A new method for finding the maximum of a general non-linear function of several variables within a constrained region is described, and shown to be efficient compared with existing methods when the required optimum lies on one or more constraints. The efficacy of using effective constraints to eliminate variables is demonstrated, and a program to achieve this easily and automatically is described. Finally, the performance of the new method (the “Complex” method) with unconstrained problems, is compared with those of the Simplex method, from which it was evolved, and Rosenbrock's method.