一般化縮減信賴域搜尋及其在多目標統計模型最佳化之應用

Abstract

一般化縮減梯度(Generalized Reduced Gradient)法是一個廣受喜愛的非線性規劃問題解法，但於具有四次目標式之多目標統計最佳化(Statistical Multi-objective Optimization)問題中，一般化縮減梯度法容易出現搜尋路徑曲折(Zigzagging)的現象。於本研究中，我們改善了信賴域(Trust Region)搜尋法，並發展了一般化縮減信賴域(Generalized Reduced Trust Region)搜尋法。此方法結合了一般化縮減梯度與信賴域搜尋法，將具有限制式的非線性規劃問題，轉化成由非基礎變數(Nonbasic variable)所構成的不具現制式的非線性規劃問題，並且在縮減空間(Reduced Space)中獲得最佳改善的方向，且於案例中克服了一般化縮減梯度法的缺點，此外，我們也結合了一般化縮減信賴域搜尋法與Zoutendijk’s搜尋法以改善搜尋效果。最後，為了驗證該演算法的成效，我們利用一個眾所皆知且具四次目標式的測試問題：Rosenbrock’s function與三個案例來測試，第一個案例是關於半導體可製造性設計(DFM)之問題，而第二個案例是半導體供應鏈穩健配置之案例，最後一個案例為半導體製造過程中，臨界尺寸均勻度(CDU)在軌道系統之曝光後烘烤(PEB)步驟下之最佳化。經由與商業套裝軟體Lingo的結果比較，我們可以在相似的計算時間內獲得同樣甚至更好的最佳解。

“Generalized Reduced Gradient” method is a popular NLP method, but it often incurs a zigzagging search path especially for the statistical multi-objective optimization (SMOO) problem where the objective function is a quartic function. In this study, we improve the “Trust Region (TR)” search method and develop the “Generalized Reduced Trust Region” (GRT) search method which combines the GRG method and the improved TR method. The GRT search transforms the constrained NLP problem to an unconstrained NLP problem consisting of only the nonbasic variables and searches the best improving direction in the reduced space. The proposed method is shown to overcome the zigzagging problem of the GRG method. To verify the performance of our methods, we study a well know test problem and three cases. The test problem is called Rosenbrock’s function which has a quartic objective function with two decision variables. The first case is a semiconductor design for manufacturing (DFM) problem. The second case is the problem to configure a robust semiconductor supply chain. The final case is the “Track System PEB CDU Optimization”. Compared against the result of the commercial software “Lingo”, the same or better solutions are obtained by our methods with comparable computation time.