Simulation-Based Optimization by New stochastic Approximation Algorithm

This paper proposes one new stochastic approximation algorithm for solving simulation-based optimization problems. It employs a weighted combination of two independent current noisy gradient measurements as the iterative direction. It can be regarded as a stochastic approximation algorithm with a special matrix step size. The almost sure convergence and the asymptotic rate of convergence of the new algorithm are established. Our numerical experiments show that it outperforms the classical Robbins–Monro (RM) algorithm and several other existing algorithms for one noisy nonlinear function minimization problem, several unconstrained optimization problems and one typical simulation-based optimization problem, i.e., (s, S)-inventory problem.

[1]  H. Robbins,et al.  A CONVERGENCE THEOREM FOR NON NEGATIVE ALMOST SUPERMARTINGALES AND SOME APPLICATIONS**Research supported by NIH Grant 5-R01-GM-16895-03 and ONR Grant N00014-67-A-0108-0018. , 1971 .

[2]  Marc S. Meketon,et al.  Optimization in simulation: a survey of recent results , 1987, WSC '87.

[3]  J. Spall.  Multivariate stochastic approximation using a simultaneous perturbation gradient approximation , 1992 .

[4]  M. Hossein Safizadeh,et al.  Optimization in simulation: Current issues and the future outlook , 1990 .

[5]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[6]  Farhad Azadivar,et al.  A tutorial on simulation optimization , 1992, WSC '92.

[7]  Zi Xu,et al.  New combinatorial direction stochastic approximation algorithms , 2013, Optim. Methods Softw..

[8]  S. Andradóttir.  A Scaled Stochastic Approximation Algorithm , 1996 .

[9]  H. Kesten.  Accelerated Stochastic Approximation , 1958 .

[10]  V. Fabian.  On Asymptotic Normality in Stochastic Approximation , 1968 .

[11]  Dai,et al.  A Stochastic Approximation Frame Algorithm with Adaptive Directions , 2008 .

[12]  James C. Spall.  Feedback and Weighting Mechanisms for Improving Jacobian Estimates in the Adaptive Simultaneous Perturbation Algorithm , 2009, IEEE Trans. Autom. Control..

[13]  Bernard Delyon,et al.  Accelerated Stochastic Approximation , 1993, SIAM J. Optim..

[14]  J. Blum.  Multidimensional Stochastic Approximation Methods , 1954 .

[15]  Michael C. Fu,et al.  Optimization via simulation: A review , 1994, Ann. Oper. Res..

[16]  James C. Spall,et al.  Adaptive stochastic approximation by the simultaneous perturbation method , 2000, IEEE Trans. Autom. Control..

[17]  Stephen M. Robinson,et al.  Sample-path optimization of convex stochastic performance functions , 1996, Math. Program..

[18]  C. Kao,et al.  A modified quasi-newton method for optimization in simulation , 1997 .

[19]  James C. Spall,et al.  Introduction to stochastic search and optimization - estimation, simulation, and control , 2003, Wiley-Interscience series in discrete mathematics and optimization.

[20]  S. Jacobson,et al.  Techniques for simulation response optimization , 1989 .

[21]  J. S. Ivey,et al.  Nelder-Mead simplex modifications for simulation optimization , 1996 .

[22]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

[23]  Stephen M. Robinson,et al.  Analysis of Sample-Path Optimization , 1996, Math. Oper. Res..

[24]  Nicol N. Schraudolph,et al.  Combining Conjugate Direction Methods with Stochastic Approximation of Gradients , 2003, AISTATS.

[25]  Zi Xu,et al.  A combined direction stochastic approximation algorithm , 2010, Optim. Lett..

[26]  J. Kiefer,et al.  Stochastic Estimation of the Maximum of a Regression Function , 1952 .

[27]  MICHAEL C. FU,et al.  Techniques for optimization via simulation: an experimental study on an (s,S) inventory system , 1997 .