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 this context. This talk will present a new weaker notion of simultaneous diagonalizability which extends the reach of the SDC property. Specifically, we say that a set of quadratic forms is d-restricted SDC (d-RSDC) if it is the restriction of an SDC set in up to d-many additional dimensions. Surprisingly, we will see that almost every pair of symmetric matrices is 1-RSDC.
We accompany our theoretical results with preliminary numerical experiments applying the RSDC property to QCQPs with a single quadratic constraint and additional linear constraints.
Based on joint work with Rujun Jiang. https://arxiv.org/abs/2101.12141