Simultaneous Diagonalization of Matrices and its Applications in Quadratically Constrained Quadratic Programming

TL;DR

This paper establishes necessary and sufficient conditions for simultaneous diagonalization of matrices and applies these results to improve understanding and solution methods for quadratically constrained quadratic programming (QCQP).

Contribution

It provides the first complete SD conditions for multiple matrices and links SD to convexity in QCQP, advancing theoretical understanding and practical solution techniques.

Findings

Derived necessary and sufficient SD conditions for two matrices.

Extended SD conditions to collections of matrices with semi-definite pencil assumption.

Applied SD conditions to verify relaxation exactness in QCQP.

Abstract

An equivalence between attainability of simultaneous diagonalization (SD) and hidden convexity in quadratically constrained quadratic programming (QCQP) stimulates us to investigate necessary and sufficient SD conditions, which is one of the open problems posted by Hiriart-Urruty (SIAM Rev., 49 (2007), pp. 255-273) nine years ago. In this paper we give a necessary and sufficient SD condition for any two real symmetric matrices and offer a necessary and sufficient SD condition for any finite collection of real symmetric matrices under the existence assumption of a semi-definite matrix pencil. Moreover, we apply our SD conditions to QCQP, especially with one or two quadratic constraints, to verify the exactness of its second-order cone programming relaxation and to facilitate the solution process of QCQP.