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T-optimal designs formulti-factor polynomial regressionmodelsvia a semidefinite relaxation method
We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models wherethe design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets.Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When theregression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtainan equivalent semidefinite program.When there are two or more factors in the models, we apply a moment relaxation techniqueand approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When therelaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and itsoptimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples
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Adaptive grid semidefinite programming for finding optimal designs
We find optimal designs for linear models using anovel algorithm that iteratively combines a semidefinite programming(SDP) approach with adaptive grid techniques.The proposed algorithm is also adapted to find locally optimaldesigns for nonlinear models. The search space is firstdiscretized, and SDP is applied to find the optimal designbased on the initial grid. The points in the next grid set arepoints that maximize the dispersion function of the SDPgeneratedoptimal design using nonlinear programming. Theprocedure is repeated until a user-specified stopping rule isreached. The proposed algorithm is broadly applicable, andwe demonstrate its flexibility using (i) models with one ormore variables and (ii) differentiable design criteria, suchas A-, D-optimality, and non-differentiable criterion like Eoptimality,including the mathematically more challengingcasewhen theminimum eigenvalue of the informationmatrixof the optimal design has geometric multiplicity larger thanΒ 1. Our algorithm is computationally efficient because it isbased on mathematical programming tools and so optimalityis assured at each stage; it also exploits the convexity of theproblems whenever possible. Using several linear and nonlinearmodelswith one or more factors, we showthe proposedalgorithm can efficiently find optimal designs
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Optimal exact designs of experiments via Mixed Integer Nonlinear Programming
Optimal exact designs are problematic to find and study because there is no unified theory for determining them and studyingtheir properties. Each has its own challenges and when a method exists to confirm the design optimality, it is invariablyapplicable to the particular problem only.We propose a systematic approach to construct optimal exact designs by incorporatingthe Cholesky decomposition of the Fisher Information Matrix in a Mixed Integer Nonlinear Programming formulation. Asexamples, we apply the methodology to find D- and A-optimal exact designs for linear and nonlinear models using global orlocal optimizers. Our examples include design problems with constraints on the locations or the number of replicates at theoptimal design points
Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming
Elfving's Theorem is a major result in the theory of optimal experimental
design, which gives a geometrical characterization of optimality. In this
paper, we extend this theorem to the case of multiresponse experiments, and we
show that when the number of experiments is finite, and optimal
design of multiresponse experiments can be computed by Second-Order Cone
Programming (SOCP). Moreover, our SOCP approach can deal with design problems
in which the variable is subject to several linear constraints.
We give two proofs of this generalization of Elfving's theorem. One is based
on Lagrangian dualization techniques and relies on the fact that the
semidefinite programming (SDP) formulation of the multiresponse optimal
design always has a solution which is a matrix of rank . Therefore, the
complexity of this problem fades.
We also investigate a \emph{model robust} generalization of optimality,
for which an Elfving-type theorem was established by Dette (1993). We show with
the same Lagrangian approach that these model robust designs can be computed
efficiently by minimizing a geometric mean under some norm constraints.
Moreover, we show that the optimality conditions of this geometric programming
problem yield an extension of Dette's theorem to the case of multiresponse
experiments.
When the number of unknown parameters is small, or when the number of linear
functions of the parameters to be estimated is small, we show by numerical
examples that our approach can be between 10 and 1000 times faster than the
classic, state-of-the-art algorithms
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