  Computing the discrete rational minimax approximation in the complex plane is
challenging. Apart from Ruttan's sufficient condition, there are few other
sufficient conditions for global optimality. The state-of-the-art rational
approximation algorithms, such as the adaptive Antoulas-Anderson (AAA),
AAA-Lawson, and the rational Krylov fitting (RKFIT) method, perform highly
efficiently, but the computed rational approximants may be near-best. In this
paper, we propose a convex programming approach, the solution of which is
guaranteed to be the rational minimax approximation under Ruttan's sufficient
condition. Furthermore, we present a new version of Lawson's iteration for
solving this convex programming problem. The computed solution can be easily
verified as the rational minimax approximant. Our numerical experiments
demonstrate that this updated version of Lawson's iteration generally converges
monotonically with respect to the objective function of the convex programming.
It is an effective competitive approach for the rational minimax problem,
compared to the highly efficient AAA, AAA-Lawson, and the stabilized
Sanathanan-Koerner iteration.
