A direct method for sparse least squares problems with lower and upper bounds

Abstract

Summary. A direct method is developed for solving linear least squares problems min ]lAx-btl2, where A is large and sparse and the solution is subject x to lower and upper bounds l<_x<_u. The problem is initially transformed to upper triangular form by a sparse QR-factorization. An active set algorithm is then used. The key step is the stable updating of the R-factor associated with the columns of A corresponding to the free variables, when the Q-factor is not available. For this a new method is developed, which uses the semi-normal equations and iterative refinement.

A comparison of block pivoting and interior-point algorithms for linear least squares problems with nonnegative variables

“…The active-set method for the solution of the NVLSQ problem has been introduced in [2]. The algorithm can be seen as a single principal pivoting algorithm, and may be stated in the following form:…”

“…The implementation of the active-set method has been discussed in [2]. In the first step of this procedure, a so-called analyze phase is performed in which a permutation of the columns of the matrix A is found by applying the minimum-degree strategy to the matrix ATA [8].…”

“…If the matrix A has full column rank (rank(A) -n), the matrix ATA is positive definite and the strictly convex program (2) and the strictly monotone LCP (3) have unique solutions for each vector b. In this paper we only deal with this case.…”

“…Owing to its equivalence to the quadratic problem (2), the unique optimal solution of the NVLSQ problem can be found by a large number of algorithms that have been designed for this type of programs. In particular, active-set methods and projected gradient algorithms can be used for this purpose.…”

“…Recently, Bierlair, Toint, and Tuyttens [1] have studied the performance of projected gradient algorithms for the solution of large-scale NVLSQ problems. In another paper [2], Björck proposes an efficient implementation of the active-set method discussed in [ 13], which is capable of dealing with large-scale NVLSQ problems.…”