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SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression

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  • Flores, Salvador

Abstract

This paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion, with an special emphasis on applications to computing the Least Trimmed Squares Estimator (LTSE) for robust regression. The computation of the LTSE is a challenging subset selection problem involving a nonlinear program with continuous and binary variables, linked in a highly nonlinear fashion. The selection of a globally optimal subset using the branch and bound (BB) algorithm is limited to problems in very low dimension, typically d ≤ 5, as the complexity of the problem increases exponentially with d. We introduce a bold pruning strategy in the BB algorithm that results in a significant reduction in computing time, at the price of a negligeable accuracy lost. The novelty of our algorithm is that the bounds at nodes of the BB tree come from pseudo-convexifications derived using a linearization technique with approximate bounds for the nonlinear terms. The approximate bounds are computed solving an auxiliary semidefinite optimization problem. We show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the LTSE problem.

Suggested Citation

  • Flores, Salvador, 2015. "SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression," European Journal of Operational Research, Elsevier, vol. 246(1), pages 44-50.
    • Handle: RePEc:eee:ejores:v:246:y:2015:i:1:p:44-50
    DOI: 10.1016/j.ejor.2015.04.024
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    References listed on IDEAS

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    1. Fred Glover, 1975. "Improved Linear Integer Programming Formulations of Nonlinear Integer Problems," Management Science, INFORMS, vol. 22(4), pages 455-460, December.
    2. Torti, Francesca & Perrotta, Domenico & Atkinson, Anthony C. & Riani, Marco, 2012. "Benchmark testing of algorithms for very robust regression: FS, LMS and LTS," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2501-2512.
    3. Mount, David M. & Netanyahu, Nathan S. & Romanik, Kathleen & Silverman, Ruth & Wu, Angela Y., 2007. "A practical approximation algorithm for the LMS line estimator," Computational Statistics & Data Analysis, Elsevier, vol. 51(5), pages 2461-2486, February.
    4. Warren P. Adams & Hanif D. Sherali, 1990. "Linearization Strategies for a Class of Zero-One Mixed Integer Programming Problems," Operations Research, INFORMS, vol. 38(2), pages 217-226, April.
    5. Nguyen, T.D. & Welsch, R., 2010. "Outlier detection and least trimmed squares approximation using semi-definite programming," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3212-3226, December.
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    Cited by:

    1. Bottmer, Lea & Croux, Christophe & Wilms, Ines, 2022. "Sparse regression for large data sets with outliers," European Journal of Operational Research, Elsevier, vol. 297(2), pages 782-794.

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