不定齐次二次优化的半定松弛界
Semidefnite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization
AI总结：
研究不定齐次二次优化问题与其半定规划松弛最优值之间的比率界，针对两种模型分别给出上界和下界，并证明结果紧致。
摘要：
本文研究齐次二次优化问题的最优值与其半定规划松弛最优值之间的关系。我们考虑两个二次优化模型：(1) $\min \{x^* C x \mid x^* A_k x \ge 1, x\in\mathbb{F}^n, k=0,1,...,m\}$；(2) $\max \{x^* C x \mid x^* A_k x \le 1, x\in\mathbb{F}^n, k=0,1,...,m\}$。如果 $A_k$ 中有一个是不定的，而其他 $A_k$ 和 $C$ 是半正定的，我们证明当 $\mathbb{F}$ 为实数域 $\mathbb{R}$ 时，(1) 的最优值与其 SDP 松弛的比率上界为 $O(m^2)$，当 $\mathbb{F}$ 为复数域 $\mathbb{C}$ 时，上界为 $O(m)$。该结果是 Luo 等人 \cite{LSTZ} 近期工作的推广。对于 (2)，我们证明当除了一个 $A_k$ 外其余均为半正定且 $C$ 可以是不定时，在实数和复数情形下该比率的下界均为 $O(1/\log m)$。该结果改进了 Ben-Tal 等人 \cite{BNR02} 的所谓近似 S-引理。我们还考虑了具有多个不定二次约束的模型 (2)，并根据问题数据和 SDP 解推导出一个一般界。在整篇论文中，我们给出例子表明所有结果本质上是紧的。
英文摘要：
In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{x^* C x \mid x^* A_k x \ge 1, x\in\mathbb{F}^n, k=0,1,...,m\}$; and (2) $\max \{x^* C x \mid x^* A_k x \le 1, x\in\mathbb{F}^n, k=0,1,...,m\}$. If \emph{one} of $A_k$'s is indefinite while others and $C$ are positive semidefinite, we prove that the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by $O(m^2)$ when $\mathbb{F}$ is the real line $\mathbb{R}$, and by $O(m)$ when $\mathbb{F}$ is the complex plane $\mathbb{C}$. This result is an extension of the recent work of Luo {\em et al.} \cite{LSTZ}. For (2), we show that the same ratio is bounded from below by $O(1/\log m)$ for both the real and complex case, whenever all but one of $A_k$'s are positive semidefinite while $C$ can be indefinite. This result improves the so-called approximate S-Lemma of Ben-Tal {\em et al.} \cite{BNR02}. We also consider (2) with multiple indefinite quadratic constraints and derive a general bound in terms of the problem data and the SDP solution. Throughout the paper, we present examples showing that all of our results are essentially tight.
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