Enhancement of the downhill simplex method of optimization
Enhancement of the downhill simplex method of optimization
复制标题
下坡单纯形优化方法的增强
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
R. John Koshel
中科院分区：
文献类型：
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作者：
R. John Koshel
单纯形优化法是一种实现函数极小化的“几何”方法。标准算法使用任意值的确定性因素，描述的“运动”的单形在价值空间。虽然它是一种鲁棒的优化方法，但收敛到局部最小值的速度相对较慢。然而，它的稳定性和缺乏使用的衍生工具，使它有用的光学设计优化，特别是在照明领域。本文介绍了优化单纯形优化器的性能的初步努力。这种增强是通过优化各种控制因素来实现的：alpha（反射）、beta（收缩）和gamma（扩展）。这一努力是通过研究最佳设计的“最终游戏”来完成的，即，品质因数空间的形状在局部最小值附近在N维上是抛物线形的。控制因子优化的品质因数是与使用标准控制因子的相同情况相比实现解决方案的迭代次数。该优化是针对N阶等于2至15的抛物线威尔斯井进行的。在这项研究中，它表明，正确选择的控制因素，可以实现高达35%的改善收敛。提出了使用梯度加权和包含附加控制因子的技术。
The downhill simplex method of optimization is a "geometric" method to achieve function minimization. The standard algorithm uses arbitrary values for the deterministic factors that describe the "movement" of the simplex in the merit space. While it is a robust method of optimization, it is relatively slow to converge to local minima. However, its stability and the lack of use of derivatives make it useful for optical design optimization, especially for the field of illumination. This paper describes preliminary efforts of optimizing the performance of the simplex optimizer. This enhancement is accomplished by optimizing the various control factors: alpha (reflection), beta (contraction), and gamma (expansion). This effort is accomplished by investigating the "end game" of optimal design, i.e., the shape of the figure of merit space is parabolic in N-dimensions near local minima. The figure of merit for the control factor optimization is the number of iterations to achieve a solution in comparison to the same case using the standard control factors. This optimization is done for parabolic wells of order N equals 2 to 15. In this study it is shown that with the correct choice of the control factors, one can achieve up to a 35% improvement in convergence. Techniques using gradient weighting and the inclusion of additional control factors are proposed.