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T-optimal designs for multi-factor polynomial regression models via a semidefinite relaxation method

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Abstract

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models where the 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 the regression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. When there are two or more factors in the models, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When the relaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and its optimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples.

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Acknowledgements

All authors gratefully acknowledge partial support from a grant from the National Institute of General Medical Sciences of the National Institutes of Health under Award Number R01GM107639. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Dr. Lieven Vandenberghe is also partially by a National Science Foundation Grant 1509789. We wish to thank Dr. Didier Henrion for helpful discussions and advice on using the software package GloptiPoly3. We also thank Dr. Fabrice Gamboa for his helpful comments on an earlier version of this manuscript.

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Authors and Affiliations

  1. Department of Biostatistics, University of California, Los Angeles, USA

    Yuguang Yue & Weng Kee Wong

  2. Department of Electrical Engineering, University of California, Los Angeles, USA

    Lieven Vandenberghe

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  1. Yuguang Yue
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  2. Lieven Vandenberghe
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  3. Weng Kee Wong
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Correspondence to Yuguang Yue.

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Appendix

Appendix

We provide here an illustrative MATLAB code for finding the moment matrix of optimal discrimination design in Example 1.

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Yue, Y., Vandenberghe, L. & Wong, W.K. T-optimal designs for multi-factor polynomial regression models via a semidefinite relaxation method. Stat Comput 29, 725–738 (2019). https://doi.org/10.1007/s11222-018-9834-2

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