Source code for picos.expressions.exp_sqnorm

# coding: utf-8

# ------------------------------------------------------------------------------
# Copyright (C) 2020 Maximilian Stahlberg
#
# This file is part of PICOS.
#
# PICOS is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# PICOS is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.
# ------------------------------------------------------------------------------

"""Implements :class:`SquaredNorm`."""

import operator
from collections import namedtuple

import cvxopt

from .. import glyphs
from ..apidoc import api_end, api_start
from ..caching import cached_property, cached_unary_operator
from ..constraints import SquaredNormConstraint
from .data import convert_and_refine_arguments, convert_operands, cvxopt_hcat
from .exp_affine import AffineExpression, ComplexAffineExpression
from .exp_quadratic import QuadraticExpression
from .expression import Expression, refine_operands, validate_prediction

_API_START = api_start(globals())
# -------------------------------


[docs]class SquaredNorm(QuadraticExpression): """A squared Euclidean or Frobenius norm. This is a lightweight wrapper around :class:`~picos.expressions.QuadraticExpression` that can handle common constraint formulations more efficiently. """ # -------------------------------------------------------------------------- # Initialization and factory methods. # --------------------------------------------------------------------------
[docs] @convert_and_refine_arguments("x") def __init__(self, x): """Create a squared Euclidean or Frobenius norm. :param x: The (complex) affine expression under the squared norm. :type affinePart: ~picos.expressions.ComplexAffineExpression """ # Validate x. if not isinstance(x, ComplexAffineExpression): raise TypeError("Can only form the squared norm of an affine " "expression, not of {}.".format(type(x).__name__)) if len(x) == 1: typeStr = "Squared Scalar" symbStr = glyphs.squared(x.string) else: typeStr = "Squared Norm" symbStr = glyphs.squared(glyphs.norm(x.string)) Expression.__init__(self, typeStr, symbStr) # TODO: Add a nonzero-vectorization to BiaffineExpression that returns # a scalar zero for an all-zero expression and the vectorization # of the expression with zero rows removed otherwise. if x.is0: self._x = AffineExpression.zero() else: # Vectorize and stack real and imaginary parts. vec = x.vec if x.isreal else x.vec.real // x.vec.imag # Remove zero rows from the vectorization. A = abs(cvxopt_hcat(vec._coefs.values())) a = cvxopt.sparse(sum(A[:, j] for j in range(A.size[1]))) nonzero = a.I nnz = len(nonzero) B = cvxopt.spmatrix([1.0]*nnz, range(nnz), nonzero, (nnz, len(x))) self._x = B*vec
# -------------------------------------------------------------------------- # Allow inheriting from QuadraticExpression. # -------------------------------------------------------------------------- @cached_property def _quadratic_form(self): """The squared norm as a pure quadratic expression. If the expression under the norm is constant, then this is :obj:`None`. """ # HACK: Make a shallow copy of self._x so that the product does not # recognize that the operation below represents a squared norm. # This only works as long as the product checks for operand # equality with the "is" keyword. result = (self._x.renamed("HACK") | self._x) if isinstance(result, AffineExpression): return None else: assert isinstance(result, QuadraticExpression) result._symbStr = self.string return result @cached_property def _quads(self): if self._quadratic_form: return self._quadratic_form._quads else: return {} @cached_property def _aff(self): if self._quadratic_form: return self._quadratic_form._aff else: refined = self.refined assert isinstance(refined, ComplexAffineExpression) return refined @cached_property def _sf(self): if len(self._x) == 1: return (self._x, self._x) else: return None # -------------------------------------------------------------------------- # Squared norm specific properties. # -------------------------------------------------------------------------- @property def argdim(self): """Number of nonzero elements of the expression under the norm.""" return len(self._x) # -------------------------------------------------------------------------- # Abstract method implementations and method overridings, except _predict. # -------------------------------------------------------------------------- @cached_unary_operator def _get_refined(self): if self._x.constant: value = self._x.value_as_matrix return AffineExpression.from_constant( value.T*value, (1, 1), self._symbStr) else: return self Subtype = namedtuple("Subtype", ("argdim", "quadratic_subtype")) @cached_unary_operator def _get_subtype(self): return self.Subtype( len(self._x), self._quadratic_form.subtype if self._quadratic_form else None) def _get_value(self): value = self._x._get_value() return value.T*value @cached_unary_operator def _get_variables(self): return self._x.variables def _is_convex(self): return True def _is_concave(self): return self._x.constant def _replace_variables(self, var_map): return self.__class__(self._x._replace_variables(var_map)) # -------------------------------------------------------------------------- # Python special method implementations, except constraint-creating ones. # -------------------------------------------------------------------------- def __len__(self): # Faster version that overrides Expression.__len__. return 1 @convert_operands(sameShape=True) @refine_operands() def __add__(self, other): """Denote addition from the right hand side.""" if isinstance(other, SquaredNorm): result = SquaredNorm(self._x // other._x) result._symbStr = glyphs.clever_add(self.string, other.string) return result else: return QuadraticExpression.__add__(self, other) @convert_operands(scalarRHS=True) @refine_operands() def __mul__(self, other): """Denote scaling from the right hand side.""" if isinstance(other, AffineExpression): if not other.constant: raise TypeError("You may only multiply a squared norm with a " "constant term.") value = other.value if value < 0: return QuadraticExpression.__mul__(self, other) result = SquaredNorm(self._x*value**0.5) result._symbStr = glyphs.clever_mul(self.string, other.string) return result else: return QuadraticExpression.__mul__(self, other) @convert_operands(scalarRHS=True) @refine_operands() def __rmul__(self, other): """Denote scaling from the left hand side.""" if isinstance(other, AffineExpression): result = self.__mul__(other) # NOTE: __mul__ always creates a fresh expression. result._symbStr = glyphs.clever_mul(other.string, self.string) return result else: return QuadraticExpression.__rmul__(self, other) @convert_operands(scalarRHS=True) @refine_operands() def __truediv__(self, other): """Denote division by a constant scalar.""" if isinstance(other, AffineExpression): if not other.constant: raise TypeError("You may only divide a squared norm by a " "constant term.") result = self.__mul__(1.0 / other.value) # NOTE: __mul__ always creates a fresh expression. result._symbStr = glyphs.div(self.string, other.string) return result else: return QuadraticExpression.__truediv__(self, other) def __div__(self, other): """Denote division by a scalar constant.""" # Python 2 uses this instead of __truediv__. return self.__truediv__(other) # -------------------------------------------------------------------------- # Method overridings for QuadraticExpression. # -------------------------------------------------------------------------- @cached_property def fullroot(self): """Affine expression whose squared norm equals the expression. Overrides :meth:`~picos.expressions.QuadraticExpression.fullroot` of :class:`~picos.expressions.QuadraticExpression`. """ return self._x.renamed(glyphs.Fn("fullroot({})")(self.string)) @property def is_squared_norm(self): """Always :obj:`True` for squared norm instances. Overrides :meth:`~picos.expressions.QuadraticExpression.is_squared_norm` of :class:`~picos.expressions.QuadraticExpression`. """ return True @property def is0(self): """Whether the expression is zero. Overrides :meth:`~picos.expressions.QuadraticExpression.is0` of :class:`~picos.expressions.QuadraticExpression`. """ return self._x.is0 # -------------------------------------------------------------------------- # Constraint-creating operators, and _predict. # -------------------------------------------------------------------------- @classmethod def _predict(cls, subtype, relation, other): assert isinstance(subtype, cls.Subtype) if relation == operator.__le__: if issubclass(other.clstype, AffineExpression) \ and other.subtype.dim == 1: return SquaredNormConstraint.make_type( subtype.argdim, other.subtype.constant) return QuadraticExpression._predict( subtype.quadratic_subtype, relation, other) @convert_operands(sameShape=True) @validate_prediction @refine_operands() def __le__(self, other): if isinstance(other, AffineExpression): return SquaredNormConstraint(self, other) # NOTE: The following should handle the case where the upper bound has a # scalar factorization efficiently by virtue of # SquaredNorm.fullroot. See ConicQuadraticConstraint. return QuadraticExpression.__le__(self, other)
# -------------------------------------- __all__ = api_end(_API_START, globals())