This paper addresses the problem of characterizing the stability of an exponentially stable, linear time invariant system connected with a single feedback monotone nonlinearity. By using the optimal multiplier approach as in [1], this problem is transformed to a constrained infinite dimensional optimization. A sequence of approximate finite dimensional problems is then constructed and three algorithms are proposed for their solutions. In applying these algorithms to a large number of examples, convergence is typically very fast, and a surprisingly many of them are found to satisfy the Kalman Conjecture.