Stochastic approximations
Manuscript in preparation
Zago, D., Capizzi, G., & Qiu, P. (2024). Optimal constrained design of control charts using stochastic approximations. Journal of Quality Technology. https://doi.org/10.1080/00224065.2024.2323585
Description
In statistical process monitoring, control charts typically depend on a set of tuning parameters $\mathbf{\zeta}$ besides its control limit(s). In a specific application, a control chart is often designed for detecting a target process distributional shift as efficiently as possible.
However, explicit solutions $\mathbf{\zeta}^*$ for such a design are unavailable for most control charts, and thus numerical optimization methods are needed. Two key problems in this setting are computational cost and scalability to multivariate tuning parameters.
In this work we present an algorithm that solves the doubly-stochastic constrained optimization problem
\[\begin{aligned} \mathbf{g}(\mathbf{\zeta}^*) &= \frac{\partial \mathbb{E}_1\{\text{RL}[\mathbf{\zeta}, h(\mathbf{\zeta})]\}} {\partial \mathbf{\zeta}}\Big|_{\mathbf{\zeta}=\mathbf{\zeta}^*} = \mathbf{0},\\ \text{s.t. } & \mathbb{E}_0\{\text{RL}[\mathbf{\zeta}^*, h(\mathbf{\zeta}^*)]\} =\text{ARL}_0. \end{aligned}\]Comparing this algorithm to more traditional optimization methods, such as grid search and Nelder-Mead optimization, shows a remarkable improvement in computational efficiency.
