雙反應最佳化法延伸應用於三反應系統最佳化之研究

Optimization of Triple Response System: Using a Dual Response Based Approach

黃嘉芬(Chia-Fen Huang)

指導教授 ： 范書楷

元智大學/工程學院/工業工程與管理學系/碩士(2008年)

https://doi.org/10.6838/YZU.2008.00214

摘要

本論文針對三反應系統最佳化問題提出一新的演算流程，並且此三反應系統不限於凸(或凹)方程的二次方程模型。在本論文所提出的三反應系統中，包含主要關切的反應值(系統中亦稱為目標方程式)及兩項次要關切的反應值(系統中亦稱為等式限制式)，此外並針對模型的投入變數限制其必須符合實驗範圍(系統中亦稱為非等式限制式)。本論文提出以雙反應最佳化法為基礎之三反應系統最佳化演算法(TRSALG)來解決三反應系統中的非退化問題。此方法利用數學規劃之Lagrangian model將三反應系統中之限制式轉換成目標式，並以求解Lagrange multipliers 的方式作為主要的演算流程。其中使用Hooke and Jeeves 的pattern search 來求解等式限制式之 Lagrange multipliers並同時以Trust Region (TR) method來求解非等式限制式之Lagrange multiplier。其中，Trust Region (TR) method利用非線性規劃問題之最佳化條件(optimality conditions)進行運算，故可保證TRSALG最後求得的解為全域最佳解。此外TRSALG亦能偵測退化問題的發生，以確保此方法所求得之解必定為非退化問題的全域最佳解。最後，利用文獻中的多反應最佳化問題延伸至本論文之三反應系統中，並以梯度(gradient)演算法來測試及比較TRSALG的演算效用。

關鍵字

三反應系統 ； 全域最佳化 ； 非線性規劃 ； Trust Region (TR) ； Hooke-Jeeves'search method

並列摘要

This paper presents a new computing procedure for the global optimization of the triple response system (TRS) where the response functions are nonconvex (nonconcave) quadratics and the input factors satisfy a radial region of interest. The TRS arising from response surface modeling can be approximated using a nonlinear mathematical program involving one primary (objective) function and two secondary (constraints) functions. An optimization algorithm named triple response surface algorithm (TRSALG) is proposed to determine the global optimum for the nondegenerate TRS. In TRSALG, the Lagrange multipliers of target (secondary) functions are computed by using the Hooke-Jeeves’ search method, and the Lagrange multiplier of the radial constraint is located by using the trust region (TR) method at the same time. To ensure global optimality that can be attained by TRSALG, included is the means for detecting the degenerate case. In the field of numerical optimization, as the family of TR approach always exhibits excellent mathematical properties during optimization steps, thus the proposed algorithm can guarantee the global optimal solution where the optimality conditions are satisfied for the nondegenerate TRS. The computing procedure is illustrated in terms of examples found in the quality literatures where the comparison results with a gradient based method are used to calibrate TRSALG.

並列關鍵字

Triple response system (TRS) ； Global optimization ； Nonlinear programming (NLP) ； Trust region (TR) ； Hooke-Jeeves'search method

參考文獻

Bazaraa, M. S., Sherali, H. D., and Shetty C. M., Nonlinear Programming: Theory and Algorithms, ( Ed.) A John Wiley & Sons, Inc., New York, 1993.

Carlyle, W. M., Montgomery, D. C., and Runger, G. C., Optimization problems and methods in quality control and improvement, Journal of Quality Technology, 32(1), 2000, pp. 1-17.

Draper, N. R., “Ridge Analysis” of response surfaces, Technometrics, 5(4), 1963, pp.469-479.

Derringer, G. and Suich, R., Simultaneous optimization of several response variables, Journal of Quality Technology, 12, 1980, pp. 214-219.

Del Castillo, E. and Montgomery, D. C., A nonlinear programming solution to the dual response problem, Journal of Quality Technology, 25, 1993, pp. 199-204.