過去的模擬最佳化問題大都以全部模擬結果之平均值作為績效指標，而本研究所探討的目標式—條件期望值也是另一個重要的績效指標。條件期望值只考慮兩個分量之間的期望值而非整個母體的平均值，現階段常被用於財務工程中來計算風險值。而本研究則將過去著重在尾端期望值的目標式轉換成在任意兩個分量之間的期望值，並發展一套演算法來進行求解。由於本研究之問題為具有隨機性，故利用蒙地卡羅模擬來估計條件期望值。而為了在有限的資源下有效率地找到最佳解，本研究以林星妤(2018)所提出的條件期望值最佳化架構—AGLS-CE(Adaptive Global and Local search for Conditional Expectation)為基礎，提出了APGHLS-CE (Adaptive Particle Global and Hyperbox Local search for Conditional Expectation)。該方法不需要估計梯度，僅需在每一次迭代決定最佳信賴區域以進行區域搜尋，並同時進行全域搜尋。在區域搜尋的部分，參考了Jie Xu et al.所提出Adaptive Hyperbox Algorithm 來決定最佳信賴區間，並在其區間進行拉丁超立方體抽樣。在全域搜尋的部分，本研究參考了粒子群演算法(Particle Swarm Optimization)，並加入跳脫區最佳解的機制以確保能不斷更新最佳解。在模擬資源的分配部分，為了節省模擬資源，本研究在搜尋法的部分結合了OCBA-1(Optimal Computing Budget Allocation-1)來將模擬資源分配給較有潛力的解。最後，我們將用不同方程式與實際案例來檢測本研究所提出的方法之效率，並與近年提出之離散型條件期望值最佳化方法來進行比較。實驗結果顯示本研究所提出之方法可行且有效，值得深入研究。

While most past research in optimization via simulation examines a mean-based performance metric, conditional expectation is another important kind of performance measure. To date, this measure has been most widely-used in the financial field. This research develops an algorithm to efficiently optimize the conditional expectation between two quantiles of a distribution. As the problem is assumed to include randomness, Monte Carlo simulation methods are applied to estimate the conditional expectation. In order to find the optimal solution efficiently, we propose a new optimization framework called APGHLS-CE (Adaptive Particle Global and Hyperbox Local search for Conditional Expectation). This method is gradient-free and does global search and local search simultaneously in each iteration based on the concept of the neighborhood to find the optimal solution. In the local search, we introduce the concept of Adaptive Hyperbox Algorithm proposed by Jie Xu et.al. to do sampling in most promising area(MPA) by Latin Hypercube Sampling(LHS) in each iteration. In the global search part, we combine the concept of Particle Swarm Optimization (PSO) and design a procedure to prevent our method from local convergence. Finally, in our numerical study, we use some different functions and case study to compare the performance of our method with other optimization framework for discrete conditional expectation proposed recently. The experiment result shows efficiency and efficacy of our method, which is worth further research.

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