Continuous Optimization Reducing quadratic programming problem to regression problem: Stepwise algorithm Dong Q. Wang a, * , Stefanka Chukova a , C.D. Lai b a School of Mathematical and Computing Sciences, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand b Institute of Information Sciences and Technology, Massey University, Palmerston North, New Zealand Received 18 December 2001; accepted 10 July 2002 Available online 9 April 2004 Abstract Quadratic programming is concerned with minimizing a convex quadratic function subject to linear inequality constraints. The variables are assumed to be nonnegative. The unique solution of quadratic programming (QP) problem (QPP) exists provided that a feasible region is non-empty (the QP has a feasible space). A method for searching for the solution to a QP is provided on the basis of statistical theory. It is shown that QPP can be reduced to an appropriately formulated least squares (LS) problem (LSP) with equality constraints and non- negative variables. This approach allows us to obtain a simple algorithm to solve QPP. The applicability of the sug- gested method is illustrated with numerical examples. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Quadratic programming; Least squares method; Linear inequality; Search algorithms 1. Introduction Regression analysis with linear equality and inequality constraints is useful data analysis tool in applied mathematics, physics, statistics, mathematical programming, economics, social science and quality control. Minimizing the sum of the absolute values (L 1 norm) of the regression errors has shown that the L 1 regression problem can be reduced to a general linear programming problem (Kiountouzis, 1973). This procedure resembles the simplex method in the quantity to minimize a sum of infeasibilities (Davies, 1967), that is, the solution of L 1 regression is obtained by making use of the simplex method of linear pro- gramming. The least squares method of regression with linear equality constraints also has been treated as a quadratic programming (QP) problem (Fang et al., 1982; Brown and Goodall, 1994). The unique solution of QP exists provided that the feasible region is non-empty (the QP has a feasible space), and its solution is given by the Kuhn–Tucker theorem (Hillier and Lieberman, 1995). * Corresponding author. Tel.: +64-4463-5275; fax: +64-4463-5045. E-mail address: dong.wang@mcs.vuw.ac.nz (D.Q. Wang). 0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2002.07.001 European Journal of Operational Research 164 (2005) 79–88 www.elsevier.com/locate/dsw