Coverage for picos/expressions/exp_sumxtr.py : 82.87%

Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
1# ------------------------------------------------------------------------------
2# Copyright (C) 2019 Maximilian Stahlberg
3# Based on the original picos.expressions module by Guillaume Sagnol.
4#
5# This file is part of PICOS.
6#
7# PICOS is free software: you can redistribute it and/or modify it under the
8# terms of the GNU General Public License as published by the Free Software
9# Foundation, either version 3 of the License, or (at your option) any later
10# version.
11#
12# PICOS is distributed in the hope that it will be useful, but WITHOUT ANY
13# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
14# A PARTICULAR PURPOSE. See the GNU General Public License for more details.
15#
16# You should have received a copy of the GNU General Public License along with
17# this program. If not, see <http://www.gnu.org/licenses/>.
18# ------------------------------------------------------------------------------
20"""Implements :class:`SumExtremes`."""
22import operator
23from collections import namedtuple
25import cvxopt
26import numpy
28from .. import glyphs
29from ..apidoc import api_end, api_start
30from ..constraints import Constraint, SumExtremesConstraint
31from .data import convert_and_refine_arguments, convert_operands, cvx2np
32from .exp_affine import AffineExpression, ComplexAffineExpression
33from .expression import Expression, refine_operands, validate_prediction
35_API_START = api_start(globals())
36# -------------------------------
39class SumExtremes(Expression):
40 r"""Sum of the :math:`k` largest or smallest elements or eigenvalues.
42 :Definition:
44 Let :math:`k \in \mathbb{Z}_{\geq 1}`.
46 1. If :math:`x` is an :math:`n`-dimensional real vector or matrix and
47 ``eigenvalues == False``, then this is the sum of the :math:`k \leq n`
48 largest or smallest scalar elements of :math:`x`, depending on the truth
49 value of ``largest``.
51 Special cases:
53 - If :math:`k = 1`, this is either the largest element
54 :math:`\max_{i = 1}^n \operatorname{vec}(x)_i` or the smallest
55 element :math:`\min_{i = 1}^n \operatorname{vec}(x)_i` of :math:`x`.
56 - If :math:`k = n`, this is the sum of all elements
57 :math:`\langle x, 1 \rangle` of :math:`x`.
59 2. If :math:`X` is an :math:`n \times n` hermitian matrix and
60 ``eigenvalues == True``, then this is the sum of the :math:`k \leq n`
61 largest or smallest eigenvalues of :math:`X`, depending on the truth
62 value of ``largest``. Recall that the eigenvalues of a hermitian matrix
63 are real.
65 Special cases:
67 - If :math:`k = 1`, this is either the largest eigenvalue
68 :math:`\lambda_{\max}(X)` or the smallest eigenvalue
69 :math:`\lambda_{\min}(X)` of :math:`X`.
70 - If :math:`k = n`, this equals the trace
71 :math:`\operatorname{tr}(X)`.
73 If the given :math:`k` exceeds the :math:`n` of either case, then :math:`k`
74 is silently clipped to :math:`n`.
75 """
77 # --------------------------------------------------------------------------
78 # Initialization and factory methods.
79 # --------------------------------------------------------------------------
81 @convert_and_refine_arguments("x")
82 def __init__(self, x, k, largest, eigenvalues=False):
83 """Construct a :class:`SumExtremes`.
85 :param x: The affine expression to take a sum over.
86 :type x: ~picos.expressions.ComplexAffineExpression
87 :param int k: Number of summands.
88 :param bool largest: Whether to sum over the largest (eigen)values as
89 opposed to the smallest.
90 :param bool eigenvalues: Whether to sum eigenvalues instead of elements.
91 """
92 largest = bool(largest)
93 eigenvalues = bool(eigenvalues)
95 lStr = "largest" if largest else "smallest"
96 eStr = "eigenvalues" if eigenvalues else "scalar elements"
97 what = "{} {}".format(lStr, eStr)
99 # Validate x.
100 if not isinstance(x, ComplexAffineExpression):
101 raise TypeError("Can only sum {} of an affine expression, not of "
102 "{}.".format(what, type(x).__name__))
104 # Further validate x.
105 if eigenvalues:
106 if not x.square:
107 raise TypeError("Cannot sum {} of {} as its shape of {} is not "
108 "square.".format(what, x.string, glyphs.shape(x.shape)))
109 elif not x.hermitian:
110 raise NotImplementedError(
111 "Summing the {0} of {1} is not supported as {1} is not "
112 "necessarily hermitian.".format(what, x.string))
113 else:
114 if not isinstance(x, AffineExpression):
115 raise TypeError("Can only sum {} of a real-valued expression "
116 "but {} is properly complex.".format(what, x.string))
118 # Validate k.
119 if int(k) != k:
120 raise ValueError(
121 "Conversion of k = {} to an integer is ambiguous.".format(k))
122 k = int(k)
123 if k < 1:
124 raise ValueError(
125 "Number of {} to sum must be positive.".format(what))
127 # Clip k to be at most n.
128 k = min(k, x.shape[0]) if eigenvalues else min(k, len(x))
130 # Find out if all (eigen)values are summed.
131 full = k == x.shape[0] if eigenvalues else k == len(x)
132 assert len(x) != 1 or full
134 self._x = x
135 self._k = k
136 self._largest = largest
137 self._eigenvalues = eigenvalues
138 self._full = full
140 s, lbd = x.string, glyphs.lambda_()
141 if full:
142 if eigenvalues:
143 typeStr = "Sum of Eigenvalues"
144 symbStr = symbStr = glyphs.trace(s)
145 else:
146 typeStr = "Sum of Elements"
147 symbStr = glyphs.sum(s)
148 elif k > 1:
149 if eigenvalues and largest:
150 typeStr = "Sum of Largest Eigenvalues"
151 symbStr = glyphs.make_function(
152 "sum_{}_largest_{}".format(k, lbd))(s)
153 elif eigenvalues and not largest:
154 typeStr = "Sum of Smallest Eigenvalues"
155 symbStr = glyphs.make_function(
156 "sum_{}_smallest_{}".format(k, lbd))(s)
157 elif not eigenvalues and largest:
158 typeStr = "Sum of Largest Elements"
159 symbStr = glyphs.make_function("sum_{}_largest".format(k))(s)
160 else:
161 typeStr = "Sum of Smallest Elements"
162 symbStr = glyphs.make_function("sum_{}_smallest".format(k))(s)
163 else:
164 if eigenvalues and largest:
165 typeStr = "Largest Eigenvalue"
166 symbStr = glyphs.make_function("{}_max".format(lbd))(s)
167 elif eigenvalues and not largest:
168 typeStr = "Smallest Eigenvalue"
169 symbStr = glyphs.make_function("{}_min".format(lbd))(s)
170 elif not eigenvalues and largest:
171 typeStr = "Largest Element"
172 symbStr = glyphs.max(s)
173 else:
174 typeStr = "Smallest Element"
175 symbStr = glyphs.min(s)
177 Expression.__init__(self, typeStr, symbStr)
179 # --------------------------------------------------------------------------
180 # Abstract method implementations and method overridings, except _predict.
181 # --------------------------------------------------------------------------
183 def _get_refined(self):
184 if self._x.constant:
185 return AffineExpression.from_constant(self.value, 1, self._symbStr)
186 elif self._full:
187 if len(self._x) == 1:
188 return self._x # Don't carry the string for an identity.
189 if self._eigenvalues:
190 return self._x.tr # Symbolic strings already match.
191 else:
192 return (1 | self._x).renamed(self._symbStr)
193 else:
194 return self
196 Subtype = namedtuple("Subtype",
197 ("argdim", "k", "largest", "eigenvalues", "complex"))
199 def _get_subtype(self):
200 return self.Subtype(len(self._x), self._k, self._largest,
201 self._eigenvalues, self._x.complex)
203 def _get_value(self):
204 value = self._x._get_value()
206 if self._eigenvalues:
207 value = sorted(numpy.linalg.eigvalsh(cvx2np(value)))
208 else:
209 value = sorted(value)
211 value = sum(value[-self._k:] if self._largest else value[:self._k])
212 value = cvxopt.matrix(value)
214 return value
216 def _get_mutables(self):
217 return self._x._get_mutables()
219 def _is_convex(self):
220 return self._largest or self._full
222 def _is_concave(self):
223 return not self._largest or self._full
225 def _replace_mutables(self, mapping):
226 return self.__class__(self._x._replace_mutables(mapping),
227 self._k, self._largest, self._eigenvalues)
229 def _freeze_mutables(self, freeze):
230 return self.__class__(self._x._freeze_mutables(freeze),
231 self._k, self._largest, self._eigenvalues)
233 # --------------------------------------------------------------------------
234 # Python special method implementations, except constraint-creating ones.
235 # --------------------------------------------------------------------------
237 @convert_operands(scalarRHS=True)
238 @refine_operands()
239 def __mul__(self, other):
240 if isinstance(other, AffineExpression):
241 if not other.constant:
242 raise NotImplementedError("You may only multiply a nonconstant "
243 "PICOS sum of extremes with a constant term.")
245 if other.value < 0:
246 raise NotImplementedError("You may only multiply a nonconstant "
247 "PICOS sum of extremes with a nonnegative term.")
249 product = SumExtremes(
250 other.value*self._x, self._k, self._largest, self._eigenvalues)
251 product._typeStr = "Scaled " + product._typeStr
252 product._symbStr = glyphs.clever_mul(self.string, other.string)
253 return product
254 else:
255 return NotImplemented
257 @convert_operands(scalarRHS=True)
258 @refine_operands()
259 def __rmul__(self, other):
260 if isinstance(other, AffineExpression):
261 product = self.__mul__(other)
262 # NOTE: __mul__ always creates a fresh expression.
263 product._symbStr = glyphs.clever_mul(other.string, self.string)
264 return product
265 else:
266 return NotImplemented
268 # --------------------------------------------------------------------------
269 # Methods and properties that return expressions.
270 # --------------------------------------------------------------------------
272 @property
273 def x(self):
274 """The expression under the sum."""
275 return self._x
277 # --------------------------------------------------------------------------
278 # Methods and properties that describe the expression.
279 # --------------------------------------------------------------------------
281 @property
282 def k(self):
283 """Number of (eigen)values to sum."""
284 return self._k
286 @property
287 def largest(self):
288 """Whether the sum concerns largest values as opposed to smallest."""
289 return self._largest
291 @property
292 def eigenvalues(self):
293 """Whether the sum concerns eigenvalues as opposed to elements."""
294 return self._eigenvalues
296 @property
297 def full(self):
298 """Whether the sum concerns *all* (eigen)values of the expression."""
299 return self._full
301 # --------------------------------------------------------------------------
302 # Constraint-creating operators, and _predict.
303 # --------------------------------------------------------------------------
305 @classmethod
306 def _predict(cls, subtype, relation, other):
307 assert isinstance(subtype, cls.Subtype)
309 n = subtype.argdim
310 k = subtype.k
311 e = subtype.eigenvalues
312 c = subtype.complex
314 kmax = int(n**0.5) if e else n
315 full = k == kmax
317 convex = subtype.largest or full
318 concave = not subtype.largest or full
320 if relation == operator.__le__:
321 if not convex:
322 return NotImplemented
324 if issubclass(other.clstype, AffineExpression) \
325 and other.subtype.dim == 1:
326 return SumExtremesConstraint.make_type(n, k, e, c)
327 elif relation == operator.__ge__:
328 if not concave:
329 return NotImplemented
331 if issubclass(other.clstype, AffineExpression) \
332 and other.subtype.dim == 1:
333 return SumExtremesConstraint.make_type(n, k, e, c)
335 return NotImplemented
337 @convert_operands(scalarRHS=True)
338 @validate_prediction
339 @refine_operands()
340 def __le__(self, other):
341 if not self.convex:
342 raise TypeError("Cannot upper-bound the nonconvex expression {}."
343 .format(self._symbStr))
345 if isinstance(other, AffineExpression):
346 return SumExtremesConstraint(self, Constraint.LE, other)
347 else:
348 return NotImplemented
350 @convert_operands(scalarRHS=True)
351 @validate_prediction
352 @refine_operands()
353 def __ge__(self, other):
354 if not self.concave:
355 raise TypeError("Cannot upper-bound the nonconcave expression {}."
356 .format(self._symbStr))
358 if isinstance(other, AffineExpression):
359 return SumExtremesConstraint(self, Constraint.GE, other)
360 else:
361 return NotImplemented
364# --------------------------------------
365__all__ = api_end(_API_START, globals())