A generalized simulated annealing method has been developed and applied to the optimization of functions (possibly constrained) having many local extrema. The method is illustrated in some difftcult pedagogical examples and used to solve a problem analyzed by Bates (Technometrics, 25, pp. 373–376, 1983) for which we identify an improved optimum. The sensitivity of the solution to changes in the constraints and in other specifications of the problem is analyzed and discussed.