Constructing QCQP Instances Equivalent to Their SDP Relaxations

TL;DR

This work addresses when a general QCQP is exactly solvable via its SDP relaxation by focusing on a class of constraints that guarantee η = ζ without restrictions on the quadratic objective. It introduces a constructive framework based on Condition (B)' and Condition (D) to generate QCQP instances (starting from 2D, then extending to higher dimensions) whose SDP relaxations are exact, and it provides explicit rank-1 extraction procedures to recover QCQP optima from SDP optima. The paper develops a 2D foundation using basic quadratic constraints and linear transforms, then presents a recursive method to build high-dimensional instances, together with a practical procedure (Ye–Zhang) to obtain rank-1 solutions from SDP solutions. The approach broadens the set of QCQPs that can be solved exactly via SDP and offers a concrete route to construct and solve such problems, with numerical demonstrations illustrating the method’s effectiveness and limitations.

Abstract

General quadratically constrained quadratic programs (QCQPs) are challenging to solve as they are known to be NP-hard. A popular approach to approximating QCQP solutions is to use semidefinite programming (SDP) relaxations. It is well-known that the optimal value $η$ of the SDP relaxation problem bounds the optimal value $ζ$ of the QCQP from below, i.e., $η\leq ζ$. The two problems are considered equivalent if $η= ζ$. In the recent paper by Arima, Kim and Kojima [arXiv:2409.07213], a class of QCQPs that are equivalent to their SDP relaxations are proposed with no condition imposed on the quadratic objective function, which can be chosen arbitrarily. In this work, we explore the construction of various QCQP instances within this class to complement the results in [arXiv:2409.07213]. Specifically, we first construct QCQP instances with two variables and then extend them to higher dimensions. We also discuss how to compute an optimal QCQP solution from the SDP relaxation.

Figures (13)

• Figure 1: The disk constraint $\hbox{\boldmath $E$}^{\rm d}(r)_\geq$: $r=1$ (left), $r=2$ (right).
• Figure 2: The hyperbola constraint $\hbox{\boldmath $E$}^{\rm h}(r)_\geq$: $r=1$ (left), $r=0$ (right).
• Figure 3: The parabola constraint $\hbox{\boldmath $E$}^{\rm p}(r)_\geq$: $r=1$ (left), $r=2$ (right).
• Figure 4: The linear constraint $\hbox{\boldmath $E$}^{\rm \ell}(r)_\geq$: $r=0$ (left), $r=1$ (right).
• Figure 5: The application of scaling, rotation and linear transformation to the parabola constraint $\hbox{\boldmath $E$}^{\rm p}(r)_\geq$ described in Section 2.1. Here $\hbox{\boldmath $B$}^0 = \hbox{$\hbox{\boldmath $E$}^{\rm p}(2)$}, \ \hbox{\boldmath $B$}^1 = \hbox{\boldmath $S$}((1,0.2))^T \hbox{\boldmath $B$}^0 \hbox{\boldmath $S$}((1,0.2)), \ \hbox{\boldmath $B$}^2 = \hbox{\boldmath $R$}(\pi/4)^T \hbox{\boldmath $B$}^1 \hbox{\boldmath $R$}(\pi/4), \ \hbox{\boldmath $B$}^3 = \hbox{\boldmath $P$}((-1,-2))^T \hbox{\boldmath $B$}^2 \hbox{\boldmath $P$}((-1,-2))$.

Theorems & Definitions (4)

• Theorem 1.1
• Remark 1.2
• Remark 3.1
• Lemma 4.1