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 than 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

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

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

A model-based framework assisting the design of vapor-liquid equilibrium experimental plans

In this paper we propose a framework for Model-based Sequential Optimal Design of Experiments to assist experimenters involved in Vapor-Liquid equilibrium characterization studies to systematically construct thermodynamically consistent models. The approach uses an initial continuous optimal design obtained via semidefinite programming, and then iterates between two stages (i) model fitting using the information available; and (ii) identification of the next experiment, so that the information content in data is maximized. The procedure stops when the number of experiments reaches the maximum for the experimental program or the dissimilarity between the parameter estimates during two consecutive iterations is below a given threshold. This methodology is exemplified with the D-optimal design of isobaric experiments, for characterizing binary mixtures using the NRTL and UNIQUAC thermodynamic models for liquid phase. Significant reductions of the confidence regions for the parameters are achieved compared with experimental plans where the observations are uniformly distributed over the domain

Rank-Two Beamforming and Power Allocation in Multicasting Relay Networks

In this paper, we propose a novel single-group multicasting relay beamforming scheme. We assume a source that transmits common messages via multiple amplify-and-forward relays to multiple destinations. To increase the number of degrees of freedom in the beamforming design, the relays process two received signals jointly and transmit the Alamouti space-time block code over two different beams. Furthermore, in contrast to the existing relay multicasting scheme of the literature, we take into account the direct links from the source to the destinations. We aim to maximize the lowest received quality-of-service by choosing the proper relay weights and the ideal distribution of the power resources in the network. To solve the corresponding optimization problem, we propose an iterative algorithm which solves sequences of convex approximations of the original non-convex optimization problem. Simulation results demonstrate significant performance improvements of the proposed methods as compared with the existing relay multicasting scheme of the literature and an algorithm based on the popular semidefinite relaxation technique