Linear Programming Using ABS Method

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.

We consider the application of the ABS procedure to the linear system arising in the primal-dual interior point method where Newton method is used to compute the 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.

2012

The main aim of this paper intends to discuss the solution of general dual fuzzy linear system (GDFLS)

SpringerPlus, 2016

Background We consider the absolute value equations (AVEs): where A ∈ R n×n , b ∈ R n , and |x| denotes a vector in R n , whose i-th component is |x i |. A more general form of the AVEs, Ax + B|x| = b, was introduced by Rohn (2004) and researched in a more general context in Mangasarian (2007a). Hu et al. (2011) proposed a generalized Newton method for solving absolute value equation Ax + B|x| = b associated with second order cones, and showed that the method is globally linearly and locally quadratically convergent under suitable assumptions. As was shown in Mangasarian and Meyer (2006) by Mangasarian, the general NP-hard linear complementarity problems (LCPs) (Cottle and Dantzing 1968; Chung 1989; Cottle et al. 1992) subsume many mathematical programming problems such as absolute value equations (AVEs) (1), which own much simpler structure than any LCP. Hence it has inspired many scholars to study AVEs. And in Mangasarian and Meyer (2006) the AVEs (1) was investigated in detail theoretically, the bilinear program and the generalized LCP were prescribed there for the special case when the singular values of A are not less than 1. Based on the LCP reformulation, sufficient conditions for the existence and nonexistence of solutions are given in this paper. Mangasarian also has used concave minimization model (Mangasarian 2007b), dual complementarity (Mangasarian 2013), linear complementarity (Mangasarian 2014a), linear programming (Mangasarian 2014b) and a hybrid algorithm (Mangasarian 2015) to solve AVEs (1). Hu and Huang reformulated a system of absolute value equations as a standard linear complementarity problem without any

Applied Mathematics and Computation, 2015

In this paper, we introduce and analyze two new methods for solving the NP-hard absolute value equations (AVE) Ax − |x| = b, where A is an arbitrary n × n real matrix and b ∈ R n , in the case, singular value of A exceeds 1. The comparison with other known methods is carried to show the effectiveness of the proposed methods for a variety of randomly generated problems. The ideas and techniques of this paper may stimulate further research.

2001

The results of computational experiments with ABS algorithms for overdetermined linear systems are reported.

GANIT, Bangladesh Mathematical Society, 2013

In this paper, we study the methodology of primal dual solutions in Linear Programming (LP) & Linear Fractional Programming (LFP) problems. A comparative study is also made on different duals of LP & LFP. We then develop an improved decomposition approach for showing the relationship of primal and dual approach of LP & LFP problems by giving algorithm. Numerical examples are given to demonstrate our method. A computer programming code is also developed for showing primal and dual decomposition approach of LP & LFP with proper instructions using AMPL. Finally, we have drawn a conclusion stating the privilege of our method of computation.

Ferdowsi University of Mashhad, 2022

In 1984, Abaffy, Broyden, and Spediacto (ABS) introduced a class of the so-called ABS algorithms to solve systems of real linear equations. Later, the scaled ABS algorithm, the extended ABS algorithm, the block ABS algorithm, and the integer ABS algorithm were introduced leading to various well-known matrix factorizations. Here, we present a generalization of ABS algorithms containing all matrix factorizations such as triangular, W Z, and ZW. We present the octant interlocking factorization and show that the generalized ABS algorithm is more general to produce the octant interlocking factorization.

Optimization Letters, 2021

In this paper, we study the absolute value equation (AVE) Ax − b = |x|. One effective approach to handle AVE is by using concave minimization methods. We propose a new method based on concave minimization methods. We establish its finite convergence under mild conditions. We also study some classes of AVEs which are polynomial time solvable. Keywords Absolute value equation • Concave minimization algorithms • Linear complementarity problem where A ∈ R n×n , b ∈ R n and | • | denotes absolute value. In general, (AVE) is an NP-hard problem [16]. Since a general linear complementarity problem can be formulated as an absolute value equation, several methods, such as Newton-like methods [3,15,31] or concave optimization methods [20,21], have been proposed for solving (AVE).

2001

The results of computational experiments with ABS algorithms for KKT linear systems are reported.

2014

This paper presents a new approach for the solution of Linear Programming Problems with the help of LU Factorization Method of matrices. This method is based on the fact that a square matrix can be factorized into the product of unit lower triangular matrix and upper triangular matrix. In this method, we get direct solution without iteration. We also show that this method is better than simplex method.