# These are mostly actually needed.
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, norm
-import options
+from .cones import CartesianProduct, IceCream, NonnegativeOrthant
+from .errors import GameUnsolvableException
+from .matrices import (append_col, append_row, eigenvalues_re, identity,
+ inner_product, norm)
+from . import options
printing.options['dformat'] = options.FLOAT_FORMAT
solvers.options['show_progress'] = options.VERBOSE
Examples
--------
- >>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(3)
>>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
>>> e1 = [1,1,1]
Lists can (and probably should) be used for every argument::
- >>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(2)
>>> L = [[1,0],[0,1]]
>>> e1 = [1,1]
>>> import cvxopt
>>> import numpy
- >>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(2)
>>> L = [[1,0],[0,1]]
>>> e1 = cvxopt.matrix([1,1])
otherwise indexed by columns::
>>> import cvxopt
- >>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(2)
>>> L = [[1,2],[3,4]]
>>> e1 = [1,1]
This example is computed in Gowda and Ravindran in the section
"The value of a Z-transformation"::
- >>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(3)
>>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
>>> e1 = [1,1,1]
The value of the following game can be computed using the fact
that the identity is invertible::
- >>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(3)
>>> L = [[1,0,0],[0,1,0],[0,0,1]]
>>> e1 = [1,2,3]
Examples
--------
- >>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(3)
>>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
>>> e1 = [1,1,1]
-def _random_square_matrix(dims):
+def _random_matrix(dims):
"""
- Generate a random square (``dims``-by-``dims``) matrix,
- represented as a list of rows. This is used only by the
+ Generate a random square (``dims``-by-``dims``) matrix. This is used
+ only by the :class:`SymmetricLinearGameTest` class.
+ """
+ return matrix([[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. This is used only by the
+ :class:`SymmetricLinearGameTest` class.
+ """
+ L = _random_matrix(dims)
+ return matrix([abs(entry) for entry in L], (dims, 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.
"""
- return [[uniform(-10, 10) for i in range(dims)] for j in range(dims)]
+ return matrix([[uniform(-10, 10)*int(i == j) for i in range(dims)]
+ for j in range(dims)])
+
+
+def _random_skew_symmetric_matrix(dims):
+ """
+ Generate a random skew-symmetrix (``dims``-by-``dims``) matrix.
+
+ Examples
+ --------
+
+ >>> A = _random_skew_symmetric_matrix(randint(1, 10))
+ >>> norm(A + A.trans()) < options.ABS_TOL
+ True
+
+ """
+ strict_ut = [[uniform(-10, 10)*int(i < j) for i in range(dims)]
+ for j in range(dims)]
+
+ strict_ut = matrix(strict_ut, (dims, dims))
+ 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.
+ """
+ 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)
+ row1 = append_col(a, b.trans())
+ row2 = append_col(b, D)
+ return append_row(row1, row2)
def _random_orthant_params():
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_square_matrix(K.dimension())
- return (L, K, e1, e2)
+ L = _random_matrix(K.dimension())
+ return (L, K, matrix(e1), matrix(e2))
def _random_icecream_params():
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_square_matrix(K.dimension())
+ L = _random_matrix(K.dimension())
- return (L, K, e1, e2)
+ return (L, K, matrix(e1), matrix(e2))
-class SymmetricLinearGameTest(TestCase):
+# Tell pylint to shut up about the large number of methods.
+class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
"""
Tests for the SymmetricLinearGame and Solution classes.
"""
Given the parameters needed to construct a SymmetricLinearGame,
ensure that that game has a solution.
"""
- 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(),
+ G = SymmetricLinearGame(L.trans(), K, e1, e2)
+ soln = G.solution()
+
+ expected = inner_product(L*soln.player1_optimal(),
soln.player2_optimal())
self.assert_within_tol(soln.game_value(), expected)
Test that scaling ``L`` by a nonnegative number scales the value
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()
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)
+ # 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()
- # 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...
+ # 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.
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)
+ # 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)
- # 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
+ # This is the "correct" representation of ``M``, but
# COLUMN indexed...
M = -L.trans()
Two orthogonality relations hold at an optimal solution, and we
check them here.
"""
- game = SymmetricLinearGame(L, K, e1, e2)
+ # 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()
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)
"""
(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()[1:]
+ L = _random_nonnegative_matrix(K.dimension())
+
+ game = SymmetricLinearGame(L, K, e1, e2)
+ self.assertTrue(game.solution().game_value() >= -options.ABS_TOL)
+
+
+ def assert_lyapunov_works(self, L, K, e1, e2):
+ """
+ Check that Lyapunov games act the way we expect.
+ """
+ game = SymmetricLinearGame(L, K, e1, e2)
+ soln = game.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:
+ # 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:
+ # 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()
+ self.assert_within_tol(dualsoln.game_value(), soln.game_value())
+
+
+ def test_lyapunov_orthant(self):
+ """
+ Test that a Lyapunov game on the nonnegative orthant works.
+ """
+ (K, e1, e2) = _random_orthant_params()[1:]
+ L = _random_diagonal_matrix(K.dimension())
+
+ self.assert_lyapunov_works(L, K, e1, e2)
+
+
+ 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())
+
+ self.assert_lyapunov_works(L, K, e1, e2)
projects / dunshire.git / blobdiff