]>
gitweb.michael.orlitzky.com - sage.d.git/blob - cone.py
332d4b24ed93ce9daac2a5779df94068d6479838
3 def is_lyapunov_like(L
,K
):
5 Determine whether or not ``L`` is Lyapunov-like on ``K``.
7 We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
8 L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
9 `\left\langle x,s \right\rangle` in the complementarity set of
10 ``K``. It is known [Orlitzky]_ that this property need only be
11 checked for generators of ``K`` and its dual.
15 - ``L`` -- A linear transformation or matrix.
17 - ``K`` -- A polyhedral closed convex cone.
21 ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
22 and ``False`` otherwise.
26 If this function returns ``True``, then ``L`` is Lyapunov-like
27 on ``K``. However, if ``False`` is returned, that could mean one
28 of two things. The first is that ``L`` is definitely not
29 Lyapunov-like on ``K``. The second is more of an "I don't know"
30 answer, returned (for example) if we cannot prove that an inner
35 M. Orlitzky. The Lyapunov rank of an improper cone.
36 http://www.optimization-online.org/DB_HTML/2015/10/5135.html
40 The identity is always Lyapunov-like in a nontrivial space::
42 sage: set_random_seed()
43 sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
44 sage: L = identity_matrix(K.lattice_dim())
45 sage: is_lyapunov_like(L,K)
48 As is the "zero" transformation::
50 sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
51 sage: R = K.lattice().vector_space().base_ring()
52 sage: L = zero_matrix(R, K.lattice_dim())
53 sage: is_lyapunov_like(L,K)
56 Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
59 sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
60 sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
64 return all([(L
*x
).inner_product(s
) == 0
65 for (x
,s
) in K
.discrete_complementarity_set()])
68 def motzkin_decomposition(K
):
70 Return the pair of components in the Motzkin decomposition of this cone.
72 Every convex cone is the direct sum of a strictly convex cone and a
73 linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones
74 such that ``P`` is strictly convex, ``S`` is a subspace, and ``K``
75 is the direct sum of ``P`` and ``S``.
79 An ordered pair ``(P,S)`` of closed convex polyhedral cones where
80 ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the
81 direct sum of ``P`` and ``S``.
85 .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
86 Optimization in Finite Dimensions I. Springer-Verlag, New
91 The nonnegative orthant is strictly convex, so it is its own
92 strictly convex component and its subspace component is trivial::
94 sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
95 sage: (P,S) = motzkin_decomposition(K)
96 sage: K.is_equivalent(P)
101 Likewise, full spaces are their own subspace components::
103 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
104 sage: K.is_full_space()
106 sage: (P,S) = motzkin_decomposition(K)
107 sage: K.is_equivalent(S)
114 A random point in the cone should belong to either the strictly
115 convex component or the subspace component. If the point is nonzero,
116 it cannot be in both::
118 sage: set_random_seed()
119 sage: K = random_cone(max_ambient_dim=8)
120 sage: (P,S) = motzkin_decomposition(K)
121 sage: x = K.random_element(ring=QQ)
122 sage: P.contains(x) or S.contains(x)
124 sage: x.is_zero() or (P.contains(x) != S.contains(x))
127 The strictly convex component should always be strictly convex, and
128 the subspace component should always be a subspace::
130 sage: set_random_seed()
131 sage: K = random_cone(max_ambient_dim=8)
132 sage: (P,S) = motzkin_decomposition(K)
133 sage: P.is_strictly_convex()
135 sage: S.lineality() == S.dim()
138 The generators of the components are obtained from orthogonal
139 projections of the original generators [Stoer-Witzgall]_::
141 sage: set_random_seed()
142 sage: K = random_cone(max_ambient_dim=8)
143 sage: (P,S) = motzkin_decomposition(K)
144 sage: A = S.linear_subspace().complement().matrix()
145 sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A
146 sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice())
147 sage: P.is_equivalent(expected_P)
149 sage: A = S.linear_subspace().matrix()
150 sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A
151 sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice())
152 sage: S.is_equivalent(expected_S)
155 # The lines() method only returns one generator per line. For a true
156 # line, we also need a generator pointing in the opposite direction.
157 S_gens
= [ direction
*gen
for direction
in [1,-1] for gen
in K
.lines() ]
158 S
= Cone(S_gens
, K
.lattice())
160 # Since ``S`` is a subspace, the rays of its dual generate its
161 # orthogonal complement.
162 S_perp
= Cone(S
.dual(), K
.lattice())
163 P
= K
.intersection(S_perp
)
168 def positive_operator_gens(K
):
170 Compute generators of the cone of positive operators on this cone.
174 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
175 Each matrix ``P`` in the list should have the property that ``P*x``
176 is an element of ``K`` whenever ``x`` is an element of
177 ``K``. Moreover, any nonnegative linear combination of these
178 matrices shares the same property.
182 Positive operators on the nonnegative orthant are nonnegative matrices::
184 sage: K = Cone([(1,)])
185 sage: positive_operator_gens(K)
188 sage: K = Cone([(1,0),(0,1)])
189 sage: positive_operator_gens(K)
191 [1 0] [0 1] [0 0] [0 0]
192 [0 0], [0 0], [1 0], [0 1]
195 The trivial cone in a trivial space has no positive operators::
197 sage: K = Cone([], ToricLattice(0))
198 sage: positive_operator_gens(K)
201 Every operator is positive on the trivial cone::
203 sage: K = Cone([(0,)])
204 sage: positive_operator_gens(K)
207 sage: K = Cone([(0,0)])
210 sage: positive_operator_gens(K)
212 [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
213 [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
216 Every operator is positive on the ambient vector space::
218 sage: K = Cone([(1,),(-1,)])
219 sage: K.is_full_space()
221 sage: positive_operator_gens(K)
224 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
225 sage: K.is_full_space()
227 sage: positive_operator_gens(K)
229 [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
230 [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
233 A non-obvious application is to find the positive operators on the
236 sage: K = Cone([(1,0),(0,1),(0,-1)])
237 sage: positive_operator_gens(K)
239 [1 0] [0 0] [ 0 0] [0 0] [ 0 0]
240 [0 0], [1 0], [-1 0], [0 1], [ 0 -1]
245 Each positive operator generator should send the generators of the
248 sage: set_random_seed()
249 sage: K = random_cone(max_ambient_dim=5)
250 sage: pi_of_K = positive_operator_gens(K)
251 sage: all([ K.contains(P*x) for P in pi_of_K for x in K ])
254 Each positive operator generator should send a random element of the
257 sage: set_random_seed()
258 sage: K = random_cone(max_ambient_dim=5)
259 sage: pi_of_K = positive_operator_gens(K)
260 sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ])
263 A random element of the positive operator cone should send the
264 generators of the cone into the cone::
266 sage: set_random_seed()
267 sage: K = random_cone(max_ambient_dim=5)
268 sage: pi_of_K = positive_operator_gens(K)
269 sage: L = ToricLattice(K.lattice_dim()**2)
270 sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
271 sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
272 sage: all([ K.contains(P*x) for x in K ])
275 A random element of the positive operator cone should send a random
276 element of the cone into the cone::
278 sage: set_random_seed()
279 sage: K = random_cone(max_ambient_dim=5)
280 sage: pi_of_K = positive_operator_gens(K)
281 sage: L = ToricLattice(K.lattice_dim()**2)
282 sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
283 sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
284 sage: K.contains(P*K.random_element(ring=QQ))
287 The lineality space of the dual of the cone of positive operators
288 can be computed from the lineality spaces of the cone and its dual::
290 sage: set_random_seed()
291 sage: K = random_cone(max_ambient_dim=5)
292 sage: pi_of_K = positive_operator_gens(K)
293 sage: L = ToricLattice(K.lattice_dim()**2)
294 sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
295 sage: actual = pi_cone.dual().linear_subspace()
296 sage: U1 = [ vector((s.tensor_product(x)).list())
297 ....: for x in K.lines()
298 ....: for s in K.dual() ]
299 sage: U2 = [ vector((s.tensor_product(x)).list())
301 ....: for s in K.dual().lines() ]
302 sage: expected = pi_cone.lattice().vector_space().span(U1 + U2)
303 sage: actual == expected
306 The lineality of the dual of the cone of positive operators
307 is known from its lineality space::
309 sage: set_random_seed()
310 sage: K = random_cone(max_ambient_dim=5)
311 sage: n = K.lattice_dim()
313 sage: l = K.lineality()
314 sage: pi_of_K = positive_operator_gens(K)
315 sage: L = ToricLattice(n**2)
316 sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
317 sage: actual = pi_cone.dual().lineality()
318 sage: expected = l*(m - l) + m*(n - m)
319 sage: actual == expected
322 The dimension of the cone of positive operators is given by the
323 corollary in my paper::
325 sage: set_random_seed()
326 sage: K = random_cone(max_ambient_dim=5)
327 sage: n = K.lattice_dim()
329 sage: l = K.lineality()
330 sage: pi_of_K = positive_operator_gens(K)
331 sage: L = ToricLattice(n**2)
332 sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
333 sage: expected = n**2 - l*(m - l) - (n - m)*m
334 sage: actual == expected
337 The lineality of the cone of positive operators is given by the
338 corollary in my paper::
340 sage: set_random_seed()
341 sage: K = random_cone(max_ambient_dim=5)
342 sage: n = K.lattice_dim()
343 sage: pi_of_K = positive_operator_gens(K)
344 sage: L = ToricLattice(n**2)
345 sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
346 sage: expected = n**2 - K.dim()*K.dual().dim()
347 sage: actual == expected
350 The cone ``K`` is proper if and only if the cone of positive
351 operators on ``K`` is proper::
353 sage: set_random_seed()
354 sage: K = random_cone(max_ambient_dim=5)
355 sage: pi_of_K = positive_operator_gens(K)
356 sage: L = ToricLattice(K.lattice_dim()**2)
357 sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
358 sage: K.is_proper() == pi_cone.is_proper()
361 # Matrices are not vectors in Sage, so we have to convert them
362 # to vectors explicitly before we can find a basis. We need these
363 # two values to construct the appropriate "long vector" space.
364 F
= K
.lattice().base_field()
367 tensor_products
= [ s
.tensor_product(x
) for x
in K
for s
in K
.dual() ]
369 # Convert those tensor products to long vectors.
370 W
= VectorSpace(F
, n
**2)
371 vectors
= [ W(tp
.list()) for tp
in tensor_products
]
373 # Create the *dual* cone of the positive operators, expressed as
375 pi_dual
= Cone(vectors
, ToricLattice(W
.dimension()))
377 # Now compute the desired cone from its dual...
378 pi_cone
= pi_dual
.dual()
380 # And finally convert its rays back to matrix representations.
381 M
= MatrixSpace(F
, n
)
382 return [ M(v
.list()) for v
in pi_cone
.rays() ]
385 def Z_transformation_gens(K
):
387 Compute generators of the cone of Z-transformations on this cone.
391 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
392 Each matrix ``L`` in the list should have the property that
393 ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
394 discrete complementarity set of ``K``. Moreover, any nonnegative
395 linear combination of these matrices shares the same property.
399 Z-transformations on the nonnegative orthant are just Z-matrices.
400 That is, matrices whose off-diagonal elements are nonnegative::
402 sage: K = Cone([(1,0),(0,1)])
403 sage: Z_transformation_gens(K)
405 [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
406 [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
408 sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
409 sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
410 ....: for i in range(z.nrows())
411 ....: for j in range(z.ncols())
415 The trivial cone in a trivial space has no Z-transformations::
417 sage: K = Cone([], ToricLattice(0))
418 sage: Z_transformation_gens(K)
421 Z-transformations on a subspace are Lyapunov-like and vice-versa::
423 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
424 sage: K.is_full_space()
426 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
427 sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
433 The Z-property is possessed by every Z-transformation::
435 sage: set_random_seed()
436 sage: K = random_cone(max_ambient_dim=6)
437 sage: Z_of_K = Z_transformation_gens(K)
438 sage: dcs = K.discrete_complementarity_set()
439 sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
440 ....: for (x,s) in dcs])
443 The lineality space of Z is LL::
445 sage: set_random_seed()
446 sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
447 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
448 sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
449 sage: z_cone.linear_subspace() == lls
452 And thus, the lineality of Z is the Lyapunov rank::
454 sage: set_random_seed()
455 sage: K = random_cone(max_ambient_dim=6)
456 sage: Z_of_K = Z_transformation_gens(K)
457 sage: L = ToricLattice(K.lattice_dim()**2)
458 sage: z_cone = Cone([ z.list() for z in Z_of_K ], lattice=L)
459 sage: z_cone.lineality() == K.lyapunov_rank()
462 The lineality spaces of pi-star and Z-star are equal:
464 sage: set_random_seed()
465 sage: K = random_cone(max_ambient_dim=5)
466 sage: pi_of_K = positive_operator_gens(K)
467 sage: Z_of_K = Z_transformation_gens(K)
468 sage: L = ToricLattice(K.lattice_dim()**2)
469 sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
470 sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
471 sage: pi_star.linear_subspace() == z_star.linear_subspace()
474 # Matrices are not vectors in Sage, so we have to convert them
475 # to vectors explicitly before we can find a basis. We need these
476 # two values to construct the appropriate "long vector" space.
477 F
= K
.lattice().base_field()
480 # These tensor products contain generators for the dual cone of
481 # the cross-positive transformations.
482 tensor_products
= [ s
.tensor_product(x
)
483 for (x
,s
) in K
.discrete_complementarity_set() ]
485 # Turn our matrices into long vectors...
486 W
= VectorSpace(F
, n
**2)
487 vectors
= [ W(m
.list()) for m
in tensor_products
]
489 # Create the *dual* cone of the cross-positive operators,
490 # expressed as long vectors..
491 Sigma_dual
= Cone(vectors
, lattice
=ToricLattice(W
.dimension()))
493 # Now compute the desired cone from its dual...
494 Sigma_cone
= Sigma_dual
.dual()
496 # And finally convert its rays back to matrix representations.
497 # But first, make them negative, so we get Z-transformations and
498 # not cross-positive ones.
499 M
= MatrixSpace(F
, n
)
500 return [ -M(v
.list()) for v
in Sigma_cone
.rays() ]
504 gens
= Z_transformation_gens(K
)
508 return Cone([ g
.list() for g
in gens
], lattice
=L
)
511 gens
= positive_operator_gens(K
)
515 return Cone([ g
.list() for g
in gens
], lattice
=L
)