Finding High-Dimensional D-Optimal Designs for Logistic Models via Differential Evolution

Abstract

D-optimal designs are frequently used in controlled experiments to obtain the most accurate estimate of model parameters at minimal cost. Finding them can be a challenging task, especially when there are many factors in a nonlinear model. As the number of factors becomes large and interacts with one another, there are many more variables to optimize and the D-optimal design problem becomes high-dimensional and non-separable. Consequently, premature convergence issues arise. Candidate solutions get trapped in the local optima, and the classical gradient-based optimization approaches to search for the D-optimal designs rarely succeed. We propose a specially designed version of differential evolution (DE), which is a representative gradient-free optimization approach to solve such high-dimensional optimization problems. The proposed specially designed DE uses a new novelty-based mutation strategy to explore the various regions in the search space. The exploration of the regions will be carried out differently from the previously explored regions, and the diversity of the population can be preserved. The proposed novelty-based mutation strategy is collaborated with two common DE mutation strategies to balance exploration and exploitation at the early or medium stage of the evolution. Additionally, we adapt the control parameters of DE as the evolution proceeds. Using the logistic models with several factors on various design spaces as examples, our simulation results show that our algorithm can find the D-optimal designs efficiently and the algorithm outperforms its competitors. As an application, we apply our algorithm and re-design a 10-factor car refueling experiment with discrete and continuous factors and selected pairwise interactions. Our proposed algorithm was able to consistently outperform the other algorithms and find a more efficient D-optimal design for the problem.

Keywords

• Approximate design
• design efficiency
• generalized linear model
• high-dimensional
• non-separable
• sensitivity function