X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=ae3ec48cddc9700d4f63ae378fc01b178dee6e3b;hb=8353d776d562e16cdbccfd10881662fc542c8d6f;hp=68edeb4a012b1bca7c4f9664970d2d89371b4926;hpb=138b60c0dad476335a09e81642bff241c91932ca;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 68edeb4..ae3ec48 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -65,118 +65,106 @@ def is_lyapunov_like(L,K): for (x,s) in K.discrete_complementarity_set()]) -def random_element(K): +def motzkin_decomposition(K): r""" - Return a random element of ``K`` from its ambient vector space. + Return the pair of components in the Motzkin decomposition of this cone. - ALGORITHM: + Every convex cone is the direct sum of a strictly convex cone and a + 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``. - The cone ``K`` is specified in terms of its generators, so that - ``K`` is equal to the convex conic combination of those generators. - To choose a random element of ``K``, we assign random nonnegative - coefficients to each generator of ``K`` and construct a new vector - from the scaled rays. + OUTPUT: - A vector, rather than a ray, is returned so that the element may - have non-integer coordinates. Thus the element may have an - arbitrarily small norm. + An ordered pair ``(P,S)`` of closed convex polyhedral cones where + ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the + direct sum of ``P`` and ``S``. - EXAMPLES: + REFERENCES: - A random element of the trivial cone is zero:: + .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and + Optimization in Finite Dimensions I. Springer-Verlag, New + York, 1970. - sage: set_random_seed() - sage: K = Cone([], ToricLattice(0)) - sage: random_element(K) - () - sage: K = Cone([(0,)]) - sage: random_element(K) - (0) - sage: K = Cone([(0,0)]) - sage: random_element(K) - (0, 0) - sage: K = Cone([(0,0,0)]) - sage: random_element(K) - (0, 0, 0) + EXAMPLES: - A random element of the nonnegative orthant should have all - components nonnegative:: + The nonnegative orthant is strictly convex, so it is its own + strictly convex component and its subspace component is trivial:: - sage: set_random_seed() sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - sage: all([ x >= 0 for x in random_element(K) ]) + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(P) + True + sage: S.is_trivial() + True + + Likewise, full spaces are their own subspace components:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(S) + True + sage: P.is_trivial() True TESTS: - Any cone should contain a random element of itself:: + A random point in the cone should belong to either the strictly + convex component or the subspace component. If the point is nonzero, + it cannot be in both:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) - sage: K.contains(random_element(K)) + sage: (P,S) = motzkin_decomposition(K) + sage: x = K.random_element() + sage: P.contains(x) or S.contains(x) True - - A strictly convex cone contains no lines, and thus no negative - multiples of any of its elements besides zero:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, strictly_convex=True) - sage: x = random_element(K) - sage: x.is_zero() or not K.contains(-x) + sage: x.is_zero() or (P.contains(x) != S.contains(x)) True - The sum of random elements of a cone lies in the cone:: + The strictly convex component should always be strictly convex, and + the subspace component should always be a subspace:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) - sage: K.contains(sum([random_element(K) for i in range(10)])) + sage: (P,S) = motzkin_decomposition(K) + sage: P.is_strictly_convex() + True + sage: S.lineality() == S.dim() True - """ - V = K.lattice().vector_space() - scaled_gens = [ V.base_field().random_element().abs()*V(r) for r in K ] - - # Make sure we return a vector. Without the coercion, we might - # return ``0`` when ``K`` has no rays. - return V(sum(scaled_gens)) - - -def pointed_decomposition(K): - """ - Every convex cone is the direct sum of a pointed cone and a linear - subspace. Return a pair ``(P,S)`` of cones such that ``P`` is - pointed, ``S`` is a subspace, and ``K`` is the direct sum of ``P`` - and ``S``. - - OUTPUT: - - An ordered pair ``(P,S)`` of closed convex polyhedral cones where - ``P`` is pointed, ``S`` is a subspace, and ``K`` is the direct sum - of ``P`` and ``S``. - - TESTS: + 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) = pointed_decomposition(K) - sage: x = random_element(K) - sage: P.contains(x) or S.contains(x) + sage: (P,S) = motzkin_decomposition(K) + 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: x.is_zero() or (P.contains(x) != S.contains(x)) + 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. @@ -191,12 +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,)]) @@ -210,6 +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,)]) @@ -227,14 +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 ]) + 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: all([K.contains(p*x) for p in pi_of_K for x in K.rays()]) + 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