Linear Programming Using ABS Method

Journal of Computational and Applied Mathematics, 2000

In this paper we review basic properties and the main achievements obtained by the class of ABS methods, developed since 1981 to solve linear and nonlinear algebraic equations and optimization problems.

2002

ABS methods are a large class of methods, based upon the Egervary rank reducing algebraic process, first introduced in 1984 by Abaffy, Broyden and Spedicato for solving linear algebraic systems, and later extended to nonlinear algebraic equations, to optimization problems and other fields; software based upon ABS methods is now under development. Current ABS literature consists of about 400 papers. ABS methods provide a unification of several classes of classical algorithms and more efficient new solvers for a number of problems. In this paper we review ABS methods for linear systems and optimization, from both the point of view of theory and the numerical performance of ABSPACK.

Key Words: Linear equations, ABS methods, ABSPACK, Quasi-Newton methods, Diophantine equations, linear programming, feasible direction methods, interior point methods. . If s i = 0 and τ = v T i r i = 0, set x i+1 = x i , H i+1 = H i and go to (6). Otherwise stop, the system has no solution.

2001

We present a review and bibliography of the main results obtained during a research on ABS (Abaffy, Broyden, Spedicato) methods.

Annali dell'Università di Ferrara. Sezione 7: Scienze matematiche

We present the main results obtained during a research on ABS methods in the framework of the project Analisi Numerica e Matematica Computazionale.

Proceedings of the 2009 International Symposium …, 2009

We formulate the NP-hard absolute value equation as linear complementary problem when the singular values of A exceed one, and we proposed a mixed integer linear programming method to absolute value equation problem. The effectiveness of the method is demonstrated by its ability to solve random problems.

Mathematical and Computer Modelling, 2007

A method, called the (I.) ABS-MPVT algorithm, for solving a system comprising linear equations and linear inequalities is presented. This method is characterized by solving the system of linear equations first via the ABS algorithms and then solving an unconstrained minimization obtained by substituting the ABS general form of solutions into the system of linear inequalities. For the unconstrained minimization problem it can be solved by a (modified) parallel algorithm. The convergence of this method is also given.

Soft Computing, 2019

gave the general compromised solution of an L R fuzzy linear system using ABS algorithm. Here, using this general solution, we solve quadratic programming problems with fuzzy L R variables. We convert fuzzy quadratic programming problem to a crisp quadratic problem by using general solution of fuzzy linear system. By using this method, the crisp optimization problem has fewer variables in comparison with other methods, specially when rank of the coefficient matrix is full. Thus, solving the fuzzy quadratic programming problem by using our proposed method is computationally easier than the solving fuzzy quadratic programming problem by using ranking function. Also, we study the fuzzy quadratic programming problem with symmetric variables. We show that, in this case, the associate quadratic programming problem is a convex problem, and thus, we able to find the global optimal.

We consider on application of the ABS procedure to the linear systems arising from the primal-dual interior point method where Newton methods is used to compute a path to the solution. When approaching the solution the linear system, which has the form of normal equations of the second kind, becomes more and more ill conditioned. We show how the use of the Huang algorithm in the ABS class can reduce the ill conditioning. Preliminary numerical experiments show that the proposed approach can provide a residual in the computed solution up to sixteen orders lower. Key words : ABS methods, normal equations of the second kind, Huang algorithm, primal-dual interior point method, Newton method.

Abaffy-Broyden-Spedicato (ABS) methods have been used extensively for solving linear and nonlinear systems of equations [cf. J. Abaffy, C. G. Broyden and E. Spedicato, Numer. Math. 45, No. 3, 361–376 (1984; Zbl 0535.65009)]. Here we use them to find explicitly all solutions of a system of m linear inequalities in n variables, m≤n, with full rank matrix. We apply these results to the linear programming (LP) problem with m≤n inequality constraints, obtaining optimality conditions and an explicit representation of all solutions.