Gradient Extrapolated Stochastic Kriging

We introduce an approach for enhancing stochastic kriging in the setting where additional direct gradient information is available (e.g., provided by techniques such as perturbation analysis or the likelihood ratio method). The new approach, called gradient extrapolated stochastic kriging (GESK), incorporates direct gradient estimates by extrapolating additional responses. For two simplified settings, we show that GESK reduces mean squared error (MSE) compared to stochastic kriging under certain conditions on step sizes. Since extrapolation step sizes are crucial to the performance of the GESK model, we propose two different approaches to determine the step sizes: maximizing penalized likelihood and minimizing integrated mean squared error. Numerical experiments are conducted to illustrate the performance of the GESK model and to compare it with alternative approaches.

[1]  P. L’Ecuyer,et al.  A Unified View of the IPA, SF, and LR Gradient Estimation Techniques , 1990 .

[2]  Paul Glasserman,et al.  Gradient Estimation Via Perturbation Analysis , 1990 .

[3]  Leyuan Shi,et al.  Optimizing discrete event dynamic systems via the gradient surface method , 1992, Discret. Event Dyn. Syst..

[4]  Charles Leake,et al.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1994 .

[5]  Jason H. Goodfriend,et al.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1995 .

[6]  J. Alonso,et al.  Using gradients to construct cokriging approximation models for high-dimensional design optimization problems , 2002 .

[7]  Weiyu Liu,et al.  Development of gradient-enhanced kriging approximations for multidisciplinary design optimization , 2003 .

[8]  Jack P. C. Kleijnen,et al.  Robustness of Kriging when interpolating in random simulation with heterogeneous variances: Some experiments , 2005, Eur. J. Oper. Res..

[9]  Runze Li,et al.  Analysis of Computer Experiments Using Penalized Likelihood in Gaussian Kriging Models , 2005, Technometrics.

[10]  Russell R. Barton,et al.  Chapter 18 Metamodel-Based Simulation Optimization , 2006, Simulation.

[11]  Qingling Zhang,et al.  Eigenvalue Inequalities for Matrix Product , 2006, IEEE Transactions on Automatic Control.

[12]  Jack P. C. Kleijnen,et al.  Customized sequential designs for random simulation experiments: Kriging metamodeling and bootstrapping , 2008, Eur. J. Oper. Res..

[13]  Barry L. Nelson,et al.  Stochastic kriging for simulation metamodeling , 2008, 2008 Winter Simulation Conference.

[14]  M. Fu.  What you should know about simulation and derivatives , 2008 .

[15]  Russell R. Barton,et al.  Simulation optimization using metamodels , 2009, Proceedings of the 2009 Winter Simulation Conference (WSC).

[16]  Jeremy Staum,et al.  Better simulation metamodeling: The why, what, and how of stochastic kriging , 2009, Proceedings of the 2009 Winter Simulation Conference (WSC).

[17]  Wei Xie,et al.  The influence of correlation functions on stochastic kriging metamodels , 2010, Proceedings of the 2010 Winter Simulation Conference.

[18]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[19]  Jun Zhu,et al.  Penalized maximum likelihood estimation and variable selection in geostatistics , 2011, 1109.0320.

[20]  Huashuai Qu,et al.  On direct gradient enhanced simulation metamodels , 2012, Proceedings Title: Proceedings of the 2012 Winter Simulation Conference (WSC).

[21]  Xi Chen,et al.  The effects of common random numbers on stochastic kriging metamodels , 2012, TOMC.

[22]  Xi Chen,et al.  Enhancing Stochastic Kriging Metamodels with Gradient Estimators , 2013, Oper. Res..

[23]  Huashuai Qu,et al.  Regression Models Augmented with Direct Stochastic Gradient Estimators , 2014, INFORMS J. Comput..