Based on the authors' previous work which established theoretical
foundations of two, conceptual, successive convex relaxation methods,
{\it i.e.}, the SSDP (Successive Semidefinite Programming) Relaxation
Method and the SSILP (Successive Semi-Infinite Linear Programming)
Relaxation Method, this paper proposes their implementable variants
for general quadratic optimization problems. These problems have a
linear objective function $\cc^T\x$ to be maximized over a nonconvex
compact feasible region $F$ described by a finite number of quadratic
inequalities. We introduce two new techniques, ``discretization'' and
``localization,'' into the SSDP and SSILP Relaxation Methods. The
discretization technique makes it possible to approximate an infinite
number of semi-infinite SDPs (or semi-infinite LPs) which appeared at
each iteration of the original methods by a finite number of standard
SDPs (or standard LPs) with a finite number of linear inequality
constraints. We establish: \vspace{-0.4cm} \begin{itemize} \item {\it
Given any open convex set $U$ containing $F$, an implementable
discretization of the SSDP (or SSILP) Relaxation Method generates a
compact convex set $C$ such that $F \subseteq C \subseteq U$ in a
finite number of iterations. } \vspace{-0.4cm} \end{itemize} The
localization technique is for the cases where we are only interested
in upper bounds on the optimal objective value (for a fixed objective
function vector $\cc$) but not in a global approximation of the convex
hull of $F$. This technique allows us to generate a convex relaxation
of $F$ that is accurate only in certain directions in a neighborhood
of the objective direction $\cc$. This cuts off redundant work to
make the convex relaxation accurate in unnecessary directions. We
establish: \vspace{-0.4cm} \begin{itemize} \item {\it Given any
positive number $\epsilon$, an implementable
localization-discretization of the SSDP (or SSILP) Relaxation Method
generates an upper bound of the objective values within $\epsilon$ of
their maximum in a finite number of iterations. }