Abstract
We develop four algorithms for simulation-based optimization under multiple inequality constraints. Both the cost and the constraint functions are considered to be long-run averages of certain state-dependent single-stage functions. We pose the problem in the simulation optimization framework by using the Lagrange multiplier method. Two of our algorithms estimate only the gradient of the Lagrangian, while the other two estimate both the gradient and the Hessian of it. In the process, we also develop various new estimators for the gradient and Hessian. All our algorithms use two simulations each. Two of these algorithms are based on the smoothed functional (SF) technique, while the other two are based on the simultaneous perturbation stochastic approximation (SPSA) method. We prove the convergence of our algorithms and show numerical experiments on a setting involving an open Jackson network. The Newton-based SF algorithm is seen to show the best overall performance.
- Bertsekas, D. P. and Tsitsiklis, J. N. 1996. Neuro-Dynamic Programming. Athena Scientific, Belmont, MA. Google ScholarDigital Library
- Bhatnagar, S. 2005. Adaptive multivariate three-timescale stochastic approximation algorithms for simulation based optimization. ACM Trans. Model. Comput. Simul. 15, 1, 74--107. Google ScholarDigital Library
- Bhatnagar, S. 2007. Adaptive Newton-based smoothed functional algorithms for simulation optimization. ACM Trans. Model. Comput. Simul. 18, 1, 27--62. Google ScholarDigital Library
- Bhatnagar, S. 2010. An actor-critic algorithm with function approximation for discounted cost constrained Markov decision processes. Syst. Control Lett. 59, 760--766.Google ScholarCross Ref
- Bhatnagar, S. and Borkar, V. S. 1998. A two time scale stochastic approximation scheme for simulation based parametric optimization. Prob. Eng. Inf. Sci. 12, 519--531.Google ScholarCross Ref
- Bhatnagar, S. and Borkar, V. S. 2003. Multiscale chaotic SPSA and smoothed functional algorithms for simulation optimization. Simulation 79, 10, 568--580.Google ScholarCross Ref
- Bhatnagar, S., Fu, M. C., Marcus, S. I., and Bhatnagar, S. 2001. Two timescale algorithms for simulation optimization of hidden Markov models. IIE Trans. 33, 3, 245--258.Google ScholarCross Ref
- Borkar, V. S. 1997. Stochastic approximation with two timescales. Syst. Control Lett. 29, 291--294. Google ScholarDigital Library
- Borkar, V. S. 2005. An actor-critic algorithm for constrained Markov decision processes. Syst. Control Lett. 54, 207--213.Google ScholarCross Ref
- Brandiere, O. 1998. Some pathological traps for stochastic approximation. SIAM J. Control Optim. 36, 1293--1314. Google ScholarDigital Library
- Cassandras, C. G. 1993. Discrete Event Systems: Modeling and Performance Analysis. Aksen Associates, Boston, MA.Google Scholar
- Chen, H. F., Duncan, T. E., and Pasik-Duncan, B. 1999. A Kiefer-Wolfowitz algorithm with randomized differences. IEEE Trans. Auto. Cont. 44, 3, 442--453.Google ScholarCross Ref
- Hirsch, M. W. 1989. Convergent activation dynamics in continuous time networks. Neural Netw. 2, 331--349. Google ScholarDigital Library
- Ho, Y. C. and Cao, X. R. 1991. Perturbation Analysis of Discrete Event Dynamical Systems. Kluwer, Boston.Google Scholar
- Katkovnik, V. Y. and Kulchitsky, Y. 1972. Convergence of a class of random search algorithms. Autom. Remote Control 8, 1321--1326.Google Scholar
- Kushner, H. J. and Yin, G. G. 1997. Stochastic Approximation Algorithms and Applications. Springer, Berlin.Google Scholar
- Mas-Colell, A., Whinston, M. D., and Green, J. R. 1995. Microeconomic Theory. Oxford University Press, Oxford, UK.Google Scholar
- Meyn, S. P. and Tweedie, R. L. 1993. Markov Chains and Stochastic Stability. (2nd ed.), Springer, Berlin.Google Scholar
- Sadegh, P. 1997. Constrained optimization via stochastic approximation with a simultaneous perturbation gradient approximation. Automatica 33, 889--892. Google ScholarDigital Library
- Spall, J. C. 1992. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control 37, 3, 332--341.Google ScholarCross Ref
- Spall, J. C. 2000. Adaptive stochastic approximation by the simultaneous perturbation method. IEEE Trans. Autom. Control 45, 1839--1853.Google ScholarCross Ref
- Styblinski, M. A. and Tang, T.-S. 1990. Experiments in nonconvex optimization: Stochastic approximation with function smoothing and simulated annealing. Neural Netw. 3, 467--483. Google ScholarDigital Library
- Tadic, V. B., Doucet, A., and Singh, S. 2003. Two time-scale stochastic approximation for constrained stochastic optimization and constrained Markov decision problems. In Proceedings of the American Control Conference, 4736--4741.Google Scholar
- Wang, I.-J. and Spall, J. C. 2008. Stochastic optimization with inequality constraints using simultaneous perturbations and penalty functions. Int. J. Control 81, 8, 1232--1238.Google ScholarCross Ref
- Whitney, J. E. and Hill, S. D. 2001. Constrained optimization over discrete sets via SPSA with application to non-separable resource allocation. In Proceedings of the Winter Simulation Conference, 313--317. Google ScholarDigital Library
Index Terms
- Stochastic approximation algorithms for constrained optimization via simulation
Recommendations
Rate of Convergence for Constrained Stochastic Approximation Algorithms
There is a large literature on the rate of convergence problem for general unconstrained stochastic approximations. Typically, one centers the iterate $\theta_n$ about the limit point $\bar\theta$ and then normalizes by dividing by the square root of ...
A strongly sub-feasible primal-dual quasi interior-point algorithm for nonlinear inequality constrained optimization
In this paper, a primal-dual quasi interior-point algorithm for inequality constrained optimization problems is presented. At each iteration, the algorithm solves only two or three reduced systems of linear equations with the same coefficient matrix. The ...
A Stochastic Semismooth Newton Method for Nonsmooth Nonconvex Optimization
In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian ...
Comments