This paper proposes a matrix splitting strategy based on the Projected Gauss-Seidel algorithm for the solution of multivariable algebraic loops arising naturally in many constrained control problems. The solution algorithm is globally convergent for a very large class of problems and is orders of magnitude faster than several existing algorithms in the literature. The Projected Gauss-Seidel algorithm is easy to implement with a very minimal computation per iteration and shows promise for real-time and embedded control applications where a fast but approximate solution is of the essence.