Finding D-optimal designs for generalized linear models (GLMs) is challenging due to the dependence of the Fisher information matrix on unknown parameters and the lack of closed-form solutions, particularly when input factors include both discrete and continuous variables. Although classical algorithms and recent metaheuristic approaches have offered partial solutions, there remains a need for robust and computationally efficient methods. In this paper, we propose a penalized Particle Swarm Optimization (PSO) approach, named $p$-PSO. Here we introduce a new, general-purpose penalty formulation for constrained optimization and demonstrate its effectiveness in optimal design problems. The formulation is algorithm-agnostic and applicable to a broad class of black-box optimization methods. Results show that the method is highly efficient, with its primary contribution being a penalty formulation that enables the direct use of an off-the-shelf PSO algorithm and extends naturally to more general constrained optimization tasks.

翻译：寻找广义线性模型（GLM）的D-最优设计因费希尔信息矩阵对未知参数的依赖性及缺乏闭合形式解而极具挑战性，尤其当输入因子包含离散与连续混合变量时。尽管经典算法与近年元启发式方法已提供部分解决方案，但稳健且计算高效的方法仍待探索。本文提出一种名为$p$-PSO的惩罚粒子群优化方法，其中引入了一种面向约束优化的新型通用惩罚公式，并验证其在最优设计问题中的有效性。该公式具有算法无关性，适用于广泛的黑箱优化方法。结果表明，该方法具有极高效率，其核心贡献在于提出一种惩罚公式，使现成的PSO算法可直接应用于约束优化任务，并能自然扩展至更一般的约束优化问题。