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On semidefinite programming characterizations of the numerical radius and its dual norm

Shmuel Friedland, Chi-Kwong Li

TL;DR

This work provides self-contained semidefinite programming characterizations for the numerical radius $r(C)$ and its dual $r^\bv(C)$ of complex matrices. It proves that, for rational data, $r(C)$ and $r^\bv(C)$ (and related norms) can be approximated in polynomial time using either the ellipsoid method or short-step interior-point methods, with explicit SDP formulations and complexity reasoning. A notable contribution is the application showing that the spectral and nuclear norms of $2\times m\times n$ real tensors can be expressed via $r(C)$ and $r^\bv(C)$, thereby enabling polynomial-time computation in this tensor regime. The results also connect to known characterizations of $r(C)$ and provide practical SDP-based routes to compute tensor norms arising in quantum information and multilinear algebra contexts.

Abstract

We state and give self contained proofs of semidefinite programming characterizations of the numerical radius and its dual norm for matrices. We show that the computation of the numerical radius and its dual norm within $\varepsilon$ precision are polynomially time computable in the data and $|\log \varepsilon |$ using either the ellipsoid method or the short step, primal interior point method. We apply our results to give a simple formula for the spectral and nuclear norm of $2\times n\times m$ real tensor in terms of the numerical radius and its dual norm.

On semidefinite programming characterizations of the numerical radius and its dual norm

TL;DR

This work provides self-contained semidefinite programming characterizations for the numerical radius and its dual of complex matrices. It proves that, for rational data, and (and related norms) can be approximated in polynomial time using either the ellipsoid method or short-step interior-point methods, with explicit SDP formulations and complexity reasoning. A notable contribution is the application showing that the spectral and nuclear norms of real tensors can be expressed via and , thereby enabling polynomial-time computation in this tensor regime. The results also connect to known characterizations of and provide practical SDP-based routes to compute tensor norms arising in quantum information and multilinear algebra contexts.

Abstract

We state and give self contained proofs of semidefinite programming characterizations of the numerical radius and its dual norm for matrices. We show that the computation of the numerical radius and its dual norm within precision are polynomially time computable in the data and using either the ellipsoid method or the short step, primal interior point method. We apply our results to give a simple formula for the spectral and nuclear norm of real tensor in terms of the numerical radius and its dual norm.
Paper Structure (11 sections, 13 theorems, 80 equations)

This paper contains 11 sections, 13 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Conic and semidefinite programming
  3. Slater condition for a conic programming problem
  4. Semidefinite programming for real and complex matrices
  5. Complexity results for semidefinite programming
  6. The dual norm of $r(\cdot)$
  7. The computation of $r(C)$ and $r^\vee(C)$ is poly-time
  8. Poly-time complexity of $r(C)$, $r^\vee(C)$, $\|C\|$ and $\|C\|^\vee$ using SDP
  9. Applications
  10. A known characterization of $r(C)$
  11. Application to computing the spectral and nuclear norms of $2\times m\times n$ real tensor

Key Result

Theorem 2.1

Assume that the feasible set of conprog contains an interior point of the cone $\mathrm{K}$, and $val$ is finite. Then the dual problem dconprog is feasible, and $val=val^\vee$.

Theorems & Definitions (19)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 9 more