  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.
