Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extending Exact Convex Relaxations of Quadratically Constrained Quadratic Programs

Masakazu Kojima, Sunyoung Kim, Naohiko Arima

TL;DR

This work addresses extending QCQPs with exact convex relaxations by adding quadratic inequality constraints under non-intersecting quadratic constraint (NIQC) conditions. The authors formulate a main theorem (C1–C4) guaranteeing that the extended QCQP retains an exact convex relaxation across SDP, CPP, and DNN relaxations for several QCQP classes, including NIQC, ROG, convex, sign-pattern, and submodular problems. The proof leverages conic-form representations, a rank-1 decomposition lemma, and KKT conditions to ensure the extended problem has a rank-1 optimal solution when the base relaxation is exact. The framework unifies and extends prior results (e.g., Joyce 2024, Yang 2018) and provides a flexible tool for guaranteeing exactness in extended QCQPs encountered in control, combinatorial optimization, and signal processing. Overall, it offers a systematic approach to preserve tractability and exactness when incorporating additional quadratic constraints into QCQPs.

Abstract

A convex relaxation of a quadratically constrained quadratic program (QCQP) is called exact if it has a rank-$1$ optimal solution that corresponds to an optimal solution of the QCQP. Given a QCQP whose convex relaxation is exact, this paper investigates the incorporation of additional quadratic inequality constraints under a non-intersecting quadratic constraint condition while maintaining the exactness of the convex relaxation of the resulting QCQP. Specifically, we extend existing exact semidefinite programming relaxation, completely positive programming relaxation and doubly nonnegative programming relaxation of various classes of QCQPs in a unified manner. Illustrative examples are included to demonstrate the applicability of the established result.

Extending Exact Convex Relaxations of Quadratically Constrained Quadratic Programs

TL;DR

This work addresses extending QCQPs with exact convex relaxations by adding quadratic inequality constraints under non-intersecting quadratic constraint (NIQC) conditions. The authors formulate a main theorem (C1–C4) guaranteeing that the extended QCQP retains an exact convex relaxation across SDP, CPP, and DNN relaxations for several QCQP classes, including NIQC, ROG, convex, sign-pattern, and submodular problems. The proof leverages conic-form representations, a rank-1 decomposition lemma, and KKT conditions to ensure the extended problem has a rank-1 optimal solution when the base relaxation is exact. The framework unifies and extends prior results (e.g., Joyce 2024, Yang 2018) and provides a flexible tool for guaranteeing exactness in extended QCQPs encountered in control, combinatorial optimization, and signal processing. Overall, it offers a systematic approach to preserve tractability and exactness when incorporating additional quadratic constraints into QCQPs.

Abstract

A convex relaxation of a quadratically constrained quadratic program (QCQP) is called exact if it has a rank- optimal solution that corresponds to an optimal solution of the QCQP. Given a QCQP whose convex relaxation is exact, this paper investigates the incorporation of additional quadratic inequality constraints under a non-intersecting quadratic constraint condition while maintaining the exactness of the convex relaxation of the resulting QCQP. Specifically, we extend existing exact semidefinite programming relaxation, completely positive programming relaxation and doubly nonnegative programming relaxation of various classes of QCQPs in a unified manner. Illustrative examples are included to demonstrate the applicability of the established result.

Paper Structure

This paper contains 17 sections, 10 theorems, 49 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A conic-form QCQP and its convex relaxation
  4. A standard-form QCQP
  5. Non-intersecting quadratic constraint (NIQC) conditions
  6. Main Theorem
  7. Some existing results related to Theorem \ref{['theorem:main1']}
  8. Proof of Theorem \ref{['theorem:main1']}
  9. Exact SDP relaxations of QCQPs
  10. QCQPs characterized by the rank-one generated property
  11. Convex QCQPs
  12. QCQPs characterized by sign pattern conditions
  13. Continuous quadratic submodular minimization problem
  14. Exact CPP and DNN relaxations of QCQPs
  15. Combinatorial QCQPs
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $\mathbb{G}, \ \mathbb{K} \subseteq \mathbb{S}^n_+$ be closed convex cones such that $\mathbb{G} \subseteq \mathbb{K}$, $\hbox{\boldmath $Q$} \in \mathbb{S}^n$, $\hbox{\boldmath $H$} \in \mathbb{S}^n$, $\hbox{$\cal B$} = \{\hbox{\boldmath $B$}^k : 1 \leq k \leq m\} \subseteq \mathbb{S}^n$ and $\

Figures (5)

  • Figure 1: Summary of NIQC conditions. $\sharp$ : Some assumptions are necessary to prove the relation, including $\mathbb{F} = \mathbb{F}_{\min}$ (the minimal face of $\mathbb{S}^n_+$ that includes $\mathbb{J}_+(\hbox{$\cal A$})$); see ARIMA2024 for details.
  • Figure 2: Example \ref{['example:Argue']}. The unshaded (white) area corresponds to the interior of the feasible region $(\hbox{$\cal A$}\cup\hbox{$\cal B$})_\geq$ of QCQP \ref{['eq:QCQPS2']} with $\hbox{$\cal C$} = \hbox{$\cal A$}\cup\hbox{$\cal B$}$ and $L_1(\mathbb{F}) = \mathbb{R}^{n-1}$.
  • Figure 3: Example \ref{['example:convex']}. The unshaded (white) area represents the interior of the feasible region $(\hbox{$\cal A$}\cup\hbox{$\cal B$})_\geq$ of QCQP \ref{['eq:QCQPS2']} with $\hbox{$\cal C$} = \hbox{$\cal A$}\cup\hbox{$\cal B$}$ and $L_1(\mathbb{F}) = \mathbb{R}^{n-1}$.
  • Figure 4: Example \ref{['example:signPattern']}. The unshaded (white) area represents the interior of the feasible region $(\hbox{$\cal A$}\cup\hbox{$\cal B$})_\geq$ of QCQP \ref{['eq:QCQPS2']} with $\hbox{$\cal C$} = \hbox{$\cal A$}\cup\hbox{$\cal B$}$ and $L_1(\mathbb{F}) = \mathbb{R}^{n-1}$.
  • Figure 5: Example \ref{['example:CPP']}: $n=3$, $\hbox{\boldmath $B$}^1_\leq = \{\hbox{\boldmath $u$} \in \mathbb{R}^2 : -u_1+(u_2-u_1)^2 + 2 \leq 0\}$, $\hbox{\boldmath $B$}^2_\leq = \{ \hbox{\boldmath $u$} \in \mathbb{R}^2 : \sum_{i=1}^{2} (u_i-1)^2 -1 \leq 0 \}$.

Theorems & Definitions (19)

  • Remark 2.1
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Remark 3.6
  • Lemma 4.1
  • Remark 4.2
  • Theorem 4.3
  • ...and 9 more