Error Analysis of Elimination Methods for Equality Constrained Quadratic Programming Problems

A backward error analysis for the orthogonal factorization method for equality constrained quadratic programming problems has been developed. Furthermore, this method has been experimentally compared with direct elimination method on a class of test problems. 1 Statement of the problem We consider the equality constrained quadratic programming (QPE) problem min Cx=d 1 2 xAx+ bx+ ν (1) where C is an m × n matrix of full rank (m ≤ n), A is an n × n symmetric matrix which is positive definite on the subspace N (C) = {x | Cx = 0}, d ∈ R, b ∈ R and ν ∈ R. It is well known that under these assumptions the problem has a unique solution. This problem is closely related to the linear equality constrained least squares (LSE) problem min Cx=d 1 2 ‖Ex− f‖ (2) where E is a k × n matrix (n ≤ k), f ∈ R and ‖ · ‖ denotes the Euclidean norm. In fact problem (2) is equivalent to (1) if we put A = EE, b = −ET f, ν = 12f f. When E has full rank, A is positive definite everywhere; when N (E) ∩ N (C) = {0}, A is positive semidefinite everywhere and positive definite on N (C). A natural way to solve these problems is to use elimination methods [2, 5]. These methods can be interpreted as having the following three stages: i) derive a lower– dimensional unconstrained problem using constraints to eliminate variables; ii) solve the derived problem; iii) transform the solution of the derived problem to obtain the solution of the original constrained problem. 107 Draft of the paper in: Numerical Methods and Error Bounds (G. Alefeld, J. Herzberger eds.), Akademie Verlag, Berlin, 1996, 107-112. ISBN: 3-05-501696-3 Series: Mathematical Research, Vol. 89. (Proceedings of the “IMACS-GAMM International Symposium on Numerical Methods and Error-Bounds”, Oldenburg, July 9-12, 1995)