An L1 estimation algorithm with degeneracy and linear constraints

  • Authors:

  • Mingren Shi;Mark A. Lukas

  • Affiliations:

  • Mathematics and Statistics, Division of Science and Engineering, Murdoch University, South Street, Murdoch, WA 6150, Perth, Australia;Mathematics and Statistics, Division of Science and Engineering, Murdoch University, South Street, Murdoch, WA 6150, Perth, Australia

  • Venue:
  • Computational Statistics & Data Analysis
  • Year:
  • 2002

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Abstract

An implementation of the reduced gradient algorithm is proposed to solve the linear L1 estimation problem (least absolute deviations regression) with linear equality or inequality constraints, including rank deficient and degenerate cases. Degenerate points are treated by solving a derived L1 problem to give a descent direction. The algorithm is a direct descent, active set method that is shown to be finite, it is geometrically motivated and simpler than the projected gradient algorithm (PGA) of Bartels, Conn and Sinclair, which uses a penalty function approach for the constrained case. Computational experiments indicate that the proposed algorithm compares favourably, both in reliability and efficiency, to the PGA, to the algorithms ACM551 and AFK (which use an LP formulation of the L1 problem) and to LPASL1 (which is based on the Huber approximation method of Madsen, Nielsen and Pinar). Although it is not as efficient as ACM552 (Barrodale-Roberts algorithm) on large scale unconstrained problems, it performs better on large scale problems with bounded variable constraints.