用内点法计算最优试验设计
Computing Optimal Experimental Designs via Interior Point Method
摘要：
本文研究了一类光滑凸最优性准则下的最优试验设计问题，包括经典的A-、D-和p次平均准则。特别地，我们提出了一个内点（IP）方法，并建立其全局收敛性。此外，通过利用上述最优性准则的Hessian矩阵的结构，我们导出了计算其秩的显式公式。利用这一结果，我们然后表明，牛顿方向产生的IP方法可以有效地计算通过谢尔曼-莫里森-伍德伯里公式时，矩矩阵的大小是相对于样本量小。最后，我们将我们的IP方法与Silvey等人[29]介绍的广泛使用的乘法算法进行比较。计算结果表明，IP方法在速度和解的质量上都优于乘性算法。
英文摘要：
In this paper, we study optimal experimental design problems with a broad class of smooth convex optimality criteria, including the classical A-, D- and p th mean criterion. In particular, we propose an interior point (IP) method for them and establish its global convergence. Furthermore, by exploiting the structure of the Hessian matrix of the aforementioned optimality criteria, we derive an explicit formula for computing its rank. Using this result, we then show that the Newton direction arising in the IP method can be computed efficiently via Sherman-Morrison-Woodbury formula when the size of the moment matrix is small relative to the sample size. Finally, we compare our IP method with the widely used multiplicative algorithm introduced by Silvey et al. [29]. The computational results show that the IP method generally outperforms the multiplicative algorithm both in speed and solution quality.
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