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 interact 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 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 logistic models with several factors on various design spaces as examples, our simulation results show our algorithm can find 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.

Production and Statistical Optimization of Paromomycin by Streptomyces rimosus NRRL 2455 in Solid State Fermentation

“…RSM has been considerably used for the optimized production of many antibiotics [18,34]. This technique comes with numerous types of designs for the optimization of important fermentation parameters, and D-optimal design is one of the most accurate ones [50]. It has been used by many investigators in optimization studies [51,52].…”

A Metaheuristic Adaptive Cubature Based Algorithm to Find Bayesian Optimal Designs for Nonlinear Models

“…maximin optimal designs for a three-parameter enzyme kinetic nonlinear model and then used information from the design structure and the equivalence theorem to obtain a formula for the optimal design. Two recent applications of using a natureinspired metaheuristic algorithm to find optimal designs are Lukemire, Mandal, and Wong (2018) and Xu et al (2018).…”

Hybrid algorithms for generating optimal designs for discriminating multiple nonlinear models under various error distributional assumptions

“…When errors are normally distributed, we use the PSO-QN algorithm to identify all the Toptimal designs for discriminating between model (19) and each of the rival models, (15), (16), (17) and (18). The left panel of Table 1 shows the T-optimal designs.…”

“…For example, consider the problem, where there are 5 competing models and the PSO-S-QN algorithm was applied to find a max-min T-optimal design. We first applied the PSO-QN algorithm to find T-optimal designs for discriminating between the assumed true model and each of the rival models (15), (16), (17) and (18). The computing time for searching the T-optimal design for each of these 4 2-model discrimination problems was 7.12, 29.02, 90.94 and 73.28 seconds, respectively.…”

“…[17] developed hybridized PSO-based algorithms to solve more complicated optimal design problems such as the standardized maximin optimal criteria, which includes the simpler minimax design problems. Most recently, [18] applied DE to find optimal approximate designs for logistic models with up to 5 factors with all pairwise interaction terms. The number of variables to optimize for such a model is at least 95 if the optimal design is minimally supported; otherwise, there will be many more variables to optimize.…”