A Rapidly Convergent Descent Method for Minimization

27 information is used. A choice then has to be made as to which is the most eecient option. Acknowledgments. We are grateful to the referees for their useful comments. We thank Robert Michael Lewis for his valuable suggestions on how best to present this material, particularly the results given in x7. Mathematical models for the predictive value of early CA125 serum levels in epithelial ovarian carcinoma.Direct Search Methods on Parallel Machines 26 a direct search method seems to be in order. Tackling this problem, however, means that we will need to rethink the original implementation of our parallel multidirectional search schemes. To begin with, our current implementation is best suited for the case when all the function evaluations require approximately the same time to complete. Thus, there is a natural synchronization that allows us to implement the algorithm without either a controlling process or any concerns for load balancing. This will not always be the case when dealing with more diicult problems. Hence, we will need an asynchronous, task-queue-based implementation with a single controlling process. Another direction of research would be to extend the parallel multidirectional search algorithm to problems with constraints. We believe it is possible to extend the parallel algorithms, with only minor modiications, to problems with bounded variables. We are also interested in handling linear constraints. If we can handle bounded variables, it should be possible to transfer many of the ideas learned during the development of interior point methods to our simplex-based method for handling problems with linear constraints. There are several other ideas we would also like to pursue. Although we have a simple, fast algorithm to generate templates for the parallel multidirectional search schemes, it is possible that there are other, perhaps better, initialization schemes we could implement. One of the few pieces of information that the basic multidirectional search algorithm carries from iteration to iteration is the size of the step taken in the previous iteration|which determines the size of the step taken in the current iteration. If an expansion step was accepted, this would indicate that the simplex is still far from a solution. If the contraction step was accepted, then either the simplex is near a solution or it is trapped in a diicult region. If we allowed mixed strategies, i.e., diierent templates depending on the type of step accepted in the previous iteration, then it seems possible that we …