ELSEVIER EUROPEAN JOURNAL OF OPERATIONAL RESEARCH European Journal of Operational Research 107 (1998) 710-719 Theory and Methodology Newton’s method for linear inequality systems Mustafa C. Pmar * Department of Industrial Engineering, Biikent Uniuersify TR-06533 Ankara, Turkey Received 13 May 1996; accepted 14 April 1997 Abstract We describe a modified Newton type algorithm for the solution of linear inequality systems in the sense of minimizing the /, norm of infeasibilities. Finite termination is proved, and numerical results are given. 0 1998 Elsevier Science B.V. Keywords: Linear inequalities; Piecewise-quadratic functions; Newton’s method; Finiteness 1. Introduction and background We consider the problem of finding a feasible point with respect to the overdetermined set of linear inequalities: ATy I c, (1) where A is an m X n real matrix with n > m and c is an n-vector. Let s:=c-A’y. (2) We would like to compute a solution y such that s 2 0. This problem has applications in image recon- struction in computerized tomography; see Herman (1980). A recent account of iterative methods for this problem along with applications can be found in the forthcoming book, Censor and Zenios (1997). Linear inequality systems are also solved as auxiliary prob- lems in linear and nonlinear optimization. In the present paper we consider a modified New- ton algorithm to find a solution that minimizes the * Fax: + 90-312-266-4126; e-mail: mustafap@bilkent.cdu.tr. /‘, norm of infeasibilities with respect to (1). The problem consists in the minimization of piecewise- quadratic objective function with discontinuous sec- ond derivatives. More precisely, we consider the following minimization problem for computing a solution that minimizes the sum of squares of infea- sibilities with respect to (1): min F(y) = $[s-ll:, Y (3) where s- is a vector whose ith component s;= min{O,s;). Clearly, if the linear inequality system (1) is consistent, a minimizer of F is also a feasible solution with respect to (1) and vice versa. The algorithm proposed here is adapted from a Newton algorithm developed by Madsen and Nielsen (1990) for the minimization of the Huber function in robust linear regression analysis. The crucial observation that motivated the present paper is that the Huber function and the E’, function are structurally identi- cal in the context of linear inequality systems. In the present paper, we apply the ideas in Madsen and Nielsen (1990) to the /‘, solution of linear inequal- ity systems. However, their finite termination analy- 0377-2217/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PII SO377-2217(97)00178-l