, Volume 47, Issue 1-3, pp 425-439

Active set algorithms for isotonic regression; A unifying framework

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Abstract

In this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms.

In this paper we consider the isotonic regression problem with respect to a complete order \(\begin{gathered} minimize\sum\limits_{i = 1}^n {w_i } (y_i - x_i )^2 \\ subject tox_1 \leqslant x_2 \leqslant \cdot \cdot \cdot \leqslant x_n \\ \end{gathered} \) where eachw i is strictly positive and eachy i is an arbitrary real number. We show that the Pool Adjacent Violators algorithm (due to Ayer et al., 1955; Miles, 1959; Kruskal, 1964), is a dual feasible active set method and that the Minimum Lower Set algorithm (due to Brunk et al., 1957) is a primal feasible active set method of computational complexity O(n 2). We present a new O(n) primal feasible active set algorithm. Finally we discuss Van Eeden's method and show that it is of worst-case exponential time complexity.

This work was supported by the National Science and Engineering Research Council of Canada under Research Grant A8189 and an Ontario Graduate Scholarship.