Coverage for picos/expressions/expression.py: 75.61%
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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"""Backend for expression type implementations."""
22import functools
23import operator
24import threading
25import warnings
26from abc import abstractmethod
27from contextlib import contextmanager
29from .. import glyphs
30from ..apidoc import api_end, api_start
31from ..caching import cached_property
32from ..constraints import ConstraintType
33from ..containers import DetailedType
34from ..legacy import deprecated
35from ..valuable import NotValued, Valuable
36from .data import convert_operands
38_API_START = api_start(globals())
39# -------------------------------
42def validate_prediction(the_operator):
43 """Validate that the constraint outcome matches the predicted outcome."""
44 @functools.wraps(the_operator)
45 def wrapper(lhs, rhs, *args, **kwargs):
46 from .set import Set
48 def what():
49 return "({}).{}({})".format(
50 lhs._symbStr, the_operator.__name__, rhs._symbStr)
52 assert isinstance(lhs, (Expression, Set)) \
53 and isinstance(rhs, (Expression, Set)), \
54 "validate_prediction must occur below convert_operands."
56 lhs_type = lhs.type
57 rhs_type = rhs.type
59 try:
60 abstract_operator = getattr(operator, the_operator.__name__)
61 except AttributeError as error:
62 raise AssertionError("validate_prediction may only decorate "
63 "standard operator implementations.") from error
65 try:
66 predictedType = lhs_type.predict(abstract_operator, rhs_type)
67 except NotImplementedError:
68 predictedType = None # No prediction was made.
69 except PredictedFailure:
70 predictedType = NotImplemented # Prediction is "not possible".
72 try:
73 outcome = the_operator(lhs, rhs, *args, **kwargs)
74 except Exception as error:
75 # Case where the prediction is positive and the outcome is negative.
76 if predictedType not in (None, NotImplemented):
77 warnings.warn(
78 "Outcome for {} was predicted {} but the operation raised "
79 "an error: \"{}\" This a noncritical error (false positive)"
80 " in PICOS' constraint outcome prediction."
81 .format(what(), predictedType, error),
82 category=RuntimeWarning, stacklevel=3)
83 raise
84 else:
85 raise
87 # Case where the prediction is negative and the outcome is positive.
88 if predictedType is NotImplemented and outcome is not NotImplemented:
89 raise AssertionError(
90 "The operation {} was predicted to fail but it produced "
91 "an output of {}.".format(what(), outcome.type))
93 # Case where no prediction was made.
94 if not predictedType:
95 return outcome
97 # Case where the outcome is try-to-reverse-the-operation.
98 if outcome is NotImplemented:
99 return outcome
101 # Case where the prediction and the outcome are positive but differ.
102 outcomeType = outcome.type
103 if not predictedType.equals(outcomeType):
104 raise AssertionError("Outcome for {} was predicted {} but is {}."
105 .format(what(), predictedType, outcomeType))
107 return outcome
108 return wrapper
111def refine_operands(stop_at_affine=False):
112 """Cast :meth:`~Expression.refined` on both operands.
114 If the left hand side operand (i.e. ``self``) is refined to an instance of a
115 different type, then, instead of the decorated method, the method with the
116 same name on the refined type is invoked with the (refined) right hand side
117 operand as its argument.
119 This decorator is supposed to be used on all constraint creating binary
120 operator methods so that degenerated instances (e.g. a complex affine
121 expression with an imaginary part of zero) can occur but are not used in
122 constraints. This speeds up many computations involving expressions as these
123 degenerate cases do not need to be detected. Note that
124 :attr:`Expression.type` also refers to the refined version of an expression.
126 :param bool stop_at_affine: Do not refine any affine expressions, in
127 particular do not refine complex affine expressions to real ones.
128 """
129 def decorator(the_operator):
130 @functools.wraps(the_operator)
131 def wrapper(lhs, rhs, *args, **kwargs):
132 from .exp_affine import ComplexAffineExpression
133 from .set import Set
135 assert isinstance(lhs, (Expression, Set)) \
136 and isinstance(rhs, (Expression, Set)), \
137 "refine_operands must occur below convert_operands."
139 if stop_at_affine and isinstance(lhs, ComplexAffineExpression):
140 lhs_refined = lhs
141 else:
142 lhs_refined = lhs.refined
144 if type(lhs_refined) is not type(lhs):
145 assert hasattr(lhs_refined, the_operator.__name__), \
146 "refine_operand transformed 'self' to another type that " \
147 "does not define an operator with the same name as the " \
148 "decorated one."
150 refined_operation = getattr(lhs_refined, the_operator.__name__)
152 return refined_operation(rhs, *args, **kwargs)
154 if stop_at_affine and isinstance(rhs, ComplexAffineExpression):
155 rhs_refined = rhs
156 else:
157 rhs_refined = rhs.refined
159 return the_operator(lhs_refined, rhs_refined, *args, **kwargs)
160 return wrapper
161 return decorator
164# TODO: Once PICOS requires Python >= 3.7, use a ContextVar instead.
165class _Refinement(threading.local):
166 allowed = True
169_REFINEMENT = _Refinement()
172@contextmanager
173def no_refinement():
174 """Context manager that disables the effect of :meth:`Expression.refined`.
176 This can be necessary to ensure that the outcome of a constraint coversion
177 is as predicted, in particular when PICOS uses overridden comparison
178 operators for constraint creation internally.
179 """
180 _REFINEMENT.allowed = False
182 try:
183 yield
184 finally:
185 _REFINEMENT.allowed = True
188class PredictedFailure(TypeError):
189 """Denotes that comparing two expressions will not form a constraint."""
191 pass
194class ExpressionType(DetailedType):
195 """The detailed type of an expression for predicting constraint outcomes.
197 This is suffcient to predict the detailed type of any constraint that can be
198 created by comparing with another expression.
199 """
201 @staticmethod
202 def _relation_str(relation):
203 if relation is operator.__eq__:
204 return "=="
205 elif relation is operator.__le__:
206 return "<="
207 elif relation is operator.__ge__:
208 return ">="
209 elif relation is operator.__lshift__:
210 return "<<"
211 elif relation is operator.__rshift__:
212 return ">>"
213 else:
214 return "??"
216 @staticmethod
217 def _swap_relation(relation):
218 if relation is operator.__eq__:
219 return operator.__eq__
220 elif relation is operator.__le__:
221 return operator.__ge__
222 elif relation is operator.__ge__:
223 return operator.__le__
224 elif relation is operator.__lshift__:
225 return operator.__rshift__
226 elif relation is operator.__rshift__:
227 return operator.__lshift__
228 else:
229 return None
231 def predict(self, relation, other):
232 """Predict the constraint outcome of comparing expressions.
234 :param relation:
235 An object from the :mod:`operator` namespace representing the
236 operation being predicted.
238 :param other:
239 Another expression type representing the right hand side operand.
240 :type other:
241 ~picos.expressions.expression.ExpressionType
243 :Example:
245 >>> import operator, picos
246 >>> a = picos.RealVariable("x") + 1
247 >>> b = picos.RealVariable("y") + 2
248 >>> (a <= b).type == a.type.predict(operator.__le__, b.type)
249 True
250 """
251 if not isinstance(other, ExpressionType):
252 raise TypeError("The 'other' argument must be another {} instance."
253 .format(self.__class__.__name__))
255 # Perform the forward prediction.
256 result = self.clstype._predict(self.subtype, relation, other)
258 # Fall back to the backward prediction.
259 if result is NotImplemented:
260 reverse = self._swap_relation(relation)
261 result = other.clstype._predict(other.subtype, reverse, self)
263 # If both fail, the prediction is "not possible".
264 if result is NotImplemented:
265 raise PredictedFailure(
266 "The statement {} {} {} is predicted to error."
267 .format(self, self._relation_str(relation), other))
268 else:
269 assert isinstance(result, ConstraintType)
270 return result
273class Expression(Valuable):
274 """Abstract base class for mathematical expressions, including mutables.
276 For mutables, this is the secondary base class, with
277 :class:`~.mutable.Mutable` or a subclass thereof being the primary one.
278 """
280 def __init__(self, typeStr, symbStr):
281 """Perform basic initialization for :class:`Expression` instances.
283 :param str typeStr: Short string denoting the expression type.
284 :param str symbStr: Algebraic string description of the expression.
285 """
286 self._typeStr = typeStr
287 """A string describing the expression type."""
289 self._symbStr = symbStr
290 """A symbolic string representation of the expression. It is always used
291 by __descr__, and it is equivalent to the value returned by __str__ when
292 the expression is not fully valued."""
294 @property
295 def string(self):
296 """Symbolic string representation of the expression.
298 Use this over Python's :class:`str` if you want to output the symbolic
299 representation even when the expression is valued.
300 """
301 return self._symbStr
303 # --------------------------------------------------------------------------
304 # Abstract method implementations for the Valuable base class.
305 # NOTE: _get_value and possibly _set_value are implemented by subclasses.
306 # --------------------------------------------------------------------------
308 def _get_valuable_string(self):
309 return "expression {}".format(self.string)
311 # --------------------------------------------------------------------------
312 # Abstract and default-implementation methods.
313 # --------------------------------------------------------------------------
315 def _get_refined(self):
316 """See :attr:`refined`."""
317 return self
319 def _get_clstype(self):
320 """Return the Python class part of the expression's detailed type."""
321 return self.__class__
323 @property
324 @abstractmethod
325 def Subtype(self):
326 """The class of which :attr:`subtype` returns an instance.
328 Instances must be hashable. By convention a
329 :func:`namedtuple <collections.namedtuple>` class.
331 .. warning::
332 This should be declared in the class body as e.g.
333 `Subtype = namedtuple(…)` and not as a property so that it's static.
334 """
335 pass
337 @abstractmethod
338 def _get_subtype(self):
339 """See :attr:`subtype`."""
340 pass
342 @classmethod
343 @abstractmethod
344 def _predict(cls, subtype, relation, other):
345 """Predict the constraint outcome of a comparison.
347 :param object subtype: An object returned by the :meth:`_get_subtype`
348 instance method of :class:`cls`.
349 :param method-wrapper relation: A function from the :mod:`operator`
350 namespace, such as :func:`operator.__le__`. See
351 :class:`ExpressionType` for what operators are defined.
352 :param ExpressionType other: The detailed type of another expression.
353 :returns: Either the :obj:`NotImplemented` token or a
354 :class:`ConstraintType` object such that an instance of :class:`cls`
355 with the given subtype, when compared with another expression with
356 the given expression type, returns a constraint with that constraint
357 type.
358 """
359 pass
361 def _get_shape(self):
362 """Return the algebraic shape of the expression."""
363 return (1, 1)
365 @abstractmethod
366 def _get_mutables(self):
367 """Return the set of mutables that are involved in the expression."""
368 pass
370 @abstractmethod
371 def _is_convex(self):
372 """Whether the expression is convex in its :attr:`variables`.
374 Method implementations may assume that the expression is refined. Thus,
375 degenerate cases affected by refinement do not need to be considered.
377 For uncertain expressions, this assumes the perturbation as constant.
378 """
379 pass
381 @abstractmethod
382 def _is_concave(self):
383 """Whether the expression is concave in its :attr:`variables`.
385 Method implementations may assume that the expression is refined. Thus,
386 degenerate cases affected by refinement do not need to be considered.
388 For uncertain expressions, this assumes the perturbation as constant.
389 """
390 pass
392 @abstractmethod
393 def _replace_mutables(self, mapping):
394 """Return a copy of the expression concerning different mutables.
396 This is the fast internal-use counterpart to :meth:`replace_mutables`.
398 The returned expression should be of the same type as ``self`` (no
399 refinement) so that it can be substituted in composite expressions.
401 :param dict mapping:
402 A mutable replacement map. The caller must ensure the following
403 properties:
405 1. This must be a complete map from existing mutables to the same
406 mutable, another mutable, or a real-valued affine expression
407 (completeness).
408 2. The shape and vectorization format of each replacement must match
409 the existing mutable. Replacing with affine expressions is only
410 allowed when the existing mutable uses the trivial
411 :class:`~vectorizations.FullVectorization` (soudness).
412 3. Mutables that appear in a replacement may be the same as the
413 mutable being replaced but may otherwise not appear in the
414 expression (freshness).
415 4. Mutables may appear at most once anywhere in the image of the map
416 (uniqueness).
418 If any property is not fulfilled, the implementation does not need
419 to raise a proper exception but may fail arbitrarily.
420 """
421 pass
423 @abstractmethod
424 def _freeze_mutables(self, subset):
425 """Return a copy with some mutables frozen to their current value.
427 This is the fast internal-use counterpart to :meth:`frozen`.
429 The returned expression should be of the same type as ``self`` (no
430 refinement) so that it can be substituted in composite expressions.
432 :param dict subset:
433 An iterable of valued :class:`mutables <.mutable.Mutable>` that
434 should be frozen. May include mutables that are not present in the
435 expression, but may not include mutables without a value.
436 """
437 pass
439 # --------------------------------------------------------------------------
440 # An interface to the abstract and default-implementation methods above.
441 # --------------------------------------------------------------------------
443 @property
444 def refined(self):
445 """A refined version of the expression.
447 The refined expression can be an instance of a different
448 :class:`Expression` subclass than the original expression, if that type
449 is better suited for the mathematical object in question.
451 The refined expression is automatically used instead of the original one
452 whenever a constraint is created, and in some other places.
454 The idea behind refined expressions is that operations that produce new
455 expressions can be executed quickly without checking for exceptionnel
456 cases. For instance, the sum of two
457 :class:`~.exp_affine.ComplexAffineExpression` instances could have the
458 complex part eliminated so that storing the result as an
459 :class:`~.exp_affine.AffineExpression` would be prefered, but checking
460 for this case on every addition would be too slow. Refinement is used
461 sparingly to detect such cases at times where it makes the most sense.
463 Refinement may be disallowed within a context with the
464 :func:`no_refinement` context manager. In this case, this property
465 returns the expression as is.
466 """
467 if not _REFINEMENT.allowed:
468 return self
470 fine = self._get_refined()
472 if fine is not self:
473 # Recursively refine until the expression doesn't change further.
474 return fine.refined
475 else:
476 return fine
478 @property
479 def subtype(self):
480 """The subtype part of the expression's detailed type.
482 Returns a hashable object that, together with the Python class part of
483 the expression's type, is sufficient to predict the constraint outcome
484 (constraint class and subtype) of any comparison operation with any
485 other expression.
487 By convention the object returned is a
488 :func:`namedtuple <collections.namedtuple>` instance.
489 """
490 return self._get_subtype()
492 @property
493 def type(self):
494 """The expression's detailed type for constraint prediction.
496 The returned value is suffcient to predict the detailed type of any
497 constraint that can be created by comparing with another expression.
499 Since constraints are created from
500 :attr:`~.expression.Expression.refined` expressions only, the Python
501 class part of the detailed type may differ from the type of the
502 expression whose :attr:`type` is queried.
503 """
504 refined = self.refined
505 return ExpressionType(refined._get_clstype(), refined._get_subtype())
507 @classmethod
508 def make_type(cls, *args, **kwargs):
509 """Create a detailed expression type from subtype parameters."""
510 return ExpressionType(cls, cls.Subtype(*args, **kwargs))
512 shape = property(
513 lambda self: self._get_shape(),
514 doc=_get_shape.__doc__)
516 size = property(
517 lambda self: self._get_shape(),
518 doc="""The same as :attr:`shape`.""")
520 @property
521 def scalar(self):
522 """Whether the expression is scalar."""
523 return self._get_shape() == (1, 1)
525 @property
526 def square(self):
527 """Whether the expression is a square matrix."""
528 shape = self._get_shape()
529 return shape[0] == shape[1]
531 mutables = property(
532 lambda self: self._get_mutables(),
533 doc=_get_mutables.__doc__)
535 @property
536 def constant(self):
537 """Whether the expression involves no mutables."""
538 return not self._get_mutables()
540 @cached_property
541 def variables(self):
542 """The set of decision variables that are involved in the expression."""
543 from .variables import BaseVariable
545 return frozenset(mutable for mutable in self._get_mutables()
546 if isinstance(mutable, BaseVariable))
548 @cached_property
549 def parameters(self):
550 """The set of parameters that are involved in the expression."""
551 from .variables import BaseVariable
553 return frozenset(mutable for mutable in self._get_mutables()
554 if not isinstance(mutable, BaseVariable))
556 @property
557 def convex(self):
558 """Whether the expression is convex."""
559 return self.refined._is_convex()
561 @property
562 def concave(self):
563 """Whether the expression is concave."""
564 return self.refined._is_concave()
566 def replace_mutables(self, replacement):
567 """Return a copy of the expression concerning different mutables.
569 New mutables must have the same shape and vectorization format as the
570 mutables that they replace. This means in particular that
571 :class:`~.variables.RealVariable`, :class:`~.variables.IntegerVariable`
572 and :class:`~.variables.BinaryVariable` of same shape are
573 interchangeable.
575 If the mutables to be replaced do not appear in the expression, then
576 the expression is not copied but returned as is.
578 :param replacement:
579 Either a map from mutables or mutable names to new mutables or an
580 iterable of new mutables to replace existing mutables of same name
581 with. See the section on advanced usage for additional options.
582 :type replacement:
583 tuple or list or dict
585 :returns Expression:
586 The new expression, refined to a more suitable type if possible.
588 :Advanced replacement:
590 It is also possible to replace mutables with real affine expressions
591 concerning pairwise disjoint sets of fresh mutables. This works only on
592 real-valued mutables that have a trivial internal vectorization format
593 (i.e. :class:`~.vectorizations.FullVectorization`). The shape of the
594 replacing expression must match the variable's. Additional limitations
595 depending on the type of expression that the replacement is invoked on
596 are possible. The ``replacement`` argument must be a dictionary.
598 :Example:
600 >>> import picos
601 >>> x = picos.RealVariable("x"); x.value = 1
602 >>> y = picos.RealVariable("y"); y.value = 10
603 >>> z = picos.RealVariable("z"); z.value = 100
604 >>> c = picos.Constant("c", 1000)
605 >>> a = x + 2*y; a
606 <1×1 Real Linear Expression: x + 2·y>
607 >>> a.value
608 21.0
609 >>> b = a.replace_mutables({y: z}); b # Replace y with z.
610 <1×1 Real Linear Expression: x + 2·z>
611 >>> b.value
612 201.0
613 >>> d = a.replace_mutables({x: 2*x + z, y: c}); d # Advanced use.
614 <1×1 Real Affine Expression: 2·x + z + 2·c>
615 >>> d.value
616 2102.0
617 """
618 from .exp_biaffine import BiaffineExpression
619 from .mutable import Mutable
620 from .vectorizations import FullVectorization
622 # Change an iterable of mutables to a map from names to mutables.
623 if not isinstance(replacement, dict):
624 if not all(isinstance(new, Mutable) for new in replacement):
625 raise TypeError("If 'replacement' is a non-dictionary iterable,"
626 " then it may only contain mutables.")
628 new_replacement = {new.name: new for new in replacement}
630 if len(new_replacement) != len(replacement):
631 raise TypeError("If 'replacement' is a non-dictionary iterable,"
632 " then the mutables within must have unique names.")
634 replacement = new_replacement
636 # Change a map from names to a map from existing mutables.
637 # Names that reference non-existing mutables are dropped.
638 old_mtbs_by_name = {mtb.name: mtb for mtb in self.mutables}
639 replacing_by_name = False
640 new_replacement = {}
641 for old, new in replacement.items():
642 if isinstance(old, Mutable):
643 new_replacement[old] = new
644 elif not isinstance(old, str):
645 raise TypeError(
646 "Keys of 'replacement' must be mutables or names thereof.")
647 else:
648 replacing_by_name = True
649 if old in old_mtbs_by_name:
650 new_replacement[old_mtbs_by_name[old]] = new
651 replacement = new_replacement
653 # Check unique naming of existing mutables if it matters.
654 if replacing_by_name and len(old_mtbs_by_name) != len(self.mutables):
655 raise RuntimeError("Cannot replace mutables by name in {} as "
656 "its mutables are not uniquely named.".format(self.string))
658 # Remove non-existing sources and identities.
659 assert all(isinstance(old, Mutable) for old in replacement)
660 replacement = {old: new for old, new in replacement.items()
661 if old is not new and old in self.mutables}
663 # Do nothing if there is nothing to replace.
664 if not replacement:
665 return self
667 # Validate individual replacement requirements.
668 for old, new in replacement.items():
669 # Replacement must be a mutable or biaffine expression.
670 if not isinstance(new, BiaffineExpression):
671 raise TypeError("Can only replace mutables with other mutables "
672 "or affine expressions thereof.")
674 # Shapes must match.
675 if old.shape != new.shape:
676 raise TypeError(
677 "Cannot replace {} with {} in {}: Differing shape."
678 .format(old.name, new.name, self.string))
680 # Special requirements when replacing with mutables or expressions.
681 if isinstance(new, Mutable):
682 # Vectorization formats must match.
683 if type(old._vec) != type(new._vec): # noqa: E721
684 raise TypeError("Cannot replace {} with {} in {}: "
685 "Differing vectorization."
686 .format(old.name, new.name, self.string))
687 else:
688 # Replaced mutable must use a trivial vectorization.
689 if not isinstance(old._vec, FullVectorization):
690 raise TypeError("Can only replace mutables using a trivial "
691 "vectorization format with affine expressions.")
693 # Replacing expression must be real-valued and affine.
694 if new._bilinear_coefs or new.complex:
695 raise TypeError("Can only replace mutables with real-valued"
696 " affine expressions.")
698 old_mtbs_set = set(replacement)
699 new_mtbs_lst = [mtb # Excludes each mutable being replaced.
700 for old, new in replacement.items()
701 for mtb in new.mutables.difference((old,))]
702 new_mtbs_set = set(new_mtbs_lst)
704 # New mutables must be fresh.
705 # It is OK to replace a mutable with itself or an affine expression of
706 # itself and other fresh mutables, though.
707 if old_mtbs_set.intersection(new_mtbs_set):
708 raise ValueError("Can only replace mutables with fresh mutables "
709 "or affine expressions of all fresh mutables (the old mutable "
710 "may appear in the expression).")
712 # New mutables must be unique.
713 if len(new_mtbs_lst) != len(new_mtbs_set):
714 raise ValueError("Can only replace multiple mutables at once if "
715 "the replacing mutables (and/or the mutables in replacing "
716 "expressions) are all unique.")
718 # Turn the replacement map into a complete map.
719 mapping = {mtb: mtb for mtb in self.mutables}
720 mapping.update(replacement)
722 # Replace recursively and refine the result.
723 return self._replace_mutables(mapping).refined
725 def frozen(self, subset=None):
726 """The expression with valued mutables frozen to their current value.
728 If all mutables of the expression are valued (and in the subset unless
729 ``subset=None``), this is the same as the inversion operation ``~``.
731 If the mutables to be frozen do not appear in the expression, then the
732 expression is not copied but returned as is.
734 :param subset:
735 An iterable of valued :class:`mutables <.mutable.Mutable>` or names
736 thereof that should be frozen. If :obj:`None`, then all valued
737 mutables are frozen to their current value. May include mutables
738 that are not present in the expression, but may not include mutables
739 without a value.
741 :returns Expression:
742 The frozen expression, refined to a more suitable type if possible.
744 :Example:
746 >>> from picos import RealVariable
747 >>> x, y = RealVariable("x"), RealVariable("y")
748 >>> f = x + y; f
749 <1×1 Real Linear Expression: x + y>
750 >>> sorted(f.mutables, key=lambda mtb: mtb.name)
751 [<1×1 Real Variable: x>, <1×1 Real Variable: y>]
752 >>> x.value = 5
753 >>> g = f.frozen(); g # g is f with x frozen at its current value of 5.
754 <1×1 Real Affine Expression: [x] + y>
755 >>> sorted(g.mutables, key=lambda mtb: mtb.name)
756 [<1×1 Real Variable: y>]
757 >>> x.value, y.value = 10, 10
758 >>> f.value # x takes its new value in f.
759 20.0
760 >>> g.value # x remains frozen at [x] = 5 in g.
761 15.0
762 >>> # If an expression is frozen to a constant, this is reversable:
763 >>> f.frozen().equals(~f) and ~f.frozen() is f
764 True
765 """
766 from .mutable import Mutable
768 # Collect mutables to be frozen in the expression.
769 if subset is None:
770 freeze = set(mtb for mtb in self.mutables if mtb.valued)
771 else:
772 if not all(isinstance(mtb, (str, Mutable)) for mtb in subset):
773 raise TypeError("Some element of the subset of mutables to "
774 "freeze is neither a mutable nor a string.")
776 subset_mtbs = set(m for m in subset if isinstance(m, Mutable))
777 subset_name = set(n for n in subset if isinstance(n, str))
779 freeze = set()
780 if subset_mtbs:
781 freeze.update(m for m in subset_mtbs if m in self.mutables)
782 if subset_name:
783 freeze.update(m for m in self.mutables if m.name in subset_name)
785 if not all(mtb.valued for mtb in freeze):
786 raise NotValued(
787 "Not all mutables in the selected subset are valued.")
789 if not freeze:
790 return self
792 if freeze == self.mutables:
793 return ~self # Allow ~self.frozen() to return self.
795 return self._freeze_mutables(freeze).refined
797 @property
798 def certain(self):
799 """Always :obj:`True` for certain expression types.
801 This can be :obj:`False` for Expression types that inherit from
802 :class:`~.uexpression.UncertainExpression` (with priority).
803 """
804 return True
806 @property
807 def uncertain(self):
808 """Always :obj:`False` for certain expression types.
810 This can be :obj:`True` for Expression types that inherit from
811 :class:`~.uexpression.UncertainExpression` (with priority).
812 """
813 return False
815 # --------------------------------------------------------------------------
816 # Python special method implementations.
817 # --------------------------------------------------------------------------
819 def __len__(self):
820 """Report the number of entries of the (multidimensional) expression."""
821 return self.shape[0] * self.shape[1]
823 def __le__(self, other):
824 """Return a constraint that the expression is upper-bounded."""
825 # Try to refine self and see if the operation is then supported.
826 # This allows e.g. a <= 0 if a is a real-valued complex expression.
827 refined = self.refined
828 if type(refined) != type(self):
829 return refined.__le__(other)
831 return NotImplemented
833 def __ge__(self, other):
834 """Return a constraint that the expression is lower-bounded."""
835 # Try to refine self and see if the operation is then supported.
836 # This allows e.g. a >= 0 if a is a real-valued complex expression.
837 refined = self.refined
838 if type(refined) != type(self):
839 return refined.__ge__(other)
841 return NotImplemented
843 def __invert__(self):
844 """Convert between a valued expression and its value.
846 The value is returned as a constant affine expression whose conversion
847 returns the original expression.
848 """
849 if hasattr(self, "_origin"):
850 return self._origin
851 elif self.constant:
852 return self
854 from .exp_affine import Constant
856 A = Constant(
857 glyphs.frozen(self.string), self.safe_value_as_matrix, self.shape)
858 A._origin = self
859 return A
861 def __contains__(self, mutable):
862 """Report whether the expression concerns the given mutable."""
863 return mutable in self.mutables
865 def __eq__(self, exp):
866 """Return an equality constraint concerning the expression."""
867 raise NotImplementedError("PICOS supports equality comparison only "
868 "between affine expressions, as otherwise the problem would "
869 "become non-convex. Choose either <= or >= if possible.")
871 def __repr__(self):
872 """Return a bracketed string description of the expression.
874 The description contains both the mathematical type and a symbolic
875 description of the expression.
876 """
877 return str(glyphs.repr2(self._typeStr, self._symbStr))
879 def __str__(self):
880 """Return a dynamic string description of the expression.
882 The description is based on whether the expression is valued. If it is
883 valued, then a string representation of the value is returned.
884 Otherwise, the symbolic description of the expression is returned.
885 """
886 value = self.value
888 if value is None:
889 return str(self._symbStr)
890 else:
891 return str(value).strip()
893 def __format__(self, format_spec):
894 """Format either the value or the symbolic string of the expression.
896 If the expression is valued, then its value is formatted, otherwise its
897 symbolic string description.
898 """
899 value = self.value
901 if value is None:
902 return self._symbStr.__format__(format_spec)
903 else:
904 return value.__format__(format_spec)
906 # Since we define __eq__, __hash__ is not inherited. Do this manually.
907 __hash__ = object.__hash__
909 # --------------------------------------------------------------------------
910 # Fallback algebraic operations: Try again with converted RHS, refined LHS.
911 # NOTE: The forward operations call the backward operations manually
912 # (instead of returning NotImplemented) so that they can be performed
913 # on a converted operand, which is always a PICOS type. The backward
914 # operations then use WeightedSum as a last fallback where applicable.
915 # --------------------------------------------------------------------------
917 def _wsum_fallback(self, summands, weights, opstring):
918 """Try to represent the result as a weighted sum."""
919 from .exp_wsum import WeightedSum
921 # NOTE: WeightedSum with an opstring set will act as a final fallback
922 # and raise a proper exception if the result can't be represented.
923 # This is handled there and not here so that also operations on
924 # existing WeightedSum instances can produce such exceptions, as
925 # they cannot fallback to Expression like other operations do.
926 return WeightedSum(summands, weights, opstring)
928 def _scalar_mult_fallback(self, lhs, rhs):
929 """Try to express scalar by scalar multiplication as a weighted sum."""
930 assert isinstance(lhs, Expression) and isinstance(rhs, Expression)
932 opstring = "a product between {} and {}".format(repr(lhs), repr(rhs))
934 if lhs.scalar and lhs.constant:
935 return self._wsum_fallback((rhs,), lhs.safe_value, opstring)
936 elif rhs.scalar and rhs.constant:
937 return self._wsum_fallback((lhs,), rhs.safe_value, opstring)
938 else:
939 # NOTE: Constant scalars are also AffineExpression but otherwise
940 # raising the default Python TypeError (stating that the two
941 # types are fully operation-incompatible) makes sense here.
942 return NotImplemented
944 @convert_operands(sameShape=True)
945 def __add__(self, other):
946 """Denote addition with another expression on the right-hand side."""
947 if type(self.refined) != type(self):
948 return self.refined.__add__(other)
949 else:
950 return other.__radd__(self)
952 @convert_operands(sameShape=True)
953 def __radd__(self, other):
954 """Denote addition with another expression on the left-hand side."""
955 if type(self.refined) != type(self):
956 return self.refined.__radd__(other)
957 else:
958 opstring = "{} plus {}".format(repr(other), repr(self))
959 return self._wsum_fallback((other, self), (1, 1), opstring)
961 @convert_operands(sameShape=True)
962 def __sub__(self, other):
963 """Denote subtraction of another expression from the expression."""
964 if type(self.refined) != type(self):
965 return self.refined.__sub__(other)
966 else:
967 return other.__rsub__(self)
969 @convert_operands(sameShape=True)
970 def __rsub__(self, other):
971 """Denote subtraction of the expression from another expression."""
972 if type(self.refined) != type(self):
973 return self.refined.__rsub__(other)
974 else:
975 opstring = "{} minus {}".format(repr(other), repr(self))
976 return self._wsum_fallback((other, self), (1, -1), opstring)
978 @convert_operands(sameShape=True)
979 def __or__(self, other):
980 r"""Denote the scalar product with another expression on the right.
982 For (complex) vectors :math:`a` and :math:`b` this is the dot product
984 .. math::
985 (a \mid b)
986 &= \langle a, b \rangle \\
987 &= a \cdot b \\
988 &= b^H a.
990 For (complex) matrices :math:`A` and :math:`B` this is the Frobenius
991 inner product
993 .. math::
994 (A \mid B)
995 &= \langle A, B \rangle_F \\
996 &= A : B \\
997 &= \operatorname{tr}(B^H A) \\
998 &= \operatorname{vec}(B)^H \operatorname{vec}(\overline{A})
1000 .. note::
1001 Write ``(A|B)`` instead of ``A|B`` for the scalar product of ``A``
1002 and ``B`` to obtain correct operator binding within a larger
1003 expression context.
1004 """
1005 if type(self.refined) != type(self):
1006 return self.refined.__or__(other)
1007 else:
1008 return other.__ror__(self)
1010 @convert_operands(sameShape=True)
1011 def __ror__(self, other):
1012 """Denote the scalar product with another expression on the left.
1014 See :meth:`__or__` for details on this operation.
1015 """
1016 if type(self.refined) != type(self):
1017 return self.refined.__ror__(other)
1018 else:
1019 return self._scalar_mult_fallback(other, self)
1021 @convert_operands(rMatMul=True)
1022 def __mul__(self, other):
1023 """Denote multiplication with another expression on the right."""
1024 if type(self.refined) != type(self):
1025 return self.refined.__mul__(other)
1026 else:
1027 return other.__rmul__(self)
1029 @convert_operands(lMatMul=True)
1030 def __rmul__(self, other):
1031 """Denote multiplication with another expression on the left."""
1032 if type(self.refined) != type(self):
1033 return self.refined.__rmul__(other)
1034 else:
1035 return self._scalar_mult_fallback(other, self)
1037 @convert_operands(sameShape=True)
1038 def __xor__(self, other):
1039 """Denote the entrywise product with another expression on the right."""
1040 if type(self.refined) != type(self):
1041 return self.refined.__xor__(other)
1042 else:
1043 return other.__rxor__(self)
1045 @convert_operands(sameShape=True)
1046 def __rxor__(self, other):
1047 """Denote the entrywise product with another expression on the left."""
1048 if type(self.refined) != type(self):
1049 return self.refined.__rxor__(other)
1050 else:
1051 return self._scalar_mult_fallback(other, self)
1053 @convert_operands()
1054 def __matmul__(self, other):
1055 """Denote the Kronecker product with another expression on the right."""
1056 if type(self.refined) != type(self):
1057 return self.refined.__matmul__(other)
1058 else:
1059 return other.__rmatmul__(self)
1061 @convert_operands()
1062 def __rmatmul__(self, other):
1063 """Denote the Kronecker product with another expression on the left."""
1064 if type(self.refined) != type(self):
1065 return self.refined.__rmatmul__(other)
1066 else:
1067 return self._scalar_mult_fallback(other, self)
1069 @convert_operands(scalarRHS=True)
1070 def __truediv__(self, other):
1071 """Denote division by another, scalar expression."""
1072 if type(self.refined) != type(self):
1073 return self.refined.__truediv__(other)
1074 else:
1075 return other.__rtruediv__(self)
1077 @convert_operands(scalarLHS=True)
1078 def __rtruediv__(self, other):
1079 """Denote scalar division of another expression."""
1080 if type(self.refined) != type(self):
1081 return self.refined.__rtruediv__(other)
1082 else:
1083 if self.constant and not self.is0:
1084 try:
1085 return other.__mul__(1 / self.safe_value)
1086 except TypeError:
1087 assert False, "Multiplication of {} by a nonzero constant" \
1088 " has unexpectedly failed; it should have produced a " \
1089 "weighted sum.".format(repr(other))
1090 else:
1091 reason = "nonconstant" if not self.constant else "zero"
1092 raise TypeError("Cannot divide {} by {}: The denominator is {}."
1093 .format(repr(other), repr(self), reason))
1095 @convert_operands(scalarRHS=True)
1096 def __pow__(self, other):
1097 """Denote exponentiation with another, scalar expression."""
1098 if type(self.refined) != type(self):
1099 return self.refined.__pow__(other)
1100 else:
1101 return other.__rpow__(self)
1103 @convert_operands(scalarLHS=True)
1104 def __rpow__(self, other):
1105 """Denote taking another expression to the power of the expression."""
1106 if type(self.refined) != type(self):
1107 return self.refined.__rpow__(other)
1108 else:
1109 return NotImplemented
1111 @convert_operands(horiCat=True)
1112 def __and__(self, other):
1113 """Denote horizontal stacking with another expression on the right."""
1114 if type(self.refined) != type(self):
1115 return self.refined.__and__(other)
1116 else:
1117 return other.__rand__(self)
1119 @convert_operands(horiCat=True)
1120 def __rand__(self, other):
1121 """Denote horizontal stacking with another expression on the left."""
1122 if type(self.refined) != type(self):
1123 return self.refined.__rand__(other)
1124 else:
1125 return NotImplemented
1127 @convert_operands(vertCat=True)
1128 def __floordiv__(self, other):
1129 """Denote vertical stacking with another expression below."""
1130 if type(self.refined) != type(self):
1131 return self.refined.__floordiv__(other)
1132 else:
1133 return other.__rfloordiv__(self)
1135 @convert_operands(vertCat=True)
1136 def __rfloordiv__(self, other):
1137 """Denote vertical stacking with another expression above."""
1138 if type(self.refined) != type(self):
1139 return self.refined.__rfloordiv__(other)
1140 else:
1141 return NotImplemented
1143 def __pos__(self):
1144 """Return the expression as-is."""
1145 return self
1147 def __neg__(self):
1148 """Denote the negation of the expression."""
1149 if type(self.refined) != type(self):
1150 return self.refined.__neg__()
1151 else:
1152 opstring = "the negation of {}".format(repr(self))
1153 return self._wsum_fallback((self,), -1, opstring)
1155 def __abs__(self):
1156 """Denote the default norm of the expression.
1158 The norm used depends on the expression's domain. It is
1160 1. the absolute value of a real scalar,
1161 2. the modulus of a complex scalar,
1162 3. the Euclidean norm of a vector, and
1163 4. the Frobenius norm of a matrix.
1164 """
1165 if type(self.refined) != type(self):
1166 return self.refined.__abs__()
1167 else:
1168 return NotImplemented
1170 # --------------------------------------------------------------------------
1171 # Turn __lshift__ and __rshift__ into a single binary relation.
1172 # This is used for both Loewner order (defining LMIs) and set membership.
1173 # --------------------------------------------------------------------------
1175 def _lshift_implementation(self, other):
1176 return NotImplemented
1178 def _rshift_implementation(self, other):
1179 return NotImplemented
1181 @convert_operands(diagBroadcast=True)
1182 @validate_prediction
1183 @refine_operands()
1184 def __lshift__(self, other):
1185 """Denote either set membership or a linear matrix inequality.
1187 If the other operand is a set, then this denotes that the expression
1188 shall be constrained to that set. Otherwise, it is expected that both
1189 expressions are square matrices of same shape and this denotes that the
1190 expression is upper-bounded by the other expression with respect to the
1191 Loewner order (i.e. ``other - self`` is positive semidefinite).
1192 """
1193 result = self._lshift_implementation(other)
1195 if result is NotImplemented:
1196 result = other._rshift_implementation(self)
1198 return result
1200 @convert_operands(diagBroadcast=True)
1201 @validate_prediction
1202 @refine_operands()
1203 def __rshift__(self, other):
1204 """Denote that the expression is lower-bounded in the Lowener order.
1206 In other words, return a constraint that ``self - other`` is positive
1207 semidefinite.
1208 """
1209 result = self._rshift_implementation(other)
1211 if result is NotImplemented:
1212 result = other._lshift_implementation(self)
1214 return result
1216 # --------------------------------------------------------------------------
1217 # Backwards compatibility methods.
1218 # --------------------------------------------------------------------------
1220 @deprecated("2.0", useInstead="~picos.valuable.Valuable.valued")
1221 def is_valued(self):
1222 """Whether the expression is valued."""
1223 return self.valued
1225 @deprecated("2.0", useInstead="~picos.valuable.Valuable.value")
1226 def set_value(self, value):
1227 """Set the value of an expression."""
1228 self.value = value
1230 @deprecated("2.0", "PICOS treats all inequalities as non-strict. Using the "
1231 "strict inequality comparison operators may lead to unexpected results "
1232 "when dealing with integer problems.")
1233 def __lt__(self, exp):
1234 return self.__le__(exp)
1236 @deprecated("2.0", "PICOS treats all inequalities as non-strict. Using the "
1237 "strict inequality comparison operators may lead to unexpected results "
1238 "when dealing with integer problems.")
1239 def __gt__(self, exp):
1240 return self.__ge__(exp)
1243# --------------------------------------
1244__all__ = api_end(_API_START, globals())