  In the current work, we have formulated the optimal bit-allocation problem
for a scalable codec of images or videos as a constrained vector-valued
optimization problem and demonstrated that there can be many optimal solutions,
called Pareto optimal points. In practice, the Pareto points are derived via
the weighted sum scalarization approach. An important question which arises is
whether all the Pareto optimal points can be derived using the scalarization
approach? The present paper provides a sufficient condition on the
rate-distortion function of each resolution of a scalable codec to address the
above question. The result indicated that if the rate-distortion function of
each resolution is strictly decreasing and convex and the Pareto points form a
continuous curve, then all the optimal Pareto points can be derived by using
the scalarization method.
