[Paper Review] Using particle swarm optimization to search for locally $D$-optimal designs for mixed factor experiments with binary response

Joshua Lukemire, Abhyuday Mandal+1 more|arXiv (Cornell University)|Feb 5, 2016

Optimal Experimental Design Methods31 references3 citations

TL;DR

This paper proposes using particle swarm optimization (PSO) to find locally D-optimal designs for generalized linear models with binary responses and mixed discrete and continuous factors—factors that challenge traditional optimal design theory. PSO efficiently identifies high-efficiency designs without requiring a predefined candidate set, outperforming implemented designs in real-world odor removal and electrostatic discharge studies, and handles constrained and irregular design spaces effectively.

ABSTRACT

Identifying optimal designs for generalized linear models with a binary response can be a challenging task, especially when there are both continuous and discrete independent factors in the model. Theoretical results rarely exist for such models, and the handful that do exist come with restrictive assumptions. This paper investigates the use of particle swarm optimization (PSO) to search for locally $D$-optimal designs for generalized linear models with discrete and continuous factors and a binary outcome and demonstrates that PSO can be an effective method. We provide two real applications using PSO to identify designs for experiments with mixed factors: one to redesign an odor removal study and the second to find an optimal design for an electrostatic discharge study. In both cases we show that the $D$-efficiencies of the designs found by PSO are much better than the implemented designs. In addition, we show PSO can efficiently find $D$-optimal designs on a prototype or an irregularly shaped design space, provide insights on the existence of minimally supported optimal designs, and evaluate sensitivity of the $D$-optimal design to mis-specifications in the link function.

Motivation & Objective

• To address the lack of theoretical and computational tools for finding optimal designs in generalized linear models with mixed discrete and continuous factors and binary responses.
• To demonstrate that PSO can efficiently generate locally D-optimal designs for such complex models where analytical solutions are unavailable.
• To evaluate PSO's performance on real-world experimental designs with both regular and irregular design spaces.
• To assess the sensitivity of D-optimal designs to mis-specifications in the link function and to investigate the existence of minimally supported designs.

Proposed method

• Adapted particle swarm optimization (PSO) to search for locally D-optimal designs without requiring a pre-specified candidate set of design points.
• Employed the equivalence theorem to verify the optimality of PSO-generated designs by checking the gradient condition at each support point.
• Modified PSO to incorporate inequality constraints, enabling optimization within irregular or bounded design spaces.
• Used default PSO parameters (25 particles, 200 iterations, convergence tolerance 0.0001) across multiple runs to ensure robustness and stability.
• Applied PSO to both synthetic and real experimental problems, including a bio-plastics odor removal study and an electrostatic discharge study.
• Evaluated D-efficiency of PSO-generated designs relative to implemented designs using the formula: D-efficiency = (det(M(D*)) / det(M(D)))^(1/p), where M(D) is the information matrix.

Experimental results

Research questions

• RQ1 Can PSO effectively generate locally D-optimal designs for generalized linear models with mixed discrete and continuous factors and binary responses when theoretical results are unavailable?
• RQ2 How does the D-efficiency of PSO-generated designs compare to that of implemented designs in real-world experiments?
• RQ3 Can PSO handle optimization on irregularly shaped or constrained design spaces without requiring a candidate set?
• RQ4 What is the sensitivity of the locally D-optimal design to mis-specifications in the link function (e.g., logit vs. probit)?
• RQ5 Do minimally supported locally D-optimal designs exist for mixed-factor models with binary responses, and can PSO identify them?

Key findings

• PSO-generated designs achieved significantly higher D-efficiencies than the implemented designs in both the odor removal and electrostatic discharge studies, with improvements exceeding 30% in some cases.
• The locally D-optimal design on the irregular design space (e.g., constrained by process limits) was found to be distinct from the unconstrained design, confirming the importance of constraint-aware optimization.
• PSO successfully identified a locally D-optimal design with support points at (450, 1100) with weight 0.334, (460, 1200) with weight 0.335, and (460, 1000) with weight 0.331, satisfying the equivalence theorem.
• The algorithm converged reliably across 100 runs with default parameters, indicating low sensitivity to tuning and strong robustness.
• PSO identified a minimally supported design with only three distinct design points for the constrained space, suggesting such designs exist and are computable.
• Sensitivity analysis showed that the D-optimal design was relatively robust to minor mis-specifications in the link function, though larger deviations reduced efficiency.