Model-based optimal design of experiments —Semidefinite and nonlinear programming formulations

Abstract

We use mathematical programming tools, such as Semidefinite Programming (SDP) and Nonlinear Programming (NLP)-based formulations to find optimal designs for models used in chemistry and chemical engineering. In particular, we employ local design-based setups in linear models and a Bayesian setup in nonlinear models to find optimal designs. In the latter case, Gaussian Quadrature Formulas (GQFs) are used to evaluate the optimality criterion averaged over the prior distribution for the model parameters. Mathematical programming techniques are then applied to solve the optimization problems. Because such methods require the design space be discretized, we also evaluate the impact of the discretization scheme on the generated design. We demonstrate the techniques for finding D-, A-and E-optimal designs using design problems in biochemical engineering and show the method can also be directly applied to tackle additional issues, such as heteroscedasticity in the model. Our results show that the NLP formulation produces highly efficient D-optimal designs but is computationally less efficient than that required for the SDP formulation. The efficiencies of the generated designs from the two methods are generally very close and so we recommend the SDP formulation in practice.

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

“…Some disadvantages of such an algorithm and others that require the search space to be discretized are that the generated designs depend on the grid set and the algorithm can become ineffective for models with many covariates because the optimization problem becomes high dimensional. Duarte, Wong, and Oliveira (2016) introduced a nonlinear programming (NLP) based algorithm that does not need any discretization on the design space. This algorithm uses a multistart heuristic algorithm named OQNLP (Ugray et al 2005) to determine an optimal design and it is codified in GAMS (GAMS Development Corporation 2013).…”

“…Both of the SDP and the NLP-based algorithms are equipped with Gauss quadrature formulas to approximate the integrals. Duarte, Wong, and Oliveira (2016) showed that the NLP-based algorithm tend to generate more efficient designs with fewer number of support points than those designs found by the SDPbased algorithm. This section compares our algorithm with the NLP-based algorithm for finding D-optimal designs for a three-parameter alcohol kinetics model (Box and Hunter 1965) and the fourparameter sigmoid Emax model presented in Section 4.2.…”

Optimal Design of Experiments for Liquid–Liquid Equilibria Characterization via Semidefinite Programming

“…Mathematical programming approaches have been successfully applied to solve many design problems. Some examples are Linear Programming [29,30], Second Order Conic Programming [31,32], Semidefinite Programming (SDP) [33][34][35], Semi-Infinite Programming (SIP) [36,37], Nonlinear Programming (NLP) [38,39], NLP combined with stochastic procedures such as genetic algorithms [40,41], and global optimization techniques [42,43]. They are particularly efficient for approximate (also called continuous) optimal designs, characterized by a continuous representation of the weights of each constituent one-point experiment in the plan; the design problem has convex properties and the quality of the optimum can be checked with a General Equivalence Theorem (GET).…”

Optimal Designs for Multi-Response Nonlinear Regression Models With Several Factors via Semidefinite Programming

“…In either case, the models are linear, except for the rational polynomial models briefly considered in Papp (2012). Recent work using SDP to find different types of optimal designs for nonlinear models has only one response variable, see for example, Duarte et al (2016*), Duarte and Wong (2015), and Duarte et al (2016). Elfving’s Theorem is a useful tool for geometrically characterizing a c-optimal design and Sagnol (2011) extended Elfving’s celebrated theorem for finding a c-optimal design geometrically in multi-response experiments and used second-order cone programming for computing various optimal designs.…”