X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=ae3ec48cddc9700d4f63ae378fc01b178dee6e3b;hb=8353d776d562e16cdbccfd10881662fc542c8d6f;hp=a1ded5270032de21f2647de8453384fde4804c72;hpb=090b2c77aa4bd371d66885451f9df44c6b6d818f;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index a1ded52..ae3ec48 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -67,12 +67,12 @@ def is_lyapunov_like(L,K): def motzkin_decomposition(K): r""" - Return the pair of components in the motzkin decomposition of this cone. + Return the pair of components in the Motzkin decomposition of this cone. Every convex cone is the direct sum of a strictly convex cone and a - linear subspace. Return a pair ``(P,S)`` of cones such that ``P`` is - strictly convex, ``S`` is a subspace, and ``K`` is the direct sum of - ``P`` and ``S``. + linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones + such that ``P`` is strictly convex, ``S`` is a subspace, and ``K`` + is the direct sum of ``P`` and ``S``. OUTPUT: @@ -80,6 +80,12 @@ def motzkin_decomposition(K): ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the direct sum of ``P`` and ``S``. + REFERENCES: + + .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and + Optimization in Finite Dimensions I. Springer-Verlag, New + York, 1970. + EXAMPLES: The nonnegative orthant is strictly convex, so it is its own @@ -129,32 +135,36 @@ def motzkin_decomposition(K): sage: S.lineality() == S.dim() True - The generators of the strictly convex component are obtained from - the orthogonal projections of the original generators onto the - orthogonal complement of the subspace component:: + The generators of the components are obtained from orthogonal + projections of the original generators [Stoer-Witzgall]_:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) sage: (P,S) = motzkin_decomposition(K) - sage: S_perp = S.linear_subspace().complement() - sage: A = S_perp.matrix().transpose() - sage: proj = A * (A.transpose()*A).inverse() * A.transpose() - sage: expected = Cone([ proj*g for g in K ], K.lattice()) - sage: P.is_equivalent(expected) + sage: A = S.linear_subspace().complement().matrix() + sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A + sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice()) + sage: P.is_equivalent(expected_P) + True + sage: A = S.linear_subspace().matrix() + sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A + sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice()) + sage: S.is_equivalent(expected_S) True """ - linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ] - linspace_gens += [ -b for b in linspace_gens ] + # The lines() method only returns one generator per line. For a true + # line, we also need a generator pointing in the opposite direction. + S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ] + S = Cone(S_gens, K.lattice()) - S = Cone(linspace_gens, K.lattice()) - - # Since ``S`` is a subspace, its dual is its orthogonal complement - # (albeit in the wrong lattice). + # Since ``S`` is a subspace, the rays of its dual generate its + # orthogonal complement. S_perp = Cone(S.dual(), K.lattice()) P = K.intersection(S_perp) return (P,S) + def positive_operator_gens(K): r""" Compute generators of the cone of positive operators on this cone. @@ -169,12 +179,6 @@ def positive_operator_gens(K): EXAMPLES: - The trivial cone in a trivial space has no positive operators:: - - sage: K = Cone([], ToricLattice(0)) - sage: positive_operator_gens(K) - [] - Positive operators on the nonnegative orthant are nonnegative matrices:: sage: K = Cone([(1,)]) @@ -188,6 +192,27 @@ def positive_operator_gens(K): [0 0], [0 0], [1 0], [0 1] ] + The trivial cone in a trivial space has no positive operators:: + + sage: K = Cone([], ToricLattice(0)) + sage: positive_operator_gens(K) + [] + + Every operator is positive on the trivial cone:: + + sage: K = Cone([(0,)]) + sage: positive_operator_gens(K) + [[1], [-1]] + + sage: K = Cone([(0,0)]) + sage: K.is_trivial() + True + sage: positive_operator_gens(K) + [ + [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1] + ] + Every operator is positive on the ambient vector space:: sage: K = Cone([(1,),(-1,)]) @@ -205,14 +230,58 @@ def positive_operator_gens(K): [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1] ] + A non-obvious application is to find the positive operators on the + right half-plane:: + + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: positive_operator_gens(K) + [ + [1 0] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [1 0], [-1 0], [0 1], [ 0 -1] + ] + TESTS: - A positive operator on a cone should send its generators into the cone:: + Each positive operator generator should send the generators of the + cone into the cone:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=5) sage: pi_of_K = positive_operator_gens(K) - sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()]) + sage: all([ K.contains(P*x) for P in pi_of_K for x in K ]) + True + + Each positive operator generator should send a random element of the + cone into the cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + sage: pi_of_K = positive_operator_gens(K) + sage: all([ K.contains(P*K.random_element()) for P in pi_of_K ]) + True + + A random element of the positive operator cone should send the + generators of the cone into the cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L) + sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list()) + sage: all([ K.contains(P*x) for x in K ]) + True + + A random element of the positive operator cone should send a random + element of the cone into the cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L) + sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list()) + sage: K.contains(P*K.random_element()) True The dimension of the cone of positive operators is given by the