  In this paper we develop and analyse convex searches for Zames--Falb
multipliers. We present two different approaches: Infinite Impulse Response
(IIR) and Finite Impulse Response (FIR) multipliers. The set of FIR multipliers
is complete in that any IIR multipliers can be phase-substituted by an
arbitrarily large order FIR multiplier. We show that searches in discrete-time
for FIR multipliers are effective even for large orders. As expected, the
numerical results provide the best $\ell_{2}$-stability results in the
literature for slope-restricted nonlinearities. Finally, we demonstrate that
the discrete-time search can provide an effective method to find suitable
continuous-time multipliers.
