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gitweb.michael.orlitzky.com - sage.d.git/blob - mjo/cone/cone.py
68edeb4a012b1bca7c4f9664970d2d89371b4926
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 random_element(K
):
70 Return a random element of ``K`` from its ambient vector space.
74 The cone ``K`` is specified in terms of its generators, so that
75 ``K`` is equal to the convex conic combination of those generators.
76 To choose a random element of ``K``, we assign random nonnegative
77 coefficients to each generator of ``K`` and construct a new vector
80 A vector, rather than a ray, is returned so that the element may
81 have non-integer coordinates. Thus the element may have an
82 arbitrarily small norm.
86 A random element of the trivial cone is zero::
88 sage: set_random_seed()
89 sage: K = Cone([], ToricLattice(0))
90 sage: random_element(K)
92 sage: K = Cone([(0,)])
93 sage: random_element(K)
95 sage: K = Cone([(0,0)])
96 sage: random_element(K)
98 sage: K = Cone([(0,0,0)])
99 sage: random_element(K)
102 A random element of the nonnegative orthant should have all
103 components nonnegative::
105 sage: set_random_seed()
106 sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
107 sage: all([ x >= 0 for x in random_element(K) ])
112 Any cone should contain a random element of itself::
114 sage: set_random_seed()
115 sage: K = random_cone(max_ambient_dim=8)
116 sage: K.contains(random_element(K))
119 A strictly convex cone contains no lines, and thus no negative
120 multiples of any of its elements besides zero::
122 sage: set_random_seed()
123 sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
124 sage: x = random_element(K)
125 sage: x.is_zero() or not K.contains(-x)
128 The sum of random elements of a cone lies in the cone::
130 sage: set_random_seed()
131 sage: K = random_cone(max_ambient_dim=8)
132 sage: K.contains(sum([random_element(K) for i in range(10)]))
136 V
= K
.lattice().vector_space()
137 scaled_gens
= [ V
.base_field().random_element().abs()*V(r
) for r
in K
]
139 # Make sure we return a vector. Without the coercion, we might
140 # return ``0`` when ``K`` has no rays.
141 return V(sum(scaled_gens
))
144 def pointed_decomposition(K
):
146 Every convex cone is the direct sum of a pointed cone and a linear
147 subspace. Return a pair ``(P,S)`` of cones such that ``P`` is
148 pointed, ``S`` is a subspace, and ``K`` is the direct sum of ``P``
153 An ordered pair ``(P,S)`` of closed convex polyhedral cones where
154 ``P`` is pointed, ``S`` is a subspace, and ``K`` is the direct sum
159 sage: set_random_seed()
160 sage: K = random_cone(max_ambient_dim=8)
161 sage: (P,S) = pointed_decomposition(K)
162 sage: x = random_element(K)
163 sage: P.contains(x) or S.contains(x)
165 sage: x.is_zero() or (P.contains(x) != S.contains(x))
168 linspace_gens
= [ copy(b
) for b
in K
.linear_subspace().basis() ]
169 linspace_gens
+= [ -b
for b
in linspace_gens
]
171 S
= Cone(linspace_gens
, K
.lattice())
173 # Since ``S`` is a subspace, its dual is its orthogonal complement
174 # (albeit in the wrong lattice).
175 S_perp
= Cone(S
.dual(), K
.lattice())
176 P
= K
.intersection(S_perp
)
180 def positive_operator_gens(K
):
182 Compute generators of the cone of positive operators on this cone.
186 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
187 Each matrix ``P`` in the list should have the property that ``P*x``
188 is an element of ``K`` whenever ``x`` is an element of
189 ``K``. Moreover, any nonnegative linear combination of these
190 matrices shares the same property.
194 The trivial cone in a trivial space has no positive operators::
196 sage: K = Cone([], ToricLattice(0))
197 sage: positive_operator_gens(K)
200 Positive operators on the nonnegative orthant are nonnegative matrices::
202 sage: K = Cone([(1,)])
203 sage: positive_operator_gens(K)
206 sage: K = Cone([(1,0),(0,1)])
207 sage: positive_operator_gens(K)
209 [1 0] [0 1] [0 0] [0 0]
210 [0 0], [0 0], [1 0], [0 1]
213 Every operator is positive on the ambient vector space::
215 sage: K = Cone([(1,),(-1,)])
216 sage: K.is_full_space()
218 sage: positive_operator_gens(K)
221 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
222 sage: K.is_full_space()
224 sage: positive_operator_gens(K)
226 [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
227 [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
232 A positive operator on a cone should send its generators into the cone::
234 sage: set_random_seed()
235 sage: K = random_cone(max_ambient_dim=5)
236 sage: pi_of_K = positive_operator_gens(K)
237 sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
240 The dimension of the cone of positive operators is given by the
241 corollary in my paper::
243 sage: set_random_seed()
244 sage: K = random_cone(max_ambient_dim=5)
245 sage: n = K.lattice_dim()
247 sage: l = K.lineality()
248 sage: pi_of_K = positive_operator_gens(K)
249 sage: L = ToricLattice(n**2)
250 sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
251 sage: expected = n**2 - l*(m - l) - (n - m)*m
252 sage: actual == expected
255 The lineality of the cone of positive operators is given by the
256 corollary in my paper::
258 sage: set_random_seed()
259 sage: K = random_cone(max_ambient_dim=5)
260 sage: n = K.lattice_dim()
261 sage: pi_of_K = positive_operator_gens(K)
262 sage: L = ToricLattice(n**2)
263 sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
264 sage: expected = n**2 - K.dim()*K.dual().dim()
265 sage: actual == expected
268 The cone ``K`` is proper if and only if the cone of positive
269 operators on ``K`` is proper::
271 sage: set_random_seed()
272 sage: K = random_cone(max_ambient_dim=5)
273 sage: pi_of_K = positive_operator_gens(K)
274 sage: L = ToricLattice(K.lattice_dim()**2)
275 sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
276 sage: K.is_proper() == pi_cone.is_proper()
279 # Matrices are not vectors in Sage, so we have to convert them
280 # to vectors explicitly before we can find a basis. We need these
281 # two values to construct the appropriate "long vector" space.
282 F
= K
.lattice().base_field()
285 tensor_products
= [ s
.tensor_product(x
) for x
in K
for s
in K
.dual() ]
287 # Convert those tensor products to long vectors.
288 W
= VectorSpace(F
, n
**2)
289 vectors
= [ W(tp
.list()) for tp
in tensor_products
]
291 # Create the *dual* cone of the positive operators, expressed as
293 pi_dual
= Cone(vectors
, ToricLattice(W
.dimension()))
295 # Now compute the desired cone from its dual...
296 pi_cone
= pi_dual
.dual()
298 # And finally convert its rays back to matrix representations.
299 M
= MatrixSpace(F
, n
)
300 return [ M(v
.list()) for v
in pi_cone
.rays() ]
303 def Z_transformation_gens(K
):
305 Compute generators of the cone of Z-transformations on this cone.
309 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
310 Each matrix ``L`` in the list should have the property that
311 ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
312 discrete complementarity set of ``K``. Moreover, any nonnegative
313 linear combination of these matrices shares the same property.
317 Z-transformations on the nonnegative orthant are just Z-matrices.
318 That is, matrices whose off-diagonal elements are nonnegative::
320 sage: K = Cone([(1,0),(0,1)])
321 sage: Z_transformation_gens(K)
323 [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
324 [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
326 sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
327 sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
328 ....: for i in range(z.nrows())
329 ....: for j in range(z.ncols())
333 The trivial cone in a trivial space has no Z-transformations::
335 sage: K = Cone([], ToricLattice(0))
336 sage: Z_transformation_gens(K)
339 Z-transformations on a subspace are Lyapunov-like and vice-versa::
341 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
342 sage: K.is_full_space()
344 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
345 sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
351 The Z-property is possessed by every Z-transformation::
353 sage: set_random_seed()
354 sage: K = random_cone(max_ambient_dim=6)
355 sage: Z_of_K = Z_transformation_gens(K)
356 sage: dcs = K.discrete_complementarity_set()
357 sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
358 ....: for (x,s) in dcs])
361 The lineality space of Z is LL::
363 sage: set_random_seed()
364 sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
365 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
366 sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
367 sage: z_cone.linear_subspace() == lls
370 And thus, the lineality of Z is the Lyapunov rank::
372 sage: set_random_seed()
373 sage: K = random_cone(max_ambient_dim=6)
374 sage: Z_of_K = Z_transformation_gens(K)
375 sage: L = ToricLattice(K.lattice_dim()**2)
376 sage: z_cone = Cone([ z.list() for z in Z_of_K ], lattice=L)
377 sage: z_cone.lineality() == K.lyapunov_rank()
380 The lineality spaces of pi-star and Z-star are equal:
382 sage: set_random_seed()
383 sage: K = random_cone(max_ambient_dim=5)
384 sage: pi_of_K = positive_operator_gens(K)
385 sage: Z_of_K = Z_transformation_gens(K)
386 sage: L = ToricLattice(K.lattice_dim()**2)
387 sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
388 sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
389 sage: pi_star.linear_subspace() == z_star.linear_subspace()
392 # Matrices are not vectors in Sage, so we have to convert them
393 # to vectors explicitly before we can find a basis. We need these
394 # two values to construct the appropriate "long vector" space.
395 F
= K
.lattice().base_field()
398 # These tensor products contain generators for the dual cone of
399 # the cross-positive transformations.
400 tensor_products
= [ s
.tensor_product(x
)
401 for (x
,s
) in K
.discrete_complementarity_set() ]
403 # Turn our matrices into long vectors...
404 W
= VectorSpace(F
, n
**2)
405 vectors
= [ W(m
.list()) for m
in tensor_products
]
407 # Create the *dual* cone of the cross-positive operators,
408 # expressed as long vectors..
409 Sigma_dual
= Cone(vectors
, lattice
=ToricLattice(W
.dimension()))
411 # Now compute the desired cone from its dual...
412 Sigma_cone
= Sigma_dual
.dual()
414 # And finally convert its rays back to matrix representations.
415 # But first, make them negative, so we get Z-transformations and
416 # not cross-positive ones.
417 M
= MatrixSpace(F
, n
)
418 return [ -M(v
.list()) for v
in Sigma_cone
.rays() ]
422 gens
= Z_transformation_gens(K
)
426 return Cone([ g
.list() for g
in gens
], lattice
=L
)
429 gens
= positive_operator_gens(K
)
433 return Cone([ g
.list() for g
in gens
], lattice
=L
)