  In many compressed sensing applications, linear programming (LP) has been
used to reconstruct a sparse signal. When observation is noisy, the LP
formulation is extended to allow an inequality constraint and the solution is
dependent on a parameter $\delta$, related to the observation noise level.
Recently, some researchers also considered quadratic programming (QP) for
compressed sensing signal reconstruction and the solution in this case is
dependent on a Lagrange multiplier $\beta$. In this work, we investigated the
relation between $\delta$ and $\beta$ and derived an upper and a lower bound on
$\beta$ in terms of $\delta$. For a given $\delta$, these bounds can be used to
approximate $\beta$. Since $\delta$ is a physically related quantity and easy
to determine for an application while there is no easy way in general to
determine $\beta$, our results can be used to set $\beta$ when the QP is used
for compressed sensing. Our results and experimental verification also provide
some insight into the solutions generated by compressed sensing.
