picos.expressions.exp_specnorm

Implements SpectralNorm.

Classes

class picos.expressions.exp_specnorm.SpectralNorm(x)[source]

Bases: picos.expressions.expression.Expression

The spectral norm of a matrix.

This class can represent the spectral norm of a matrix-affine expression (real- or complex valued). The spectral norm is convex, so we can form expressions of the form SpectralNorm(X) <= t which are typically reformulated as LMIs that can be handled by SDP solvers.

Definition

If the normed expression is a matrix X, then its spectral norm is

\|X\|_2 = \max \{ \|Xu\|_2 : \|u\| \leq 1\} = \sqrt{\lambda_{\max}(XX^*)},

where \lambda_{\max}(\cdot) denotes the largest eigenvalue of a matrix, and X^* denotes the adjoint matrix of X (i.e., the transposed matrix X^T if X is real-valued).

Special cases:

  • If X is scalar, then \|X\|_2 reduces to the the absolute value (or modulus) |X|.

  • If X is scalar, then \|X\|_2 coincides with the Euclidean norm of X.

__init__(x)[source]

Construct a SpectralNorm.

Parameters

x (ComplexAffineExpression) – The affine expression to take the norm of.

__le__(other)[source]

Return a constraint that the expression is upper-bounded.

__mul__(other)[source]

Denote multiplication with another expression on the right.

__rmul__(other)[source]

Denote multiplication with another expression on the left.

property x

Real expression whose norm equals that of the original expression.