]>
gitweb.michael.orlitzky.com - sage.d.git/blob - cone/cone.py
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()
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 linspace_gens
= [ copy(b
) for b
in K
.linear_subspace().basis() ]
156 linspace_gens
+= [ -b
for b
in linspace_gens
]
158 S
= Cone(linspace_gens
, K
.lattice())
160 # Since ``S`` is a subspace, its dual is its orthogonal complement
161 # (albeit in the wrong lattice).
162 S_perp
= Cone(S
.dual(), K
.lattice())
163 P
= K
.intersection(S_perp
)
167 def positive_operator_gens(K
):
169 Compute generators of the cone of positive operators on this cone.
173 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
174 Each matrix ``P`` in the list should have the property that ``P*x``
175 is an element of ``K`` whenever ``x`` is an element of
176 ``K``. Moreover, any nonnegative linear combination of these
177 matrices shares the same property.
181 The trivial cone in a trivial space has no positive operators::
183 sage: K = Cone([], ToricLattice(0))
184 sage: positive_operator_gens(K)
187 Positive operators on the nonnegative orthant are nonnegative matrices::
189 sage: K = Cone([(1,)])
190 sage: positive_operator_gens(K)
193 sage: K = Cone([(1,0),(0,1)])
194 sage: positive_operator_gens(K)
196 [1 0] [0 1] [0 0] [0 0]
197 [0 0], [0 0], [1 0], [0 1]
200 Every operator is positive on the ambient vector space::
202 sage: K = Cone([(1,),(-1,)])
203 sage: K.is_full_space()
205 sage: positive_operator_gens(K)
208 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
209 sage: K.is_full_space()
211 sage: positive_operator_gens(K)
213 [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
214 [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
219 A positive operator on a cone should send its generators into the cone::
221 sage: set_random_seed()
222 sage: K = random_cone(max_ambient_dim=5)
223 sage: pi_of_K = positive_operator_gens(K)
224 sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
227 The dimension of the cone of positive operators is given by the
228 corollary in my paper::
230 sage: set_random_seed()
231 sage: K = random_cone(max_ambient_dim=5)
232 sage: n = K.lattice_dim()
234 sage: l = K.lineality()
235 sage: pi_of_K = positive_operator_gens(K)
236 sage: L = ToricLattice(n**2)
237 sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
238 sage: expected = n**2 - l*(m - l) - (n - m)*m
239 sage: actual == expected
242 The lineality of the cone of positive operators is given by the
243 corollary in my paper::
245 sage: set_random_seed()
246 sage: K = random_cone(max_ambient_dim=5)
247 sage: n = K.lattice_dim()
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).lineality()
251 sage: expected = n**2 - K.dim()*K.dual().dim()
252 sage: actual == expected
255 The cone ``K`` is proper if and only if the cone of positive
256 operators on ``K`` is proper::
258 sage: set_random_seed()
259 sage: K = random_cone(max_ambient_dim=5)
260 sage: pi_of_K = positive_operator_gens(K)
261 sage: L = ToricLattice(K.lattice_dim()**2)
262 sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
263 sage: K.is_proper() == pi_cone.is_proper()
266 # Matrices are not vectors in Sage, so we have to convert them
267 # to vectors explicitly before we can find a basis. We need these
268 # two values to construct the appropriate "long vector" space.
269 F
= K
.lattice().base_field()
272 tensor_products
= [ s
.tensor_product(x
) for x
in K
for s
in K
.dual() ]
274 # Convert those tensor products to long vectors.
275 W
= VectorSpace(F
, n
**2)
276 vectors
= [ W(tp
.list()) for tp
in tensor_products
]
278 # Create the *dual* cone of the positive operators, expressed as
280 pi_dual
= Cone(vectors
, ToricLattice(W
.dimension()))
282 # Now compute the desired cone from its dual...
283 pi_cone
= pi_dual
.dual()
285 # And finally convert its rays back to matrix representations.
286 M
= MatrixSpace(F
, n
)
287 return [ M(v
.list()) for v
in pi_cone
.rays() ]
290 def Z_transformation_gens(K
):
292 Compute generators of the cone of Z-transformations on this cone.
296 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
297 Each matrix ``L`` in the list should have the property that
298 ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
299 discrete complementarity set of ``K``. Moreover, any nonnegative
300 linear combination of these matrices shares the same property.
304 Z-transformations on the nonnegative orthant are just Z-matrices.
305 That is, matrices whose off-diagonal elements are nonnegative::
307 sage: K = Cone([(1,0),(0,1)])
308 sage: Z_transformation_gens(K)
310 [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
311 [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
313 sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
314 sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
315 ....: for i in range(z.nrows())
316 ....: for j in range(z.ncols())
320 The trivial cone in a trivial space has no Z-transformations::
322 sage: K = Cone([], ToricLattice(0))
323 sage: Z_transformation_gens(K)
326 Z-transformations on a subspace are Lyapunov-like and vice-versa::
328 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
329 sage: K.is_full_space()
331 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
332 sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
338 The Z-property is possessed by every Z-transformation::
340 sage: set_random_seed()
341 sage: K = random_cone(max_ambient_dim=6)
342 sage: Z_of_K = Z_transformation_gens(K)
343 sage: dcs = K.discrete_complementarity_set()
344 sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
345 ....: for (x,s) in dcs])
348 The lineality space of Z is LL::
350 sage: set_random_seed()
351 sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
352 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
353 sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
354 sage: z_cone.linear_subspace() == lls
357 And thus, the lineality of Z is the Lyapunov rank::
359 sage: set_random_seed()
360 sage: K = random_cone(max_ambient_dim=6)
361 sage: Z_of_K = Z_transformation_gens(K)
362 sage: L = ToricLattice(K.lattice_dim()**2)
363 sage: z_cone = Cone([ z.list() for z in Z_of_K ], lattice=L)
364 sage: z_cone.lineality() == K.lyapunov_rank()
367 The lineality spaces of pi-star and Z-star are equal:
369 sage: set_random_seed()
370 sage: K = random_cone(max_ambient_dim=5)
371 sage: pi_of_K = positive_operator_gens(K)
372 sage: Z_of_K = Z_transformation_gens(K)
373 sage: L = ToricLattice(K.lattice_dim()**2)
374 sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
375 sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
376 sage: pi_star.linear_subspace() == z_star.linear_subspace()
379 # Matrices are not vectors in Sage, so we have to convert them
380 # to vectors explicitly before we can find a basis. We need these
381 # two values to construct the appropriate "long vector" space.
382 F
= K
.lattice().base_field()
385 # These tensor products contain generators for the dual cone of
386 # the cross-positive transformations.
387 tensor_products
= [ s
.tensor_product(x
)
388 for (x
,s
) in K
.discrete_complementarity_set() ]
390 # Turn our matrices into long vectors...
391 W
= VectorSpace(F
, n
**2)
392 vectors
= [ W(m
.list()) for m
in tensor_products
]
394 # Create the *dual* cone of the cross-positive operators,
395 # expressed as long vectors..
396 Sigma_dual
= Cone(vectors
, lattice
=ToricLattice(W
.dimension()))
398 # Now compute the desired cone from its dual...
399 Sigma_cone
= Sigma_dual
.dual()
401 # And finally convert its rays back to matrix representations.
402 # But first, make them negative, so we get Z-transformations and
403 # not cross-positive ones.
404 M
= MatrixSpace(F
, n
)
405 return [ -M(v
.list()) for v
in Sigma_cone
.rays() ]
409 gens
= Z_transformation_gens(K
)
413 return Cone([ g
.list() for g
in gens
], lattice
=L
)
416 gens
= positive_operator_gens(K
)
420 return Cone([ g
.list() for g
in gens
], lattice
=L
)