Least trimmed squares regression, least median squares regression, and mathematical programming

Authors:
A. Giloni;M. Padberg
Affiliations:
Sy Syms School of Business, Yeshiva University New York, NY 10033, U.S.A.;Operations Management Department Stern School of Business, New York University New York, NY 10012, U.S.A.
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
2002

Citing 7
Cited 0

Methods for global concave minimization: A bibliographic survey

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Computing the exact least median of squares estimate and stability diagnostics in multiple linear regression

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The feasible set algorithm for least median of squares regression

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The feasible solution algorithm for least trimmed squares regression

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Convergence of the feasible solution algorithm for least median of squares regression

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Abstract

In this paper, we study LTS and LMS regression, two high breakdown regression estimators, from an optimization point of view. We show that LTS regression is a nonlinear optimization problem that can be treated as a concave minimization problem over a polytope. We derive several important properties of the corresponding objective function that can be used to obtain algorithms for the exact solution of LTS regression problems, i.e., to find a global optimum to the problem. Because of today's limited problem-solving capabilities in exact concave minimization, we give an easy-to-implement pivoting algorithm to determine regression parameters corresponding to local optima of the LTS regression problem. For the LMS regression problem, we briefly survey the existing solution methods which are all based on enumeration. We formulate the LMS regression problem as a mixed zero-one linear programming problem which we analyze in depth to obtain theoretical insights required for future algorithmic and computational work.