Adaptive grid semidefinite programming for finding optimal designs
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Date
2016
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Abstract
We find optimal designs for linear models using a novel algorithm that iteratively combines a Semidefinite
Programming (SDP) approach with adaptive grid (AG) techniques. The search space is first discretized
and SDP is applied to find the optimal design based on the initial grid. The points in the next grid set are
points that maximize the dispersion function of the SDP-generated optimal design using Nonlinear Programming
(NLP). The procedure is repeated until a user-specified stopping rule is reached. The proposed
algorithm is broadly applicable and we demonstrate its flexibility using (i) models with one or more variables,
and (ii) differentiable design criteria, such as A-, D-optimality, and non-differentiable criterion like
E-optimality, including the mathematically more challenging case when the minimum eigenvalue of the
information matrix of the optimal design has geometric multiplicity larger than 1. Our algorithm is computationally
efficient because it is based on mathematical programming tools and so optimality is assured at
each stage; it also exploits the convexity of the problems whenever possible. Using several linear models,
we show the proposed algorithm can efficiently find both old and new optimal designs.
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Keywords
adaptive grid, semidefinite programming, nonlinear programming, model-based optimal design, continuous design