Locally optimal designs for nonlinear models require a single set of nominal values for the unknown parameters. An alternative is the maximin approach that allows the user to specify a range of values for each parameter of interest. However, the maximin approach is difficult because we first have to determine the locally optimal design for each set of nominal values before maximin types of optimal designs can be found via a nested optimization process. We show that particle swarm optimization (PSO) techniques can solve such complex optimization problems effectively. We demonstrate numerical results from PSO can help find, for the first time, formulae for standardized maximin D-optimal designs for nonlinear model with 3 or 4 parameters on the compact and nonnegative design space. Additionally, we show locally and standardized maximin D-optimal designs for inhibition models are not necessarily supported at a minimum number of points. To facilitate use of such designs, we create a web-based tool for practitioners to find tailor-made locally and standardized maximin optimal designs.

Finding a model-based optimal design that can optimally discriminate among a class of plausible models is a difficult task because the design criterion is non-differentiable and requires 2 or more layers of nested optimization. We propose hybrid algorithms based on particle swarm optimization (PSO) to solve such optimization problems, including cases when the optimal design is singular, the mean response of some models are not fully specified and problems that involve 4 layers of nested optimization. Using several classical examples, we show that the proposed PSO-based algorithms are not models or criteria specific, and with a few repeated runs, can produce either an optimal design or a highly efficient design. They are also generally faster than the current algorithms, which are generally slow and work for only specific models or discriminating criteria. As an application, we apply our techniques to find optimal discriminating designs for a dose-response study in toxicology with 5 possible models and compare their performances with traditional and a recently proposed algorithm. In the supplementary material, we provide a R package to generate different types of discriminating designs and evaluate efficiencies of competing designs so that the user can implement an informed design.

In this work, we consider a manufactory process which can be described by a multiple-instance logistic regression model. In order to compute the maximum likelihood estimation of the unknown coefficient, an expectation-maximization algorithm is proposed, and the proposed modeling approach can be extended to identify the important covariates by adding the coefficient penalty term into the likelihood function. In addition to essential technical details, we demonstrate the usefulness of the proposed method by simulations and real examples.

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