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

Alex L. Wang, Rujun Jiang

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 paper extends the reach of the SDC property by studying two new related but 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. Our main contributions are a complete characterization of the ASDC pairs and the nonsingular ASDC triples, as well as a sufficient condition for the 1-RSDC property for pairs of quadratic forms. Surprisingly, we show that every singular pair is ASDC and that almost every pair is 1-RSDC. We accompany our theoretical results with preliminary numerical experiments applying the RSDC property to QCQPs with a single quadratic constraint.

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 this context. This paper extends the reach of the SDC property by studying two new related but 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. Our main contributions are a complete characterization of the ASDC pairs and the nonsingular ASDC triples, as well as a sufficient condition for the 1-RSDC property for pairs of quadratic forms. Surprisingly, we show that every singular pair is ASDC and that almost every pair is 1-RSDC. We accompany our theoretical results with preliminary numerical experiments applying the RSDC property to QCQPs with a single quadratic constraint.

Paper Structure

This paper contains 37 sections, 28 theorems, 94 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main contributions and outline
  4. Related work
  5. Branch-and-bound methods for QCQPs
  6. The SDC property for sets of quadratic forms and SDC algorithms.
  7. The almost SDS property.
  8. Notation
  9. Preliminaries
  10. Characterization of SDC
  11. The ASDC property of symmetric pairs
  12. A canonical form for symmetric pairs
  13. The nonsingular case
  14. The singular case
  15. Case 1.
  16. ...and 22 more sections

Key Result

proposition 1

Let ${\cal A}\subseteq{\mathbb{S}}^n$ and suppose $S\in\mathop{\mathrm{span}}\nolimits({\cal A})$ is nonsingular. Then, ${\cal A}$ is SDC if and only if $S^{-1}{\cal A}$ is a commuting set of diagonalizable matrices with real eigenvalues.

Figures (3)

  • Figure 1: Comparison of SDPBB, SDCBB and eigBB for the case with $k=0$.
  • Figure 2: Comparison of initial bound and time between SDP and SOCP relaxations for instances of different dimensions.
  • Figure 3: Comparison of SDPBB, k-RSDCBB and eigBB for non-SDC instances.

Theorems & Definitions (57)

  • remark 1
  • definition 1
  • remark 2
  • definition 2
  • definition 3
  • definition 4
  • remark 3
  • proposition 1
  • proposition 2
  • lemma 1
  • ...and 47 more