TY - JOUR

T1 - Adaptive grid semidefinite programming for finding optimal designs

AU - Duarte, Belmiro P.M.

AU - Wong, Weng Kee

AU - Dette, Holger

N1 - Publisher Copyright:
© 2017, Springer Science+Business Media New York.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - We find optimal designs for linear models using a novel algorithm that iteratively combines a semidefinite programming (SDP) approach with adaptive grid techniques. The proposed algorithm is also adapted to find locally optimal designs for nonlinear models. 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. 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 and nonlinear models with one or more factors, we show the proposed algorithm can efficiently find optimal designs.

AB - We find optimal designs for linear models using a novel algorithm that iteratively combines a semidefinite programming (SDP) approach with adaptive grid techniques. The proposed algorithm is also adapted to find locally optimal designs for nonlinear models. 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. 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 and nonlinear models with one or more factors, we show the proposed algorithm can efficiently find optimal designs.

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U2 - 10.1007/s11222-017-9741-y

DO - 10.1007/s11222-017-9741-y

M3 - Article

AN - SCOPUS:85015973328

SN - 0960-3174

VL - 28

SP - 441

EP - 460

JO - Statistics and Computing

JF - Statistics and Computing

IS - 2

ER -