Source code for picos.expressions.exp_nucnorm

# ------------------------------------------------------------------------------
# Copyright (C) 2020 Guillaume Sagnol
# 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
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# ------------------------------------------------------------------------------

"""Implements :class:`NuclearNorm`."""

import operator
from collections import namedtuple

import cvxopt
import numpy

from .. import glyphs
from ..apidoc import api_end, api_start
from ..caching import cached_unary_operator
from ..constraints import AbsoluteValueConstraint, NuclearNormConstraint
from .data import convert_and_refine_arguments, convert_operands, cvx2np
from .exp_affine import AffineExpression, ComplexAffineExpression
from .exp_norm import Norm
from .expression import Expression, refine_operands, validate_prediction

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

[docs]class NuclearNorm(Expression): r"""The nuclear norm of a matrix. This class can represent the nuclear norm of a matrix-affine expression (real- or complex valued). The nuclear norm is convex, so we can form expressions of the form ``NuclearNorm(X) <= t`` which are typically reformulated as LMIs that can be handled by SDP solvers. :Definition: If the normed expression is a matrix :math:`X`, then its nuclear norm is .. math:: \|X\|_* = \operatorname{trace}\ (X^*X)^{1/2} = \sum_{i=1}^{\min(n,m)} \sigma_i(X) where the :math:`\sigma_i(X)` denote the singular values of a :math:`X`, and :math:`X^*` denotes the adjoint matrix of :math:`X` (i.e., the transposed matrix :math:`X^T` if :math:`X` is real-valued). Special cases: - If :math:`X` is scalar, then :math:`\|X\|_*` reduces to the the absolute value (or modulus) :math:`|X|`. - If :math:`X` is scalar, then :math:`\|X\|_*` coincides with the Euclidean norm of :math:`X`. """
[docs] @convert_and_refine_arguments("x") def __init__(self, x): """Construct a :class:`NuclearNorm`. :param x: The affine expression to take the norm of. :type x: ~picos.expressions.ComplexAffineExpression """ # Validate x. if not isinstance(x, ComplexAffineExpression): raise TypeError("Can only form the nuclear norm of an affine " "expression, not of {}.".format(type(x).__name__)) complex = not isinstance(x, AffineExpression) # Build the string representations. if len(x) == 1: typeStr = "Modulus" if complex else "Absolute Value" symbStr = glyphs.abs(x.string) elif 1 in x.shape: typeStr = "Euclidean Norm" symbStr = glyphs.norm(x.string) else: typeStr = "Nuclear Norm" symbStr = glyphs.ncnorm(x.string) if complex: typeStr = "Complex " + typeStr self._x = x self._complex = complex Expression.__init__(self, typeStr, symbStr)
# -------------------------------------------------------------------------- # Abstract method implementations and method overridings, except _predict. # -------------------------------------------------------------------------- @cached_unary_operator def _get_refined(self): if self._x.constant: return AffineExpression.from_constant(self.value, 1, self.string) elif len(self._x) == 1 or (1 in self._x.shape): return Norm(self._x) else: return self Subtype = namedtuple("Subtype", ("argshape", "complex", "hermitian")) def _get_subtype(self): return self.Subtype(self._x.shape, self._complex, self._x.hermitian) def _get_value(self): value = self._x._get_value() value = cvx2np(value) value = numpy.linalg.norm(value, 'nuc') return cvxopt.matrix(value) def _get_mutables(self): return self._x._get_mutables() def _is_convex(self): return True def _is_concave(self): return False def _replace_mutables(self, mapping): return self.__class__(self._x._replace_mutables(mapping)) def _freeze_mutables(self, freeze): return self.__class__(self._x._freeze_mutables(freeze)) # -------------------------------------------------------------------------- # Python special method implementations, except constraint-creating ones. # -------------------------------------------------------------------------- @classmethod def _mul(cls, self, other, forward): if isinstance(other, AffineExpression) and other.constant: factor = other.safe_value if not factor: return elif factor == 1: return self elif factor > 0: if forward: string = glyphs.clever_mul(self.string, other.string) else: string = glyphs.clever_mul(other.string, self.string) norm = cls(other*self._x) norm._typeStr = "Scaled " + norm._typeStr norm._symbStr = string return norm if forward: return Expression.__mul__(self, other) else: return Expression.__rmul__(self, other)
[docs] @convert_operands(scalarRHS=True) @refine_operands() def __mul__(self, other): return NuclearNorm._mul(self, other, True)
[docs] @convert_operands(scalarRHS=True) @refine_operands() def __rmul__(self, other): return NuclearNorm._mul(self, other, False)
# -------------------------------------------------------------------------- # Methods and properties that return modified copies. # -------------------------------------------------------------------------- @property def x(self): """Real expression whose norm equals that of the original expression.""" return self._x # -------------------------------------------------------------------------- # Constraint-creating operators, and _predict. # -------------------------------------------------------------------------- @classmethod def _predict(cls, subtype, relation, other): assert isinstance(subtype, cls.Subtype) arg_shape, arg_complex, arg_hermitian = subtype xLen = arg_shape[0] * arg_shape[1] if relation == operator.__le__: if issubclass(other.clstype, AffineExpression) \ and other.subtype.dim == 1: if xLen == 1: return AbsoluteValueConstraint.make_type() elif 1 in arg_shape: assert False, "Unexpected case (should have been refined)" else: return NuclearNormConstraint.make_type( arg_shape, arg_complex, arg_hermitian) elif relation == operator.__ge__: return NotImplemented # Not concave. return NotImplemented
[docs] @convert_operands(scalarRHS=True) @validate_prediction @refine_operands() def __le__(self, other): assert self.convex if isinstance(other, AffineExpression): if len(self._x) == 1: return AbsoluteValueConstraint(self._x, other) elif 1 in self._x.shape: assert False, "Unexpected case (should have been refined)" else: return NuclearNormConstraint(self, other) else: return NotImplemented
# -------------------------------------- __all__ = api_end(_API_START, globals())