Parameter optimization methods have recently proved to be an efficient tool for the design of structurally constrained multivariate regulators, such as decentralized and PID. However, most of the underlying numerical algorithms require preliminarily selection of a regulator ensuring closed-loop asymptotic stability. Solving this problem is particularly intriguing whenever robust zero-error regulation must be guaranteed, since the regulator cannot, in this case, be asymptotically stable. In this paper, constructive sufficient conditions are given for the existence of structurally constrained high-gain regulators ensuring both closed-loop asymptotic stability and robust zero-error asymptotic regulation in the face of polynomial exogenous signals.