Abstract
We consider stochastic discrete optimization problems where the decision variables are nonnegative integers. We propose and analyze an online control scheme which transforms the problem into a surrogate continuous optimization problem and proceeds to solve the latter using standard gradient-based approaches, while simultaneously updating both the actual and surrogate system states. It is shown that the solution of the original problem is recovered as an element of the discrete state neighborhood of the optimal surrogate state. For the special case of separable cost functions, we show that this methodology becomes particularly efficient. Finally, convergence of the proposed algorithm is established under standard technical conditions; numerical results are included in the paper to illustrate the fast convergence of this approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cassandras, C. G., and Panayiotou, C. G., Concurrent Sample Path Analysis of Discrete Event Systems, Journal of Discrete Event Dynamic Systems: Theory and Applications, Vol. 9, pp. 171–195, 1999.
Ho, Y. C., Eyler, A., and Chien, D., A Gradient Technique for General Buffer Storage Design in a Serial Production Line, International Journal of Production Research, Vol. 17, pp. 557–580, 1979.
Yan, H., Zhou, X., and Yin, G., Finding the Optimal Number of Kanbans in a Manufacturing System via Stochastic Approximation and Perturbation Analysis, Proceedings of the 11th International Conference on Analysis and Optimization of Systems, pp. 572–578, 1994.
Cassandras, C. G., and Julka, V., Scheduling Policies Using Marked / Phantom Slot Algorithms, Queueing Systems: Theory and Applications, Vol. 20, pp. 207–254, 1995.
Barnhart, C. M., Wieselthier, J. E., and Ephremides, A., Admission Control Policies for Multihop Wireless Networks, Wireless Networks, Vol. 1, pp. 373–387, 1995.
Parker, R., and Rardin, R., Discrete Optimization, Academic Press, New York, NY, 1988.
Ibaraki, T., and Katoh, N., Resource Allocation Problems: Algorithmic Approaches, MIT Press, Cambridge, Massachusetts, 1988.
Aarts, E., and Korst, J., Simulated Annealing and Boltzmann Machines, Wiley, New York, NY, 1989.
Holland, J., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Michigan, 1975.
Kiefer, J., and Wolfowitz, J., Stochastic Estimation of the Maximum of a Regression Function, Annals of Mathematical Statistics, Vol. 23, pp. 462–466, 1952.
Robbins, H., and Monro, S., A Stochastic Approximation Method, Annals of Mathematical Statistics, Vol. 22, pp. 400–407, 1951.
Yan, D., and Mukai, H., Stochastic Discrete Optimization, SIAM Journal on Control and Optimization, Vol. 30, pp. 549–612, 1992.
Gong, W. B., Ho, Y. C., and Zhai, W., Stochastic Comparison Algorithm for Discrete Optimization with Estimation, Proceedings of 31st IEEE Conference on Decision and Control, pp. 795–800, 1992.
Shi, L., and Olafsson, S., Nested Partitions Method for Global Optimization, Operations Research, Vol. 48, pp. 390–407, 2000.
Ho, Y. C., Sreenivas, R. S., and Vakili, P., Ordinal Optimization in Discrete Event Dynamic Systems, Journal of Discrete Event Dynamic Systems: Theory and Applications, Vol. 2, pp. 61–88, 1992.
Cassandras, C. G., Dai, L., and Panayiotou, C. G., Ordinal Optimization for Deterministic and Stochastic Resource Allocation, IEEE Transactions on Automatic Control, Vol. 43, pp. 881–900, 1998.
Dai, L., Cassandras, C. G., and Panayiotou, C. G., On the Convergence Rate of Ordinal Optimization for a Class of Stochastic Discrete Resource Allocation Problems, IEEE Transactions on Automatic Control, Vol. 45, pp. 588–591, 2000.
Dai, L., Convergence Properties of Ordinal Comparison in the Simulation of Discrete Event Dynamic Systems, Journal of Optimization Theory and Applications, Vol. 91, pp. 363–388, 1996.
Cassandras, C. G. and Julka, V., A New Approach for Some Combinatorially Hard Stochastic Optimization Problems, Proceedings of the 31st Annual Allerton Conference on Communication, Control, and Computing, pp. 667–676, 1993.
Gokbayrak, K., and Cassandras, C. G., Stochastic Discrete Optimization Using a Surrogate Problem Methodology, Proceedings of the 38th IEEE Conference on Decision and Control, pp. 1779–1784, 1999.
Kushner, H., and Clark, D., Stochastic Approximation for Constrained and Unconstrained Systems, Springer Verlag, Berlin, Germany, 1978.
Ho, Y. C., and Cao, X., Perturbation Analysis of Discrete Event Dynamic Systems, Kluwer Academic Publishers, Dordrecht, Holland, 1991.
Cassandras, C. G., Discrete Event Systems: Modeling and Performance Analysis, Irwin Publications, Homewood, Illinois, 1993.
Bertsimas, D., and Tsitsiklis, J. N., Introduction to Linear Optimization, Athena Scientific, Belmont, Massachusetts, 1997.
Wieselthier, J., Barnhart, C., and Ephremides, A., Standard Clock Simulation and Ordinal Optimization Applied to Admission Control in Integrated Communication Networks, Journal of Discrete Event Dynamic Systems, Vol. 5, pp. 243–279, 1995.
Gokbayrak, K., and Cassandras, C. G., Adaptive Call Admission Control in Circuit Switched Networks (submitted).
Cassandras, C. G., and Yu, R., A 'Surrogate Problem' Approach for Lot Size Optimization in Manufacturing Systems, Proceedings of the 2000 American Control Conference, pp. 3279–3283, 2000.
Gokbayrak, K., and Cassandras, C. G., Stochastic Discrete Resource Allocation Using a Surrogate Method, Technical Report, Department of Manufacturing Engineering, Boston University, 1999.
Gladyshev, E. G., On Stochastic Approximation, Theory of Probability and Its Applications, Vol. 10, pp. 275–278, 1965.
Ljung, L., Analysis of Recursive Stochastic Algorithms, IEEE Transactions on Automatic Control, Vol. 22, pp. 551–575, 1977.
Chong, E. K. P., and Ramadge, P. J., Convergence of Recursive Optimization Algorithms Using IPA Derivative Estimates, Journal of Discrete Event Dynamic Systems: Theory and Applications, Vol. 1, pp. 339–372, 1992.
Kushner, H., and Yin, G., Stochastic Approximation Algorithms and Applications, Springer Verlag, New York, NY, 1997.
Bertsekas, D. P., Nonlinear Programming, Athena Scientific, Belmont, Massachusetts, 1995.
Panayiotou, C. G., and Cassandras, C. G., Optimization of Kanban-Based Manufacturing Systems, Automatica, Vol. 35, pp. 1521–1533, 1999.
Rights and permissions
About this article
Cite this article
Gokbayrak, K., Cassandras, C.G. Online Surrogate Problem Methodology for Stochastic Discrete Resource Allocation Problems. Journal of Optimization Theory and Applications 108, 349–376 (2001). https://doi.org/10.1023/A:1026490318131
Issue Date:
DOI: https://doi.org/10.1023/A:1026490318131