]>
gitweb.michael.orlitzky.com - sage.d.git/blob - mjo/cone/cone.py
f3543a147ad8da3c5000015f3c53e837781180e5
1 # Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
2 # have to explicitly mangle our sitedir here so that "mjo.cone"
4 from os
.path
import abspath
5 from site
import addsitedir
6 addsitedir(abspath('../../'))
11 def _basically_the_same(K1
, K2
):
13 Test whether or not ``K1`` and ``K2`` are "basically the same."
15 This is a hack to get around the fact that it's difficult to tell
16 when two cones are linearly isomorphic. We have a proposition that
17 equates two cones, but represented over `\mathbb{Q}`, they are
18 merely linearly isomorphic (not equal). So rather than test for
19 equality, we test a list of properties that should be preserved
20 under an invertible linear transformation.
24 ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
29 Any proper cone with three generators in `\mathbb{R}^{3}` is
30 basically the same as the nonnegative orthant::
32 sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)])
33 sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)])
34 sage: _basically_the_same(K1, K2)
37 Negating a cone gives you another cone that is basically the same::
39 sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)])
40 sage: _basically_the_same(K, -K)
45 Any cone is basically the same as itself::
47 sage: K = random_cone(max_ambient_dim = 8)
48 sage: _basically_the_same(K, K)
51 After applying an invertible matrix to the rows of a cone, the
52 result should be basically the same as the cone we started with::
54 sage: K1 = random_cone(max_ambient_dim = 8)
55 sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
56 sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
57 sage: _basically_the_same(K1, K2)
61 if K1
.lattice_dim() != K2
.lattice_dim():
64 if K1
.nrays() != K2
.nrays():
67 if K1
.dim() != K2
.dim():
70 if K1
.lineality() != K2
.lineality():
73 if K1
.is_solid() != K2
.is_solid():
76 if K1
.is_strictly_convex() != K2
.is_strictly_convex():
79 if len(LL(K1
)) != len(LL(K2
)):
82 C_of_K1
= discrete_complementarity_set(K1
)
83 C_of_K2
= discrete_complementarity_set(K2
)
84 if len(C_of_K1
) != len(C_of_K2
):
87 if len(K1
.facets()) != len(K2
.facets()):
94 def _restrict_to_space(K
, W
):
96 Restrict this cone a subspace of its ambient space.
100 - ``W`` -- The subspace into which this cone will be restricted.
104 A new cone in a sublattice corresponding to ``W``.
108 When this cone is solid, restricting it into its own span should do
111 sage: K = Cone([(1,)])
112 sage: _restrict_to_space(K, K.span()) == K
115 A single ray restricted into its own span gives the same output
116 regardless of the ambient space::
118 sage: K2 = Cone([(1,0)])
119 sage: K2_S = _restrict_to_space(K2, K2.span()).rays()
123 sage: K3 = Cone([(1,0,0)])
124 sage: K3_S = _restrict_to_space(K3, K3.span()).rays()
133 The projected cone should always be solid::
135 sage: set_random_seed()
136 sage: K = random_cone(max_ambient_dim = 8)
137 sage: _restrict_to_space(K, K.span()).is_solid()
140 And the resulting cone should live in a space having the same
141 dimension as the space we restricted it to::
143 sage: set_random_seed()
144 sage: K = random_cone(max_ambient_dim = 8)
145 sage: K_P = _restrict_to_space(K, K.dual().span())
146 sage: K_P.lattice_dim() == K.dual().dim()
149 This function should not affect the dimension of a cone::
151 sage: set_random_seed()
152 sage: K = random_cone(max_ambient_dim = 8)
153 sage: K.dim() == _restrict_to_space(K,K.span()).dim()
156 Nor should it affect the lineality of a cone::
158 sage: set_random_seed()
159 sage: K = random_cone(max_ambient_dim = 8)
160 sage: K.lineality() == _restrict_to_space(K, K.span()).lineality()
163 No matter which space we restrict to, the lineality should not
166 sage: set_random_seed()
167 sage: K = random_cone(max_ambient_dim = 8)
168 sage: S = K.span(); P = K.dual().span()
169 sage: K.lineality() >= _restrict_to_space(K,S).lineality()
171 sage: K.lineality() >= _restrict_to_space(K,P).lineality()
174 If we do this according to our paper, then the result is proper::
176 sage: set_random_seed()
177 sage: K = random_cone(max_ambient_dim = 8)
178 sage: K_S = _restrict_to_space(K, K.span())
179 sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
180 sage: K_SP.is_proper()
182 sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
183 sage: K_SP.is_proper()
186 Test the proposition in our paper concerning the duals and
187 restrictions. Generate a random cone, then create a subcone of
188 it. The operation of dual-taking should then commute with
191 sage: set_random_seed()
192 sage: J = random_cone(max_ambient_dim = 8)
193 sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
194 sage: K_W_star = _restrict_to_space(K, J.span()).dual()
195 sage: K_star_W = _restrict_to_space(K.dual(), J.span())
196 sage: _basically_the_same(K_W_star, K_star_W)
200 # First we want to intersect ``K`` with ``W``. The easiest way to
201 # do this is via cone intersection, so we turn the subspace ``W``
203 W_cone
= Cone(W
.basis() + [-b
for b
in W
.basis()], lattice
=K
.lattice())
204 K
= K
.intersection(W_cone
)
206 # We've already intersected K with the span of K2, so every
207 # generator of K should belong to W now.
208 K_W_rays
= [ W
.coordinate_vector(r
) for r
in K
.rays() ]
210 L
= ToricLattice(W
.dimension())
211 return Cone(K_W_rays
, lattice
=L
)
215 def discrete_complementarity_set(K
):
217 Compute a discrete complementarity set of this cone.
219 A discrete complementarity set of `K` is the set of all orthogonal
220 pairs `(x,s)` such that `x \in G_{1}` and `s \in G_{2}` for some
221 generating sets `G_{1}` of `K` and `G_{2}` of its dual. Polyhedral
222 convex cones are input in terms of their generators, so "the" (this
223 particular) discrete complementarity set corresponds to ``G1
224 == K.rays()`` and ``G2 == K.dual().rays()``.
228 A list of pairs `(x,s)` such that,
230 * Both `x` and `s` are vectors (not rays).
231 * `x` is one of ``K.rays()``.
232 * `s` is one of ``K.dual().rays()``.
233 * `x` and `s` are orthogonal.
237 .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
238 Improper Cone. Work in-progress.
242 The discrete complementarity set of the nonnegative orthant consists
243 of pairs of standard basis vectors::
245 sage: K = Cone([(1,0),(0,1)])
246 sage: discrete_complementarity_set(K)
247 [((1, 0), (0, 1)), ((0, 1), (1, 0))]
249 If the cone consists of a single ray, the second components of the
250 discrete complementarity set should generate the orthogonal
251 complement of that ray::
253 sage: K = Cone([(1,0)])
254 sage: discrete_complementarity_set(K)
255 [((1, 0), (0, 1)), ((1, 0), (0, -1))]
256 sage: K = Cone([(1,0,0)])
257 sage: discrete_complementarity_set(K)
258 [((1, 0, 0), (0, 1, 0)),
259 ((1, 0, 0), (0, -1, 0)),
260 ((1, 0, 0), (0, 0, 1)),
261 ((1, 0, 0), (0, 0, -1))]
263 When the cone is the entire space, its dual is the trivial cone, so
264 the discrete complementarity set is empty::
266 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
267 sage: discrete_complementarity_set(K)
270 Likewise when this cone is trivial (its dual is the entire space)::
272 sage: L = ToricLattice(0)
273 sage: K = Cone([], ToricLattice(0))
274 sage: discrete_complementarity_set(K)
279 The complementarity set of the dual can be obtained by switching the
280 components of the complementarity set of the original cone::
282 sage: set_random_seed()
283 sage: K1 = random_cone(max_ambient_dim=6)
285 sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
286 sage: actual = discrete_complementarity_set(K1)
287 sage: sorted(actual) == sorted(expected)
290 The pairs in the discrete complementarity set are in fact
293 sage: set_random_seed()
294 sage: K = random_cone(max_ambient_dim=6)
295 sage: dcs = discrete_complementarity_set(K)
296 sage: sum([x.inner_product(s).abs() for (x,s) in dcs])
300 V
= K
.lattice().vector_space()
302 # Convert rays to vectors so that we can compute inner products.
303 xs
= [V(x
) for x
in K
.rays()]
305 # We also convert the generators of the dual cone so that we
306 # return pairs of vectors and not (vector, ray) pairs.
307 ss
= [V(s
) for s
in K
.dual().rays()]
309 return [(x
,s
) for x
in xs
for s
in ss
if x
.inner_product(s
) == 0]
314 Compute the space `\mathbf{LL}` of all Lyapunov-like transformations
319 A list of matrices forming a basis for the space of all
320 Lyapunov-like transformations on the given cone.
324 The trivial cone has no Lyapunov-like transformations::
326 sage: L = ToricLattice(0)
327 sage: K = Cone([], lattice=L)
331 The Lyapunov-like transformations on the nonnegative orthant are
332 simply diagonal matrices::
334 sage: K = Cone([(1,)])
338 sage: K = Cone([(1,0),(0,1)])
345 sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
348 [1 0 0] [0 0 0] [0 0 0]
349 [0 0 0] [0 1 0] [0 0 0]
350 [0 0 0], [0 0 0], [0 0 1]
353 Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
354 `L^{3}_{\infty}` cones [Rudolf et al.]_::
356 sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
364 sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
372 If our cone is the entire space, then every transformation on it is
375 sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
376 sage: M = MatrixSpace(QQ,2)
377 sage: M.basis() == LL(K)
382 The inner product `\left< L\left(x\right), s \right>` is zero for
383 every pair `\left( x,s \right)` in the discrete complementarity set
386 sage: set_random_seed()
387 sage: K = random_cone(max_ambient_dim=8)
388 sage: C_of_K = discrete_complementarity_set(K)
389 sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
390 sage: sum(map(abs, l))
393 The Lyapunov-like transformations on a cone and its dual are related
394 by transposition, but we're not guaranteed to compute transposed
395 elements of `LL\left( K \right)` as our basis for `LL\left( K^{*}
398 sage: set_random_seed()
399 sage: K = random_cone(max_ambient_dim=8)
400 sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
401 sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
402 sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
403 sage: LL2_vecs = [ V(m.list()) for m in LL2 ]
404 sage: V.span(LL1_vecs) == V.span(LL2_vecs)
408 V
= K
.lattice().vector_space()
410 C_of_K
= discrete_complementarity_set(K
)
412 tensor_products
= [ s
.tensor_product(x
) for (x
,s
) in C_of_K
]
414 # Sage doesn't think matrices are vectors, so we have to convert
415 # our matrices to vectors explicitly before we can figure out how
416 # many are linearly-indepenedent.
418 # The space W has the same base ring as V, but dimension
419 # dim(V)^2. So it has the same dimension as the space of linear
420 # transformations on V. In other words, it's just the right size
421 # to create an isomorphism between it and our matrices.
422 W
= VectorSpace(V
.base_ring(), V
.dimension()**2)
424 # Turn our matrices into long vectors...
425 vectors
= [ W(m
.list()) for m
in tensor_products
]
427 # Vector space representation of Lyapunov-like matrices
428 # (i.e. vec(L) where L is Luapunov-like).
429 LL_vector
= W
.span(vectors
).complement()
431 # Now construct an ambient MatrixSpace in which to stick our
433 M
= MatrixSpace(V
.base_ring(), V
.dimension())
435 matrix_basis
= [ M(v
.list()) for v
in LL_vector
.basis() ]
441 def lyapunov_rank(K
):
443 Compute the Lyapunov rank (or bilinearity rank) of this cone.
445 The Lyapunov rank of a cone can be thought of in (mainly) two ways:
447 1. The dimension of the Lie algebra of the automorphism group of the
450 2. The dimension of the linear space of all Lyapunov-like
451 transformations on the cone.
455 A closed, convex polyhedral cone.
459 An integer representing the Lyapunov rank of the cone. If the
460 dimension of the ambient vector space is `n`, then the Lyapunov rank
461 will be between `1` and `n` inclusive; however a rank of `n-1` is
462 not possible (see [Orlitzky/Gowda]_).
466 The codimension formula from the second reference is used. We find
467 all pairs `(x,s)` in the complementarity set of `K` such that `x`
468 and `s` are rays of our cone. It is known that these vectors are
469 sufficient to apply the codimension formula. Once we have all such
470 pairs, we "brute force" the codimension formula by finding all
471 linearly-independent `xs^{T}`.
475 .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of a proper
476 cone and Lyapunov-like transformations, Mathematical Programming, 147
479 .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
480 Improper Cone. Work in-progress.
482 .. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
483 optimality constraints for the cone of positive polynomials,
484 Mathematical Programming, Series B, 129 (2011) 5-31.
488 The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
491 sage: positives = Cone([(1,)])
492 sage: lyapunov_rank(positives)
494 sage: quadrant = Cone([(1,0), (0,1)])
495 sage: lyapunov_rank(quadrant)
497 sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
498 sage: lyapunov_rank(octant)
501 The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
504 sage: R5 = VectorSpace(QQ, 5)
505 sage: gs = R5.basis() + [ -r for r in R5.basis() ]
507 sage: lyapunov_rank(K)
510 The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
513 sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
514 sage: lyapunov_rank(L31)
517 Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_::
519 sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
520 sage: lyapunov_rank(L3infty)
523 A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
524 + 1` [Orlitzky/Gowda]_::
526 sage: K = Cone([(1,0,0,0,0)])
527 sage: lyapunov_rank(K)
529 sage: K.lattice_dim()**2 - K.lattice_dim() + 1
532 A subspace (of dimension `m`) in `n` dimensions should have a
533 Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_::
535 sage: e1 = (1,0,0,0,0)
536 sage: neg_e1 = (-1,0,0,0,0)
537 sage: e2 = (0,1,0,0,0)
538 sage: neg_e2 = (0,-1,0,0,0)
539 sage: z = (0,0,0,0,0)
540 sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
541 sage: lyapunov_rank(K)
543 sage: K.lattice_dim()**2 - K.dim()*K.codim()
546 The Lyapunov rank should be additive on a product of proper cones
549 sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
550 sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
551 sage: K = L31.cartesian_product(octant)
552 sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
555 Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_.
556 The cone ``K`` in the following example is isomorphic to the nonnegative
557 octant in `\mathbb{R}^{3}`::
559 sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
560 sage: lyapunov_rank(K)
563 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
564 itself [Rudolf et al.]_::
566 sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
567 sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
572 The Lyapunov rank should be additive on a product of proper cones
575 sage: set_random_seed()
576 sage: K1 = random_cone(max_ambient_dim=8,
577 ....: strictly_convex=True,
579 sage: K2 = random_cone(max_ambient_dim=8,
580 ....: strictly_convex=True,
582 sage: K = K1.cartesian_product(K2)
583 sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
586 The Lyapunov rank is invariant under a linear isomorphism
589 sage: K1 = random_cone(max_ambient_dim = 8)
590 sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
591 sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
592 sage: lyapunov_rank(K1) == lyapunov_rank(K2)
595 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
596 itself [Rudolf et al.]_::
598 sage: set_random_seed()
599 sage: K = random_cone(max_ambient_dim=8)
600 sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
603 The Lyapunov rank of a proper polyhedral cone in `n` dimensions can
604 be any number between `1` and `n` inclusive, excluding `n-1`
605 [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the
606 trivial cone in a trivial space as well. However, in zero dimensions,
607 the Lyapunov rank of the trivial cone will be zero::
609 sage: set_random_seed()
610 sage: K = random_cone(max_ambient_dim=8,
611 ....: strictly_convex=True,
613 sage: b = lyapunov_rank(K)
614 sage: n = K.lattice_dim()
615 sage: (n == 0 or 1 <= b) and b <= n
620 In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
621 Lyapunov rank `n-1` in `n` dimensions::
623 sage: set_random_seed()
624 sage: K = random_cone(max_ambient_dim=8)
625 sage: b = lyapunov_rank(K)
626 sage: n = K.lattice_dim()
630 The calculation of the Lyapunov rank of an improper cone can be
631 reduced to that of a proper cone [Orlitzky/Gowda]_::
633 sage: set_random_seed()
634 sage: K = random_cone(max_ambient_dim=8)
635 sage: actual = lyapunov_rank(K)
636 sage: K_S = _restrict_to_space(K, K.span())
637 sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
638 sage: l = K.lineality()
640 sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2
641 sage: actual == expected
644 The Lyapunov rank of any cone is just the dimension of ``LL(K)``::
646 sage: set_random_seed()
647 sage: K = random_cone(max_ambient_dim=8)
648 sage: lyapunov_rank(K) == len(LL(K))
651 We can make an imperfect cone perfect by adding a slack variable
652 (a Theorem in [Orlitzky/Gowda]_)::
654 sage: set_random_seed()
655 sage: K = random_cone(max_ambient_dim=8,
656 ....: strictly_convex=True,
658 sage: L = ToricLattice(K.lattice_dim() + 1)
659 sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L)
660 sage: lyapunov_rank(K) >= K.lattice_dim()
671 # K is not solid, restrict to its span.
672 K
= _restrict_to_space(K
, K
.span())
674 # Non-solid reduction lemma.
678 # K is not pointed, restrict to the span of its dual. Uses a
679 # proposition from our paper, i.e. this is equivalent to K =
680 # _rho(K.dual()).dual().
681 K
= _restrict_to_space(K
, K
.dual().span())
683 # Non-pointed reduction lemma.
691 def is_lyapunov_like(L
,K
):
693 Determine whether or not ``L`` is Lyapunov-like on ``K``.
695 We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
696 L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
697 `\left\langle x,s \right\rangle` in the complementarity set of
698 ``K``. It is known [Orlitzky]_ that this property need only be
699 checked for generators of ``K`` and its dual.
703 - ``L`` -- A linear transformation or matrix.
705 - ``K`` -- A polyhedral closed convex cone.
709 ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
710 and ``False`` otherwise.
714 If this function returns ``True``, then ``L`` is Lyapunov-like
715 on ``K``. However, if ``False`` is returned, that could mean one
716 of two things. The first is that ``L`` is definitely not
717 Lyapunov-like on ``K``. The second is more of an "I don't know"
718 answer, returned (for example) if we cannot prove that an inner
723 .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an
724 improper cone (preprint).
735 return all([(L
*x
).inner_product(s
) == 0
736 for (x
,s
) in discrete_complementarity_set(K
)])