these floats directly: we compute
+ the condition number as the ratio of one nonnegative real number
+ to a smaller nonnegative real number.
+ """
+ G = random_orthant_game()
+ self.assertTrue(G._condition() >= 1.0)
+ G = random_icecream_game()
+ self.assertTrue(G._condition() >= 1.0)
+
+
def test_solution_exists_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.
"""
- (L, K, e1, e2) = _random_orthant_params()
- self.assert_solution_exists(L, K, e1, e2)
+ G = random_orthant_game()
+ self.assert_solution_exists(G)
def test_solution_exists_icecream(self):
Like :meth:`test_solution_exists_nonnegative_orthant`, except
over the ice cream cone.
"""
- (L, K, e1, e2) = _random_icecream_params()
- self.assert_solution_exists(L, K, e1, e2)
+ G = random_icecream_game()
+ self.assert_solution_exists(G)
def test_negative_value_z_operator(self):
self.assertTrue(G.solution().game_value() < -options.ABS_TOL)
- def assert_scaling_works(self, L, K, e1, e2):
+ def assert_scaling_works(self, game1):
"""
Test that scaling ``L`` by a nonnegative number scales the value
of the game by the same number.
"""
- game1 = SymmetricLinearGame(L, K, e1, e2)
value1 = game1.solution().game_value()
alpha = uniform(0.1, 10)
- game2 = SymmetricLinearGame(alpha*L, K, e1, e2)
+ game2 = SymmetricLinearGame(alpha*game1._L.trans(),
+ game1._K,
+ game1._e1,
+ game1._e2)
+
+ while game2._condition() > MAX_COND:
+ # Loop until the condition number of game2 doesn't suck.
+ alpha = uniform(0.1, 10)
+ game2 = SymmetricLinearGame(alpha*game1._L.trans(),
+ game1._K,
+ game1._e1,
+ game1._e2)
+
value2 = game2.solution().game_value()
self.assert_within_tol(alpha*value1, value2)
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)
+ G = random_orthant_game()
+ self.assert_scaling_works(G)
def test_scaling_icecream(self):
The same test as :meth:`test_nonnegative_scaling_orthant`,
except over the ice cream cone.
"""
- (L, K, e1, e2) = _random_icecream_params()
- self.assert_scaling_works(L, K, e1, e2)
+ G = random_icecream_game()
+ self.assert_scaling_works(G)
- def assert_translation_works(self, L, K, e1, e2):
+ def assert_translation_works(self, game1):
"""
Check that translating ``L`` by alpha*(e1*e2.trans()) increases
the value of the associated game by alpha.
"""
# We need to use ``L`` later, so make sure we transpose it
# before passing it in as a column-indexed matrix.
- game1 = SymmetricLinearGame(L.trans(), K, e1, e2)
soln1 = game1.solution()
value1 = soln1.game_value()
x_bar = soln1.player1_optimal()
y_bar = soln1.player2_optimal()
-
- alpha = uniform(-10, 10)
- tensor_prod = e1*e2.trans()
+ tensor_prod = game1._e1*game1._e2.trans()
# This is the "correct" representation of ``M``, but COLUMN
# indexed...
- M = L + alpha*tensor_prod
+ alpha = uniform(-10, 10)
+ M = game1._L + alpha*tensor_prod
# so we have to transpose it when we feed it to the constructor.
- game2 = SymmetricLinearGame(M.trans(), K, e1, e2)
+ game2 = SymmetricLinearGame(M.trans(), game1._K, game1._e1, game1._e2)
+ while game2._condition() > MAX_COND:
+ # Loop until the condition number of game2 doesn't suck.
+ alpha = uniform(-10, 10)
+ M = game1._L + alpha*tensor_prod
+ game2 = SymmetricLinearGame(M.trans(),
+ game1._K,
+ game1._e1,
+ game1._e2)
+
value2 = game2.solution().game_value()
self.assert_within_tol(value1 + alpha, value2)
"""
Test that translation works over the nonnegative orthant.
"""
- (L, K, e1, e2) = _random_orthant_params()
- self.assert_translation_works(L, K, e1, e2)
+ G = random_orthant_game()
+ self.assert_translation_works(G)
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)
+ G = random_icecream_game()
+ self.assert_translation_works(G)
- def assert_opposite_game_works(self, L, K, e1, e2):
+ def assert_opposite_game_works(self, game1):
"""
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.
"""
- # We need to use ``L`` later, so make sure we transpose it
- # before passing it in as a column-indexed matrix.
- game1 = SymmetricLinearGame(L.trans(), K, e1, e2)
-
# This is the "correct" representation of ``M``, but
# COLUMN indexed...
- M = -L.trans()
+ M = -game1._L.trans()
# so we have to transpose it when we feed it to the constructor.
- game2 = SymmetricLinearGame(M.trans(), K, e2, e1)
+ # Note: the condition number of game2 should be comparable to game1.
+ game2 = SymmetricLinearGame(M.trans(), game1._K, game1._e2, game1._e1)
soln1 = game1.solution()
x_bar = soln1.player1_optimal()
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)
+ G = random_orthant_game()
+ self.assert_opposite_game_works(G)
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)
+ G = random_icecream_game()
+ self.assert_opposite_game_works(G)
- def assert_orthogonality(self, L, K, e1, e2):
+ def assert_orthogonality(self, G):
"""
Two orthogonality relations hold at an optimal solution, and we
check them here.
"""
- # We need to use ``L`` later, so make sure we transpose it
- # before passing it in as a column-indexed matrix.
- game = SymmetricLinearGame(L.trans(), K, e1, e2)
- soln = game.solution()
+ soln = G.solution()
x_bar = soln.player1_optimal()
y_bar = soln.player2_optimal()
value = soln.game_value()
- ip1 = inner_product(y_bar, L*x_bar - value*e1)
+ ip1 = inner_product(y_bar, G._L*x_bar - value*G._e1)
self.assert_within_tol(ip1, 0)
- ip2 = inner_product(value*e2 - L.trans()*y_bar, x_bar)
+ ip2 = inner_product(value*G._e2 - G._L.trans()*y_bar, x_bar)
self.assert_within_tol(ip2, 0)
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)
+ G = random_orthant_game()
+ self.assert_orthogonality(G)
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)
+ G = random_icecream_game()
+ self.assert_orthogonality(G)
def test_positive_operator_value(self):
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()[1:]
- L = _random_nonnegative_matrix(K.dimension())
+ G = random_orthant_game()
+ L = random_nonnegative_matrix(G._K.dimension())
+
+ # Replace the totally-random ``L`` with the random nonnegative one.
+ G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
+
+ while G._condition() > MAX_COND:
+ # Try again until the condition number is satisfactory.
+ G = random_orthant_game()
+ L = random_nonnegative_matrix(G._K.dimension())
+ G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
- game = SymmetricLinearGame(L, K, e1, e2)
- self.assertTrue(game.solution().game_value() >= -options.ABS_TOL)
+ self.assertTrue(G.solution().game_value() >= -options.ABS_TOL)
- def assert_lyapunov_works(self, L, K, e1, e2):
+ def assert_lyapunov_works(self, G):
"""
Check that Lyapunov games act the way we expect.
"""
- game = SymmetricLinearGame(L, K, e1, e2)
- soln = game.solution()
+ soln = G.solution()
# We only check for positive/negative stability if the game
# value is not basically zero. If the value is that close to
# zero, we just won't check any assertions.
- eigs = eigenvalues_re(L)
- if soln.game_value() > options.ABS_TOL:
+ #
+ # See :meth:`assert_within_tol` for an explanation of the
+ # fudge factors.
+ eigs = eigenvalues_re(G._L)
+ epsilon = 2*2*RANDOM_MAX*options.ABS_TOL
+ if soln.game_value() > epsilon:
# L should be positive stable
positive_stable = all([eig > -options.ABS_TOL for eig in eigs])
self.assertTrue(positive_stable)
- elif soln.game_value() < -options.ABS_TOL:
+ elif soln.game_value() < -epsilon:
# L should be negative stable
negative_stable = all([eig < options.ABS_TOL for eig in eigs])
self.assertTrue(negative_stable)
# The dual game's value should always equal the primal's.
- dualsoln = game.dual().solution()
+ dualsoln = G.dual().solution()
self.assert_within_tol(dualsoln.game_value(), soln.game_value())
"""
Test that a Lyapunov game on the nonnegative orthant works.
"""
- (K, e1, e2) = _random_orthant_params()[1:]
- L = _random_diagonal_matrix(K.dimension())
+ G = random_orthant_game()
+ L = random_diagonal_matrix(G._K.dimension())
+
+ # Replace the totally-random ``L`` with random Lyapunov-like one.
+ G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
+
+ while G._condition() > MAX_COND:
+ # Try again until the condition number is satisfactory.
+ G = random_orthant_game()
+ L = random_diagonal_matrix(G._K.dimension())
+ G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
- self.assert_lyapunov_works(L, K, e1, e2)
+ self.assert_lyapunov_works(G)
def test_lyapunov_icecream(self):
"""
Test that a Lyapunov game on the ice-cream cone works.
"""
- (K, e1, e2) = _random_icecream_params()[1:]
- L = _random_lyapunov_like_icecream(K.dimension())
+ G = random_icecream_game()
+ L = random_lyapunov_like_icecream(G._K.dimension())
+
+ # Replace the totally-random ``L`` with random Lyapunov-like one.
+ G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
+
+ while G._condition() > MAX_COND:
+ # Try again until the condition number is satisfactory.
+ G = random_icecream_game()
+ L = random_lyapunov_like_icecream(G._K.dimension())
+ G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
- self.assert_lyapunov_works(L, K, e1, e2)
+ self.assert_lyapunov_works(G)
projects / dunshire.git / blobdiff