Solving Linear Inequalities in a Least Squares Sense

Abstract

In 1980, Han 6] described a nitely terminating algorithm for solving a system Ax b of linear inequalities in a least squares sense. The algorithm uses a singular value decomposition of a submatrix of A on each iteration, making it impractical for all but the smallest problems. This paper shows that a modi cation of Han's algorithm allows the iterates to be computed using QR factorization with column pivoting, which signi cantly reduces the computational cost and allows e cient updating/downdating techniques to be used. The e ectiveness of this modi cation is demonstrated, implementation details are given, and the behaviour of the algorithm discussed. Theoretical and numerical results are shown from the application of the algorithm to linear separability problems.

A hybrid algorithm for solving linear inequalities in a least squares sense

“…In this section we briefly discuss the use of Newton's method for solving (1.4), as proposed in [4,31,48]. For general discussion of Newton's method see, for example, [3,23,26,28].…”

“…This approach was proposed by Han [31], who noted that the resulting active-set algorithm is essentially Newton's method. Recent papers that advocate the use of Newton's method for solving (1.4) have been published by Bramley and Winnicka [4] and Pinar [48]. In this paper we present, analyze and test a new iteration for solving (1.11).…”

“…The problem of calculating a hyperplane that separates between two sets of points in R n has a central role in these fields. The mathematical formulation of the last problem gives rise to least deviation problems, e.g., [1,4,41,43,44].…”

“…Yet, in contrast to this requirement, the objective function of (1.11) is not strictly convex, and the related Hessian is not invertible. The last fact is likely to cause difficulties in the computation of search directions, and in the rules for dropping constraints from the active set, see [4]. A similar difficulty stems from the observation that the solutions to (1.11) are usually not unique, see [31].…”

Newton's method for linear inequality systems

“…While this work was under review, we came across a recent reference Bramley and Winnicka (1996) where an algorithm very similar to ours was independently proposed for the solution of linear inequalities. This algorithm is motivated from the theory of least squares and is apparently an extension of an unpublished work by Han (1980).…”