Parameter optimization methods are known to be an efficient tool for synthesizing multivariable regulators, especially when constraints are present on both their structure and the robustness of the control system. However, most of the underlying numerical algorithms first require the selection of a regulator satisfying all the assigned constraints, which is often non-trivial. This paper aims at presenting some new results on this subproblem. Specifically, constructive sufficient conditions are given for the existence of simple low-gain centralized and decentralized regulators supplying the closed-loop system with asymptotic stability as well as robust zero-error regulation in the face of ramp-type exogenous signals. Next, solutions are presented to two design problems incorporating constraints on the robustness of the control system with respect to large plant parameter variations.