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geometric_mean_hypo_cone

CVXPY constraints for the matrix geometric mean hypograph cone.

geometric_mean_hypo_cone 

geometric_mean_hypo_cone(
    A: Expression,
    B: Expression,
    T: Expression,
    t: float = 1 / 2,
    fullhyp: bool = True,
    *,
    hermitian: bool = False,
) -> list[Constraint]

Return CVX constraints for matrix geometric-mean hypograph cone 1.

The set of matrices that satisfy the constraints are A, B, T triples such that

\[ G_t(A, B) \geq T \]

where G_t(A, B) is the matrix geometric mean function.

Parameters:

  • A (Expression) –

    A cvxpy expression representing a matrix.

  • B (Expression) –

    A cvxpy expression representing a matrix.

  • T (Expression) –

    A cvxpy expression representing a matrix.

  • t (float, default: 1 / 2 ) –

    The weight in the range [0, 1].

  • fullhyp (bool, default: True ) –

    Whether to use the full hypograph or the restricted hypograph.

  • hermitian (bool, default: False ) –

    Whether the matrices are Hermitian or symmetric.

Raises:

  • ValueError –

    If the weight is not in the range [0, 1].

  • ValueError –

    If the matrices are not the same size.

  • ValueError –

    If the matrices are not 2D or not square.

Returns:

  • list[Constraint] –

    A list of CVX constraints.

References

1 Fawzi, Hamza and Saunderson, James. Lieb's concavity theorem, matrix geometric means and semidefinite optimization. (2015). link.

Source code in toqito/cones/geometric_mean_hypo_cone.py 
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def geometric_mean_hypo_cone(
    A: cvxpy.Expression,
    B: cvxpy.Expression,
    T: cvxpy.Expression,
    t: float = 1 / 2,
    fullhyp: bool = True,
    *,
    hermitian: bool = False,
) -> list[cvxpy.Constraint]:
    r"""Return CVX constraints for matrix geometric-mean hypograph cone [@fawzi2015matrixgeometric].

    The set of matrices that satisfy the constraints are `A`, `B`, `T` triples such
    that

    \[
    G_t(A, B) \geq T
    \]

    where `G_t(A, B)` is the matrix geometric mean function.

    Args:
        A: A cvxpy expression representing a matrix.
        B: A cvxpy expression representing a matrix.
        T: A cvxpy expression representing a matrix.
        t: The weight in the range `[0, 1]`.
        fullhyp: Whether to use the full hypograph or the restricted hypograph.
        hermitian: Whether the matrices are Hermitian or symmetric.

    Raises:
        ValueError: If the weight is not in the range `[0, 1]`.
        ValueError: If the matrices are not the same size.
        ValueError: If the matrices are not 2D or not square.

    Returns:
        A list of CVX constraints.

    """
    if t < 0 or t > 1:
        raise ValueError("The weight must be in the range [0, 1].")

    if A.shape != B.shape or B.shape != T.shape:
        raise ValueError("The matrices must be the same size.")
    _require_square_2d(A, "The matrices")

    if fullhyp:
        dim = A.shape[0]
        W = _symmetric_like_variable(dim, hermitian=hermitian)
        hypo_w = _geometric_mean_cone_recursion(A, B, W, float(t), hermitian=hermitian)
        return [*hypo_w, W >> T]
    return _geometric_mean_cone_recursion(A, B, T, float(t), hermitian=hermitian)