A Study of Nelder-Mead Simplex Search Method for Solving Unconstrained and Stochastic Optimization Problems

• Main Authors: Erwie Zahara, 何怡偉
• Other Authors: Shu-Kai S. Fan
• Format: Others
• Language: en_US
• Published: 2003
• Online Access: http://ndltd.ncl.edu.tw/handle/64436961342254831346

博士 === 元智大學 === 工業工程與管理學系 === 92 === The first topic of this dissertation is concerned with a hybrid, unconstrained optimization algorithm based on Nelder-Mead simplex method (NM) and particle swarm optimization (PSO). The hybrid NM-PSO incorporates concepts from the NM and PSO, and is very easy to implement in practice and does not require gradient computation. A modification made to both Nelder-Mead simplex method and particle swarm optimization succeeds in producing faster and more accurate convergence, as born out by empirical evaluations of a suit of 20 test functions. The new algorithm proves to be extremely effective and efficient at locating best-practice optimal solutions. Therefore, the hybrid optimization scheme is applied to problems involving multiple thresholding by the criteria of (1) Otsu’s minimum within-group variance and (2) Gaussian function fitting. Six example images are used to test and illustrate the three different methods: the Otsu’s method; the NM-PSO-Otsu method and the NM-PSO-Curve method. The experimental results show that the NM-PSO-Otsu could expedite the Otsu’s method efficiently to a great extent in the case of multilevel thresholding, and that the NM-PSO-Curve method could provide better effectiveness than the Otsu’s method in the context of visualization, object size and image contrast. The second topic, an enhanced NM, is proposed in this dissertation to explore the terrains of empirical (or experimental) optimization adaptively where the known response surface function is contaminated by white-noise errors. Modifications to basic operations of NM are made primarily according to some statistical process control (SPC) statistics used in estimating response variation and confidence bands for mean responses. A series of graphical illustrations are presented to give insight into the way the new simplex-search-type approach accurately anchors the true optimum point in noisy environments. As evidenced by a wide variety of simulation studies on the published response functions, the new method proves to perform much better than two recent modifications of NM in solution quality when applied to the stochastic response surface optimization problems. As such, the new method could serve as a useful tool for process recipe optimization in noisy semiconductor manufacturing environments. Finally, the chemical mechanical planarization (CMP) process, a turnkey process during semiconductor fabrication, is simulated from batch to batch based on the real-data equipment model and the presented algorithm is employed to seek the optimal recipe profile while processing each batch of wafers sequentially through the CMP tool.