We develop a family of computer search methods for finding optimum designs, that generalize and improve upon Mitchell's exchange and DETMAK algorithms. Our emphasis is on time- and space-saving considerations that permit us to handle larger problems, or to run each search “try” for the optimum more rapidly, than was previously possible. This last means that more tries can be attempted for a given total cost, with consequent greater chance of finding an optimum or near-optimum design. Indeed, we have found a number of new optimum or improved designs using these methods. Fork= 6 to 12 parameters and withnobservations andk≤n≤ 2k, our methods are typically 15 to 50 times faster than DETMAX (more asnandkincrease), with comparable success rates. Numerical studies in linear and quadratic regression examples treat also the effect of amount of initial randomization on the success of a try.