X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=0bfc3b6a779147b0f0a30799bfe2c149cb818169;hb=6251a6e0461374bf915ec3695f62cb0349f379ae;hp=21f9862c24a9e9e3d9cffc52a1e789018f59a153;hpb=c9ab469a53dc691d2646e22bdbc0ea0b3f3961c1;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 21f9862..0bfc3b6 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -10,6 +10,12 @@ def is_lyapunov_like(L,K): ``K``. It is known [Orlitzky]_ that this property need only be checked for generators of ``K`` and its dual. + There are faster ways of checking this property. For example, we + could compute a `lyapunov_like_basis` of the cone, and then test + whether or not the given matrix is contained in the span of that + basis. The value of this function is that it works on symbolic + matrices. + INPUT: - ``L`` -- A linear transformation or matrix. @@ -65,114 +71,6 @@ def is_lyapunov_like(L,K): for (x,s) in K.discrete_complementarity_set()]) -def motzkin_decomposition(K): - r""" - 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 [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: - - 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``. - - 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 - strictly convex component and its subspace component is trivial:: - - sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - 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: - - 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: (P,S) = motzkin_decomposition(K) - sage: x = K.random_element(ring=QQ) - sage: P.contains(x) or S.contains(x) - True - sage: x.is_zero() or (P.contains(x) != S.contains(x)) - True - - 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: (P,S) = motzkin_decomposition(K) - sage: P.is_strictly_convex() - True - sage: S.lineality() == S.dim() - True - - A strictly convex cone should be equal to its strictly convex component:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, strictly_convex=True) - sage: (P,_) = motzkin_decomposition(K) - sage: K.is_equivalent(P) - True - - 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: 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 - """ - # 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(), check=False) - - # Since ``S`` is a subspace, the rays of its dual generate its - # orthogonal complement. - S_perp = Cone(S.dual(), K.lattice(), check=False) - 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. @@ -185,6 +83,17 @@ def positive_operator_gens(K): ``K``. Moreover, any nonnegative linear combination of these matrices shares the same property. + REFERENCES: + + .. [Orlitzky-Pi-Z] + M. Orlitzky. + Positive and Z-operators on closed convex cones. + + .. [Tam] + B.-S. Tam. + Some results of polyhedral cones and simplicial cones. + Linear and Multilinear Algebra, 4:4 (1977) 281--284. + EXAMPLES: Positive operators on the nonnegative orthant are nonnegative matrices:: @@ -460,6 +369,33 @@ def positive_operator_gens(K): ....: check=False) sage: actual.is_equivalent(expected) True + + A transformation is positive on a cone if and only if its adjoint is + positive on the dual of that cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: F = K.lattice().vector_space().base_field() + sage: n = K.lattice_dim() + sage: L = ToricLattice(n**2) + sage: W = VectorSpace(F, n**2) + sage: pi_of_K = positive_operator_gens(K) + sage: pi_of_K_star = positive_operator_gens(K.dual()) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: pi_star = Cone([p.list() for p in pi_of_K_star], + ....: lattice=L, + ....: check=False) + sage: M = MatrixSpace(F, n) + sage: L = M(pi_cone.random_element(ring=QQ).list()) + sage: pi_star.contains(W(L.transpose().list())) + True + + sage: L = W.random_element() + sage: L_star = W(M(L.list()).transpose().list()) + sage: pi_cone.contains(L) == pi_star.contains(L_star) + True """ # Matrices are not vectors in Sage, so we have to convert them # to vectors explicitly before we can find a basis. We need these @@ -474,14 +410,12 @@ def positive_operator_gens(K): vectors = [ W(tp.list()) for tp in tensor_products ] check = True - if K.is_solid() or K.is_strictly_convex(): - # The lineality space of either ``K`` or ``K.dual()`` is - # trivial and it's easy to show that our generating set is - # minimal. I would love a proof that this works when ``K`` is - # neither pointed nor solid. - # - # Note that in that case we can get *duplicates*, since the - # tensor product of (x,s) is the same as that of (-x,-s). + if K.is_proper(): + # All of the generators involved are extreme vectors and + # therefore minimal [Tam]_. If this cone is neither solid nor + # strictly convex, then the tensor product of ``s`` and ``x`` + # is the same as that of ``-s`` and ``-x``. However, as a + # /set/, ``tensor_products`` may still be minimal. check = False # Create the dual cone of the positive operators, expressed as @@ -496,98 +430,104 @@ def positive_operator_gens(K): return [ M(v.list()) for v in pi_cone ] -def Z_transformation_gens(K): +def Z_operator_gens(K): r""" - Compute generators of the cone of Z-transformations on this cone. + Compute generators of the cone of Z-operators on this cone. OUTPUT: A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. Each matrix ``L`` in the list should have the property that - ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the - discrete complementarity set of ``K``. Moreover, any nonnegative - linear combination of these matrices shares the same property. + ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element of + this cone's :meth:`discrete_complementarity_set`. Moreover, any + conic (nonnegative linear) combination of these matrices shares the + same property. + + REFERENCES: + + M. Orlitzky. + Positive and Z-operators on closed convex cones. EXAMPLES: - Z-transformations on the nonnegative orthant are just Z-matrices. + Z-operators on the nonnegative orthant are just Z-matrices. That is, matrices whose off-diagonal elements are nonnegative:: sage: K = Cone([(1,0),(0,1)]) - sage: Z_transformation_gens(K) + sage: Z_operator_gens(K) [ [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0] [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1] ] sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]) - sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K) + sage: all([ z[i][j] <= 0 for z in Z_operator_gens(K) ....: for i in range(z.nrows()) ....: for j in range(z.ncols()) ....: if i != j ]) True - The trivial cone in a trivial space has no Z-transformations:: + The trivial cone in a trivial space has no Z-operators:: sage: K = Cone([], ToricLattice(0)) - sage: Z_transformation_gens(K) + sage: Z_operator_gens(K) [] - Every operator is a Z-transformation on the ambient vector space:: + Every operator is a Z-operator on the ambient vector space:: sage: K = Cone([(1,),(-1,)]) sage: K.is_full_space() True - sage: Z_transformation_gens(K) + sage: Z_operator_gens(K) [[-1], [1]] sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) sage: K.is_full_space() True - sage: Z_transformation_gens(K) + sage: Z_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] ] - A non-obvious application is to find the Z-transformations on the + A non-obvious application is to find the Z-operators on the right half-plane:: sage: K = Cone([(1,0),(0,1),(0,-1)]) - sage: Z_transformation_gens(K) + sage: Z_operator_gens(K) [ [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0] [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1] ] - Z-transformations on a subspace are Lyapunov-like and vice-versa:: + Z-operators on a subspace are Lyapunov-like and vice-versa:: sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) sage: K.is_full_space() True sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) - sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ]) + sage: zs = span([ vector(z.list()) for z in Z_operator_gens(K) ]) sage: zs == lls True TESTS: - The Z-property is possessed by every Z-transformation:: + The Z-property is possessed by every Z-operator:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) - sage: Z_of_K = Z_transformation_gens(K) + sage: Z_of_K = Z_operator_gens(K) sage: dcs = K.discrete_complementarity_set() sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K ....: for (x,s) in dcs]) True - The lineality space of the cone of Z-transformations is the space of - Lyapunov-like transformations:: + The lineality space of the cone of Z-operators is the space of + Lyapunov-like operators:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) sage: L = ToricLattice(K.lattice_dim()**2) - sage: Z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ], + sage: Z_cone = Cone([ z.list() for z in Z_operator_gens(K) ], ....: lattice=L, ....: check=False) sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ] @@ -595,12 +535,12 @@ def Z_transformation_gens(K): sage: Z_cone.linear_subspace() == lls True - The lineality of the Z-transformations on a cone is the Lyapunov + The lineality of the Z-operators on a cone is the Lyapunov rank of that cone:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) - sage: Z_of_K = Z_transformation_gens(K) + sage: Z_of_K = Z_operator_gens(K) sage: L = ToricLattice(K.lattice_dim()**2) sage: Z_cone = Cone([ z.list() for z in Z_of_K ], ....: lattice=L, @@ -608,15 +548,14 @@ def Z_transformation_gens(K): sage: Z_cone.lineality() == K.lyapunov_rank() True - The lineality spaces of the duals of the positive operator and - Z-transformation cones are equal. From this it follows that the - dimensions of the Z-transformation cone and positive operator cone - are equal:: + The lineality spaces of the duals of the positive and Z-operator + cones are equal. From this it follows that the dimensions of the + Z-operator cone and positive operator cone are equal:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) sage: pi_of_K = positive_operator_gens(K) - sage: Z_of_K = Z_transformation_gens(K) + sage: Z_of_K = Z_operator_gens(K) sage: L = ToricLattice(K.lattice_dim()**2) sage: pi_cone = Cone([p.list() for p in pi_of_K], ....: lattice=L, @@ -639,7 +578,7 @@ def Z_transformation_gens(K): sage: K.is_trivial() True sage: L = ToricLattice(n^2) - sage: Z_of_K = Z_transformation_gens(K) + sage: Z_of_K = Z_operator_gens(K) sage: Z_cone = Cone([z.list() for z in Z_of_K], ....: lattice=L, ....: check=False) @@ -649,7 +588,7 @@ def Z_transformation_gens(K): sage: K = K.dual() sage: K.is_full_space() True - sage: Z_of_K = Z_transformation_gens(K) + sage: Z_of_K = Z_operator_gens(K) sage: Z_cone = Cone([z.list() for z in Z_of_K], ....: lattice=L, ....: check=False) @@ -657,29 +596,55 @@ def Z_transformation_gens(K): sage: actual == n^2 True sage: K = Cone([(1,0),(0,1),(0,-1)]) - sage: Z_of_K = Z_transformation_gens(K) + sage: Z_of_K = Z_operator_gens(K) sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False) sage: Z_cone.dim() == 3 True - The Z-transformations of a permuted cone can be obtained by - conjugation:: + The Z-operators of a permuted cone can be obtained by conjugation:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) sage: L = ToricLattice(K.lattice_dim()**2) sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix() sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False) - sage: Z_of_pK = Z_transformation_gens(pK) + sage: Z_of_pK = Z_operator_gens(pK) sage: actual = Cone([t.list() for t in Z_of_pK], ....: lattice=L, ....: check=False) - sage: Z_of_K = Z_transformation_gens(K) + sage: Z_of_K = Z_operator_gens(K) sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K], ....: lattice=L, ....: check=False) sage: actual.is_equivalent(expected) True + + An operator is a Z-operator on a cone if and only if its + adjoint is a Z-operator on the dual of that cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: F = K.lattice().vector_space().base_field() + sage: n = K.lattice_dim() + sage: L = ToricLattice(n**2) + sage: W = VectorSpace(F, n**2) + sage: Z_of_K = Z_operator_gens(K) + sage: Z_of_K_star = Z_operator_gens(K.dual()) + sage: Z_cone = Cone([p.list() for p in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: Z_star = Cone([p.list() for p in Z_of_K_star], + ....: lattice=L, + ....: check=False) + sage: M = MatrixSpace(F, n) + sage: L = M(Z_cone.random_element(ring=QQ).list()) + sage: Z_star.contains(W(L.transpose().list())) + True + + sage: L = W.random_element() + sage: L_star = W(M(L.list()).transpose().list()) + sage: Z_cone.contains(L) == Z_star.contains(L_star) + True """ # Matrices are not vectors in Sage, so we have to convert them # to vectors explicitly before we can find a basis. We need these @@ -688,7 +653,7 @@ def Z_transformation_gens(K): n = K.lattice_dim() # These tensor products contain generators for the dual cone of - # the cross-positive transformations. + # the cross-positive operators. tensor_products = [ s.tensor_product(x) for (x,s) in K.discrete_complementarity_set() ] @@ -697,14 +662,12 @@ def Z_transformation_gens(K): vectors = [ W(m.list()) for m in tensor_products ] check = True - if K.is_solid() or K.is_strictly_convex(): - # The lineality space of either ``K`` or ``K.dual()`` is - # trivial and it's easy to show that our generating set is - # minimal. I would love a proof that this works when ``K`` is - # neither pointed nor solid. - # - # Note that in that case we can get *duplicates*, since the - # tensor product of (x,s) is the same as that of (-x,-s). + if K.is_proper(): + # All of the generators involved are extreme vectors and + # therefore minimal. If this cone is neither solid nor + # strictly convex, then the tensor product of ``s`` and ``x`` + # is the same as that of ``-s`` and ``-x``. However, as a + # /set/, ``tensor_products`` may still be minimal. check = False # Create the dual cone of the cross-positive operators, @@ -715,14 +678,19 @@ def Z_transformation_gens(K): Sigma_cone = Sigma_dual.dual() # And finally convert its rays back to matrix representations. - # But first, make them negative, so we get Z-transformations and + # But first, make them negative, so we get Z-operators and # not cross-positive ones. M = MatrixSpace(F, n) return [ -M(v.list()) for v in Sigma_cone ] +def LL_cone(K): + gens = K.lyapunov_like_basis() + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) + def Z_cone(K): - gens = Z_transformation_gens(K) + gens = Z_operator_gens(K) L = ToricLattice(K.lattice_dim()**2) return Cone([ g.list() for g in gens ], lattice=L, check=False)