Abs Algorithms for Linear Systems and Optimization: a Review and a Bibliography

In this paper, we 1. The scaled ABS class. ABS methods have been introduced by Abaffy, Broyden and Spedicato (1984), in a paper where the solution of linear algebraic equations was considered via the so called ABS unscaled or basic class. That class was later generalized to the so called scaled ABS class and then applied also to the solution of linear least squares, nonlinear algebraic equations and optimization problems. For a full presentation of the main results in the ABS field see the monographs of Abaffy and Spedicato (1989) and of Zhang, Xia and Feng (1999). There are presently almost 400 papers dealing with ABS methods, see for a partial list the bibliography in the Appendix 2. In this review we restrict our attention to ABS methods for linear determined or underdetermined algebraic systems and their application to some optimization problems. We begin by giving our notation and the steps of the scaled ABS class. Consider the following general linear system (determined or underdetermined), where the rank of A ∈ R is arbitrary Ax = b x ∈ R, b ∈ R, m ≤ n (1) or ai x− bi = 0, i = 1, . . . ,m (2)