隨機最佳化問題存在於許多領域上，舉凡財金、醫療、電子、製藥產業等，所謂的隨機問題為目標式內含著不確定性，由於不確定性的存在使得在搜尋全域性最佳解的過程中十分地困難，因此本研究目的為發展新的演算法以解決隨機最佳化問題。發展之演算法為使用全域性的搜索方式找出最佳解，其架構為建構序列性的克利金反應曲面以預測目標值，並結合了核密度估計的方式以建構最佳解所在位置之機率密度函數讓我們可以有效地搜尋到最佳解。數值實驗以及個案實證結果均顯示此方法能夠有效地解決隨機最佳化的問題。

Stochastic global optimization refers an iterative procedure in attempt to find the global optima in the parameter space when the objective function can be estimated with noise. Due to the noise inherent in the objective value, the problem is difficult to be solved, especially when the time given to solve the problem is limited, which is usually the case in practice. In this research, we propose a framework that allows the stochastic global optimization problem to be solved efficiently. The proposed framework sequentially builds a Kriging metamodel based on kernel density estimation for predicting the functional behavior of the objective function and solves for the optimal solution of the metamodel. Numerical experiments show that its efficiency is satisfactory.

Ahmed, M. Y. M., & Qin, N. (2009). Comparison of response surface and kriging surrogates in aerodynamic design optimization of hypersonic spiked blunt bodies. 13th International Conference on Aerospace Sciences and Aviation Technology.
Anderson, E. J., & Ferris, M. C. (2001). A direct search algorithm for optimization with noisy function evaluations. SIAM Journal on Optimization, 11(3), 837-857.
Angün, E., & Kleijnen, J. (2012). An asymptotic test of optimality conditions in multiresponse simulation optimization. INFORMS Journal on Computing, 24(1), 53-65.
Ankenman, B., Nelson, B. L., & Staum, J. (2010). Stochastic kriging for simulation metamodeling. Operations Research, 58(2), 371-382.
Bäck, T. (1996). Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press.
Bansal, S. (2002). Promise and problems of simulation technology in SCM domain. Proceedings of the 2002 Winter Simulation Conference, 2, 1831-1837.
Bierens, H. J. (1987). Kernel estimators of regression functions. Advances in Econometrics: Fifth World Congress, 1, 99-144.
Booker, A. J., Dennis Jr, J. E., Frank, P. D., Serafini, D. B., & Torczon, V. (1998). Optimization using surrogate objectives on a helicopter test example. Computational Methods for Optimal Design and Control, 49-58.
Buttrill, Carey S., P. Douglas Arbuckle, & Keith D. Hoffler. (1992). Simulation model of a twin-tail, high performance airplane. National Aeronautics and Space Administration, Langley Research Center.
Chang, K. H. (2012). Stochastic Nelder–Mead simplex method–a new globally convergent direct search method for simulation optimization. European Journal of Operational Research, 220(3), 684-694.
Cressie, N. A., & Cassie, N. A. (1993). Statistics for spatial data. New York: Wiley.
Elgammal, A., Duraiswami, R., Harwood, D., & Davis, L. S. (2002). Background and foreground modeling using nonparametric kernel density estimation for visual surveillance. Proceedings of the IEEE, 90(7), 1151-1163.
Fu, M. C. (2002). Optimization for simulation: theory vs. practice. INFORMS Journal on Computing, 14(3), 192-215.
Fu, M. C. (2006). Gradient estimation. Handbooks in Operations Research and Management Science, 13, 575-616.
Fang, K. T., Li, R., & Sudjianto, A. (2005). Design and modeling for computer experiments. CRC Press.
Ghosh, G. S., & Carriazo-Osorio, F. (2007). Bayesian and frequentist approaches to hedonic modeling in a geo-statistical framework. Selected Paper Presented at the American Agricultural Economics Association Annual Meetings.
Genschel, U., & Meeker, W. Q. (2010). A comparison of maximum likelihood and median-rank regression for Weibull estimation. Quality Engineering, 22(4), 236-255.
Glover, F. (1986). Future paths for integer programming and links to artificial intelligence. Computers and Operations Research, 13(5), 533-549.
Goldberg, D. E., & Holland, J. H. (1988). Genetic algorithms and machine learning. Machine Learning, 3(2), 95-99.
Goovaerts, P. (1997). Geostatistics for natural resources evaluation. Oxford University Press.
Gu, L. (2001). A comparison of polynomial based regression models in vehicle safety analysis. ASME Design Engineering Technical Conferences, Paper No.: DETC/DAC-21083.
Hooke, R., & Jeeves, T. A. (1961). Direct search solution of numerical and statistical problems. Journal of the ACM, 8(2), 212-229.
Huang, D., Allen, T. T., Notz, W. I., & Zeng, N. (2006). Global optimization of stochastic black-box systems via sequential kriging meta-models. Journal of Global Optimization, 34(3), 441-466.
Jansson, C., & Knüppel, O. (1994). Numerical results for a self-validating global optimization method. Technical Report 94.1, Technical University of Hamburg.
Jones, D. R., Schonlau, M., & Welch, W. J. (1998). Efficient global optimization of expensive black-box functions. Journal of Global Optimization, 13(4), 455-492.
Kenndy, J., & Eberhart, R. C. (1995). Particle swarm optimization. Proceedings of IEEE International Conference on Neural Networks, 4, 1942-1948.
Kirkpatrick, S., & Vecchi, M. P. (1983). Optimization by simmulated annealing. Science, 220(4598), 671-680.
Koehler, J. R., & Owen, A. B. (1996). Computer experiments. Handbook of Statistics, 13(13), 261-308.
Kolda, T. G., Lewis, R. M., & Torczon, V. (2003). Optimization by direct search: new perspectives on some classical and modern methods. SIAM Review, 45(3), 385-482.
Matheron, G. (1963). Principles of geostatistics. Economic Geology, 58(8), 1246-1266.
Moré, J. J., Garbow, B. S., & Hillstrom, K. E. (1981). Testing unconstrained optimization software. ACM Transactions on Mathematical Software, 7(1), 17-41.
Myers, R. H., Montgomery, D. C., & Anderson-Cook, C. M. (2009). Response surface methodology: process and product optimization using designed experiments. John Wiley & Sons.
Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. The Computer Journal, 7(4), 308-313.
Parzen, E. (1962). On estimation of a probability density function and mode. The Annals of Mathematical Statistics, 1065-1076.
Ripley, B. D. (2005). Spatial statistics. John Wiley & Sons.
Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics, 27(3), 832-837.
Santner, T. J., Williams, B. J., & Notz, W. (2003). The design and analysis of computer experiments. Springer Science & Business Media.
Scott, D. W. (2009). Multivariate density estimation: theory, practice, and visualization. John Wiley & Sons.
Shang, Y. W., & Qiu, Y. H. (2006). A note on the extended Rosenbrock function. Evolutionary Computation, 14(1), 119-126.
Shapiro, A., & Dentcheva, D. (2014). Lectures on stochastic programming: modeling and theory. SIAM.
Silverman, B. W. (1986). Density estimation for statistics and data analysis. CRC press.
Simpson, T. W., Booker, A. J., Ghosh, D., Giunta, A. A., Koch, P. N., & Yang, R. J. (2004). Approximation methods in multidisciplinary analysis and optimization: a panel discussion. Structural and Multidisciplinary Optimization, 27(5), 302-313.
Spall, J. C. (1987). A stochastic approximation technique for generating maximum likelihood parameter estimates. American Control Conference, 1161-1167.
Spall, J. C. (1992). Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. Automatic Control, IEEE Transactions on, 37(3), 332-341.
Spendley, W. G. R. F. R., Hext, G. R., & Himsworth, F. R. (1962). Sequential application of simplex designs in optimisation and evolutionary operation. Technometrics, 4(4), 441-461.
Stein, M. L. (1999). Interpolation of spatial data: some theory for kriging. Springer Science & Business Media.
Subbotin, Y. N. (2001). Spline interpolation. Encyclopedia of Mathematics. Available at: http://www.encyclopediaofmath.org/index.php/Spline_interpolation. Accessed June 09, 2015
Tekin, E., & Sabuncuoglu, I. (2004). Simulation optimization: a comprehensive review on theory and applications. IIE Transactions, 36(11), 1067-1081.
Terrell, G. R., & Scott, D. W. (1992). Variable kernel density estimation. The Annals of Statistics, 1236-1265.
Van Beers, W. C., & Kleijnen, J. P. (2003). Kriging for interpolation in random simulation. Journal of the Operational Research Society, 54(3), 255-262.