Efficient computational algorithms for approximate optimal designs

TL;DR

This paper introduces two efficient algorithms for computing approximate D- and A-optimal designs in multi-dimensional linear regression, demonstrating their convergence and superior performance through numerical comparisons.

Contribution

The paper presents new computational algorithms for approximate D- and A-optimal designs, with proven convergence properties and improved efficiency over existing methods.

Findings

Algorithms converge to the optimal designs under certain conditions.

Proposed methods outperform some existing algorithms in numerical tests.

Approximate designs converge to continuous optimal designs.

Abstract

In this paper, we propose two simple yet efficient computational algorithms to obtain approximate optimal designs for multi-dimensional linear regression on a large variety of design spaces. We focus on the two commonly used optimal criteria, $D$- and $A$-optimal criteria. For $D$-optimality, we provide an alternative proof for the monotonic convergence for $D$-optimal criterion and propose an efficient computational algorithm to obtain the approximate $D$-optimal design. We further show that the proposed algorithm converges to the $D$-optimal design, and then prove that the approximate $D$-optimal design converges to the continuous $D$-optimal design under certain conditions. For $A$-optimality, we provide an efficient algorithm to obtain approximate $A$-optimal design and conjecture the monotonicity of the proposed algorithm. Numerical comparisons suggest that the proposed algorithms perform well and they are comparable or superior to some existing algorithms.