New notions of simultaneous diagonalizability of quadratic forms with applications to QCQPs

Abstract

A set of quadratic forms is simultaneously diagonalizable via congruence (SDC) if there exists a basis under which each of the quadratic forms is diagonal. This property appears naturally when analyzing quadratically constrained quadratic programs (QCQPs) and has important implications in globally solving such problems using branch-and-bound methods. This paper extends the reach of the SDC property by studying two new weaker notions of simultaneous diagonalizability. Specifically, we say that a set of quadratic forms is almost SDC (ASDC) if it is the limit of SDC sets and d-restricted SDC (d-RSDC) if it is the restriction of an SDC set in up to d-many additional dimensions. In the context of QCQPs, these properties correspond to problems that may be diagonalized after arbitrarily small perturbations or after the introduction of d additional variables. Our main contributions are complete characterizations of the ASDC pairs and nonsingular triples of symmetric matrices, as well as a sufficient condition for the 1-RSDC property for pairs of symmetric matrices. Surprisingly, we show that every singular symmetric pair is ASDC and that almost every symmetric pair is 1-RSDC. We accompany our theoretical results with preliminary numerical experiments applying these constructions to solve QCQPs within branch-and-bound schemes.