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Play around with positive operators and Z-transformations. Add a new test.
[sage.d.git] / mjo / cone / cone.py
1 from sage.all import *
2
3 def is_lyapunov_like(L,K):
4 r"""
5 Determine whether or not ``L`` is Lyapunov-like on ``K``.
6
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.
12
13 INPUT:
14
15 - ``L`` -- A linear transformation or matrix.
16
17 - ``K`` -- A polyhedral closed convex cone.
18
19 OUTPUT:
20
21 ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
22 and ``False`` otherwise.
23
24 .. WARNING::
25
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
31 product is zero.
32
33 REFERENCES:
34
35 M. Orlitzky. The Lyapunov rank of an improper cone.
36 http://www.optimization-online.org/DB_HTML/2015/10/5135.html
37
38 EXAMPLES:
39
40 The identity is always Lyapunov-like in a nontrivial space::
41
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)
46 True
47
48 As is the "zero" transformation::
49
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)
54 True
55
56 Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
57 on ``K``::
58
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() ])
61 True
62
63 """
64 return all([(L*x).inner_product(s) == 0
65 for (x,s) in K.discrete_complementarity_set()])
66
67
68 def random_element(K):
69 r"""
70 Return a random element of ``K`` from its ambient vector space.
71
72 ALGORITHM:
73
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
78 from the scaled rays.
79
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.
83
84 EXAMPLES:
85
86 A random element of the trivial cone is zero::
87
88 sage: set_random_seed()
89 sage: K = Cone([], ToricLattice(0))
90 sage: random_element(K)
91 ()
92 sage: K = Cone([(0,)])
93 sage: random_element(K)
94 (0)
95 sage: K = Cone([(0,0)])
96 sage: random_element(K)
97 (0, 0)
98 sage: K = Cone([(0,0,0)])
99 sage: random_element(K)
100 (0, 0, 0)
101
102 TESTS:
103
104 Any cone should contain an element of itself::
105
106 sage: set_random_seed()
107 sage: K = random_cone(max_rays = 8)
108 sage: K.contains(random_element(K))
109 True
110
111 """
112 V = K.lattice().vector_space()
113 F = V.base_ring()
114 coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
115 vector_gens = map(V, K.rays())
116 scaled_gens = [ coefficients[i]*vector_gens[i]
117 for i in range(len(vector_gens)) ]
118
119 # Make sure we return a vector. Without the coercion, we might
120 # return ``0`` when ``K`` has no rays.
121 v = V(sum(scaled_gens))
122 return v
123
124
125 def positive_operator_gens(K):
126 r"""
127 Compute generators of the cone of positive operators on this cone.
128
129 OUTPUT:
130
131 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
132 Each matrix ``P`` in the list should have the property that ``P*x``
133 is an element of ``K`` whenever ``x`` is an element of
134 ``K``. Moreover, any nonnegative linear combination of these
135 matrices shares the same property.
136
137 EXAMPLES:
138
139 The trivial cone in a trivial space has no positive operators::
140
141 sage: K = Cone([], ToricLattice(0))
142 sage: positive_operator_gens(K)
143 []
144
145 Positive operators on the nonnegative orthant are nonnegative matrices::
146
147 sage: K = Cone([(1,)])
148 sage: positive_operator_gens(K)
149 [[1]]
150
151 sage: K = Cone([(1,0),(0,1)])
152 sage: positive_operator_gens(K)
153 [
154 [1 0] [0 1] [0 0] [0 0]
155 [0 0], [0 0], [1 0], [0 1]
156 ]
157
158 Every operator is positive on the ambient vector space::
159
160 sage: K = Cone([(1,),(-1,)])
161 sage: K.is_full_space()
162 True
163 sage: positive_operator_gens(K)
164 [[1], [-1]]
165
166 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
167 sage: K.is_full_space()
168 True
169 sage: positive_operator_gens(K)
170 [
171 [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
172 [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
173 ]
174
175 TESTS:
176
177 A positive operator on a cone should send its generators into the cone::
178
179 sage: K = random_cone(max_ambient_dim = 6)
180 sage: pi_of_K = positive_operator_gens(K)
181 sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
182 True
183
184 The dimension of the cone of positive operators is given by the
185 corollary in my paper::
186
187 sage: K = random_cone(max_ambient_dim = 6)
188 sage: n = K.lattice_dim()
189 sage: m = K.dim()
190 sage: l = K.lineality()
191 sage: pi_of_K = positive_operator_gens(K)
192 sage: actual = Cone([p.list() for p in pi_of_K]).dim()
193 sage: expected = n**2 - l*(n - l) - (n - m)*m
194 sage: actual == expected
195 True
196
197 """
198 # Matrices are not vectors in Sage, so we have to convert them
199 # to vectors explicitly before we can find a basis. We need these
200 # two values to construct the appropriate "long vector" space.
201 F = K.lattice().base_field()
202 n = K.lattice_dim()
203
204 tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
205
206 # Convert those tensor products to long vectors.
207 W = VectorSpace(F, n**2)
208 vectors = [ W(tp.list()) for tp in tensor_products ]
209
210 # Create the *dual* cone of the positive operators, expressed as
211 # long vectors..
212 pi_dual = Cone(vectors, ToricLattice(W.dimension()))
213
214 # Now compute the desired cone from its dual...
215 pi_cone = pi_dual.dual()
216
217 # And finally convert its rays back to matrix representations.
218 M = MatrixSpace(F, n)
219 return [ M(v.list()) for v in pi_cone.rays() ]
220
221
222 def Z_transformation_gens(K):
223 r"""
224 Compute generators of the cone of Z-transformations on this cone.
225
226 OUTPUT:
227
228 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
229 Each matrix ``L`` in the list should have the property that
230 ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
231 discrete complementarity set of ``K``. Moreover, any nonnegative
232 linear combination of these matrices shares the same property.
233
234 EXAMPLES:
235
236 Z-transformations on the nonnegative orthant are just Z-matrices.
237 That is, matrices whose off-diagonal elements are nonnegative::
238
239 sage: K = Cone([(1,0),(0,1)])
240 sage: Z_transformation_gens(K)
241 [
242 [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
243 [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
244 ]
245 sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
246 sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
247 ....: for i in range(z.nrows())
248 ....: for j in range(z.ncols())
249 ....: if i != j ])
250 True
251
252 The trivial cone in a trivial space has no Z-transformations::
253
254 sage: K = Cone([], ToricLattice(0))
255 sage: Z_transformation_gens(K)
256 []
257
258 Z-transformations on a subspace are Lyapunov-like and vice-versa::
259
260 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
261 sage: K.is_full_space()
262 True
263 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
264 sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
265 sage: zs == lls
266 True
267
268 TESTS:
269
270 The Z-property is possessed by every Z-transformation::
271
272 sage: set_random_seed()
273 sage: K = random_cone(max_ambient_dim = 6)
274 sage: Z_of_K = Z_transformation_gens(K)
275 sage: dcs = K.discrete_complementarity_set()
276 sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
277 ....: for (x,s) in dcs])
278 True
279
280 The lineality space of Z is LL::
281
282 sage: set_random_seed()
283 sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
284 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
285 sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
286 sage: z_cone.linear_subspace() == lls
287 True
288
289 """
290 # Matrices are not vectors in Sage, so we have to convert them
291 # to vectors explicitly before we can find a basis. We need these
292 # two values to construct the appropriate "long vector" space.
293 F = K.lattice().base_field()
294 n = K.lattice_dim()
295
296 # These tensor products contain generators for the dual cone of
297 # the cross-positive transformations.
298 tensor_products = [ s.tensor_product(x)
299 for (x,s) in K.discrete_complementarity_set() ]
300
301 # Turn our matrices into long vectors...
302 W = VectorSpace(F, n**2)
303 vectors = [ W(m.list()) for m in tensor_products ]
304
305 # Create the *dual* cone of the cross-positive operators,
306 # expressed as long vectors..
307 Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
308
309 # Now compute the desired cone from its dual...
310 Sigma_cone = Sigma_dual.dual()
311
312 # And finally convert its rays back to matrix representations.
313 # But first, make them negative, so we get Z-transformations and
314 # not cross-positive ones.
315 M = MatrixSpace(F, n)
316 return [ -M(v.list()) for v in Sigma_cone.rays() ]