在財務工程的領域，條件風險值(Conditional Value at Risk)時常被用來量測及管理風險。本研究考慮含有條件風險值限制式之最佳化問題，由於條件風險值相當具有隨機性與複雜性，該限制式只能利用隨機模擬來估計，同時增加了這種問題的解題難度。在這篇論文中，我們提出了一個稱為適應性模式搜尋(Adaptive Pattern Search)的新演算法，此方法是以模式搜尋法為基礎，並在結構上進行部分修改，比如決定前進方向的方式，以確保能夠更有效率地解決問題。此外，我們應用拉丁超立方體抽樣(Latin Hypercube Sampling)來決定一組起始解，以確保演算法得以從較佳的出發點來進行後續最佳解的搜尋。在本研究的數值研究中也顯示這是一個可行且有效的方法值得深入研究。

Conditional value at risk (CVaR) is often used to measure and manage risks in financial engineering. In this paper, we consider optimization problems with CVaR constraints. Due to the profound randomness and complexities, the CVaR constraints can only be estimated by stochastic simulation. We propose a new algorithm, called adaptive pattern search (APS) based on the pattern search method in the literature but further incorporates efficient modifications, including the determination of the moving directions, to enable the problem to be solved efficiently. Moreover, we applied Latin hypercube sampling (LHS) to determine a set of solutions for the algorithm to get started for better search of the optimal solution. A numerical study shows that the proposed algorithm is efficient and is worthy of further investigation.

摘要 I
Abstract II
圖目錄 V
表目錄 VI
第一章　緒論 1
1.1研究背景與動機 1
1.2研究目的 2
1.3論文架構 3
第二章　文獻回顧 5
2.1條件風險值 5
2.2無微分最佳化方法 9
第三章　數學模型 12
3.1 問題定義 12
3.2一般化數學模型 13
第四章　求解方法 14
4.1條件風險值的估計 15
　　4.1.1原始蒙地卡羅模擬法(Crude Monte Carlo Simulation Method) 15
4.2模式搜尋法(Pattern Search Method) 17
　　4.2.1 探索步(Exploratory Move) 19
　　4.2.2 模式步(Pattern Move) 19
4.3適應性模式搜尋法(Adaptive Pattern Search Method) 21
　　4.3.1拉丁超立方體抽樣(Latin Hypercube Sampling) 21
　　4.3.2改良版探索步(Revised Exploratory Move) 22
　　4.3.3適應性模式搜尋法整體架構與說明 25
第五章　數值實驗 27
5.1 測試函數 27
5.2 比較指標 29
5.3 數值結果 29
第六章　例題驗證 35
6.1 數學模型 35
6.2 例題結果 36
第七章　結論與未來研究 41
參考文獻 42

Andersson, F., Mausser, H., Rosen, D., and Uryasev, S. (2001). Credit risk optimization with conditional value at risk criterion. Mathematical Programming, vol. 89, no.2, pp. 273–291.
Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (1999). Coherent measures of risk. Mathematical Finance, vol. 9, no. 3, pp. 203–228.
Avramidis, A.N., and Wilson, J. R. (1998). Correlation-induction techniques for estimating quantiles in simulation experiments. Operations Research, vol. 46, no. 4, pp. 574–591.
Bahadur, R. (1966). A note on quantiles in large samples. Annals of Mathematical Statistics, vol. 37, pp. 577–580.
Basova, H. G., Rockafellar, R. T., and Royset, J. O. (2011). A computational study of the buffered failure probability in reliability-based design optimization. in Proceedings of the 11th Conference on Application of Statistics and Probability in Civil Engineering.
Chow, Y., and Ghavamzadeh, M. (2014). Algorithms for CVaR optimization in MDPs. Advances in Neural Information Processing Systems, vol. 27, pp. 3509–3517.
Conn, A. R., Gould, N. I. M., and Toint, Ph. L. (2000). Trust-Region Methods. MPS-SIAM series on optimization, Philadelphia: Society for Industrial Mathematics.
Gilmore, P., and Kelley, C. T. (1995). An implicit filtering algorithm for optimization of functions with many local minima. SIAM Journal on Optimization, vol. 5, no. 2, pp. 269–285.
Glasserman, P., Heidelberger, P., and Shahabuddin, P. (2000). Variance reduction techniques for estimating value-at-risk. Management Science, vol. 46, no. 10, pp. 1349–1364.
Glasserman, P., Heidelberger, P., and Shahabuddin, P. (2002). Portfolio value-at-risk with heavy-tailed risk factors. Mathematical Finance, vol. 12, no. 3, pp. 239–269.
Glynn, P. W. (1996). Importance sampling for Monte Carlo estimation of quantiles. in Proceedings of 1996 Second International Workshop on Mathematical Methods in Stochastic Simulation and Experimental Design, pp. 180–185.
Hesterberg, T. C., and Nelson, B. L. (1998). Control variates for probability and quantile estimation. Management Science, vol. 44, pp. 1295–1312.
Hong, L. J., and Liu, G. (2009). Simulating sensitivities of conditional value-at-risk. Management Science, vol. 55, no. 2, pp. 281–293.
Hong, L. J., Hu, Z., and Liu, G. (2014). Monte Carlo methods for value-at-risk and conditional value-at-risk: a review. ACM Transactions on Modeling and Computer Simulation, vol. 24, no. 4, article no. 22.
Hooke-Robert and Jeeves, T. A. (1961). "Direct Search" Solution of numerical and statistical problems. Journal of the ACM, vol. 8, no. 2, pp. 212-229.
Hsu, J. C., and Nelson, B. L. (1990). Control variates for quantile estimation. Management Science, vol. 36, pp. 835–851.
Iyengar, G., and Ma, A. K. C. (2013). Fast gradient descent method for mean-CVaR optimization. Annals of Operations Research, vol. 205, pp. 203–212.
Jorion, P. (1997). Value at risk, New York: McGraw-Hill.
Krokhmal, P., Palmquist, J., and Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, vol. 4, no. 2, pp. 11–27.
Lim, C., Sherali, H. D., and Uryasev, S. (2010). Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Computational Optimization and Applications, vol. 46, pp. 391–415.
Mckay, M. D., Beckman, R. J., and Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, vol. 21, no. 2, pp. 239-245.
Mojtaba Meftahi and Saeed Heydarzadeh Jazi (2012). A new hybrid algorithm of pattern search and ABC for optimization. in 16th CSI International Symposium on Artificial Intelligence and Signal Processing, pp. 403-406.
More, J.J., Garbow, B.S., Hillstrom, K.E., 1981. Testing unconstrainted optimization software. ACM Transactions on Mathematical Software, vol. 7, no. 1, pp. 17-41.
Nelder, J. A., and Mead, R. (1965). A simplex method for function minimization. The Computer Journal, vol. 7, no. 4, pp. 308–313.
Nocedal, J., and Wright, Stephen J. (2006). Numerical optimization, 2nd ed. New York: Springer, ch. 9, pp. 229-234.
Pflug, G. (2000). Some remarks on the value-at-risk and conditional value-at-risk. Probabilistic Constrained Optimization: Methodology and Applications, pp. 272–281.
Prashanth, L. A. (2014). Policy gradients for CVaR-constrained MDPs. in Proceedings of 25th International Conference on Algorithmic Learning Theory, pp. 155-169.
Rockafellar, R. T., and Royset, J. O. (2010). On buffered failure probability in design and optimization of structures. Reliability Engineering & System Safety, vol. 95, no. 5, pp. 499–510.
Rockafellar, R. T., and Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, vol. 2, no. 3, pp. 21–41.
Smith, C. S. (1962). The automatic computation of maximum likelihood estimates. N.C.B. Sci. Dept. Report, SC846/MR/40.
Sun, L., and Hong, L. J. (2010). Asymptotic representations for importance-sampling estimators of value-at-risk and conditional value-at-risk. Operations Research Letters, vol. 38, pp. 246–251.
Trindade, A. A., Uryasev, S., Shapiro, A., and Zrazhevsky, G. (2007). Financial prediction with constrained tail risk. Journal of Banking and Finance, vol. 31, pp. 3524–3538.