CVXSADes: a stochastic algorithm for constructing optimal exact regression designs with single or multiple objectives

TL;DR

This paper introduces CVXSADes, a stochastic algorithm that efficiently constructs optimal exact regression designs from approximate designs, applicable to single or multiple objectives, ensuring high statistical efficiency and adaptability for various design sizes.

Contribution

The authors develop a unified, systematic method combining convex optimization and simulated annealing to construct optimal exact designs for any run size, improving over rounding methods and prior ad hoc approaches.

Findings

Design efficiency approaches 100% as the number of points increases

Method effectively transforms approximate designs into exact ones

Algorithm performs well across various design problems

Abstract

We propose an algorithm to construct optimal exact designs (EDs). Most of the work in the optimal regression design literature focuses on the approximate design (AD) paradigm due to its desired properties, including the optimality verification conditions derived by Kiefer (1959, 1974). ADs may have unbalanced weights, and practitioners may have difficulty implementing them with a designated run size $n$. Some EDs are constructed using rounding methods to get an integer number of runs at each support point of an AD, but this approach may not yield optimal results. To construct EDs, one may need to perform new combinatorial constructions for each $n$, and there is no unified approach to construct them. Therefore, we develop a systematic way to construct EDs for any given $n$. Our method can transform ADs into EDs while retaining high statistical efficiency in two steps. The first step involves constructing an AD by utilizing the convex nature of many design criteria. The second step employs a simulated annealing algorithm to search for the ED stochastically. Through several applications, we demonstrate the utility of our method for various design problems. Additionally, we show that the design efficiency approaches unity as the number of design points increases.