Exterior Point Simplex Algorithm Research Articles

Improving a primal–dual simplex-type algorithm using interior point methods

ABSTRACTInterior point methods and simplex-type algorithms are the most widely-used algorithms for solving linear programming problems. The simplex algorithm has many important applications. Hence, even small improvements in simplex-type algorithms could result in noticeable practical impact. This paper presents a hybrid algorithm that combines the strengths of interior point methods and exterior point simplex algorithms. It applies an interior point method for a few iterations leading to significant improvement of the objective function value. At this point, the proposed algorithm uses an exterior point simplex algorithm to find an optimal solution. A crucial point is the selection of the interior point that will be used by the exterior point simplex algorithm to calculate a direction to the feasible region. The goal of the proposed implementation is twofold: (i) improve the performance of the exterior point algorithm, and (ii) find an optimal basic solution starting from an interior point (purification process). The latter goal is very important since an optimal basic solution can be used to solve closely related linear programming problems (warm-start) and linear programming relaxations of integer programming problems. Computational results on a set of benchmark problems (Netlib, Kennington, Mészáros) are presented to demonstrate the efficiency of the proposed hybrid algorithm. The results show that the proposed algorithm is faster than the exterior point simplex algorithm.

Computational complexity of the exterior point simplex algorithm

In this paper, we investigate the computational behavior of the exterior point simplex algorithm. Up until now, there has been a major difference observed between the theoretical worst case complexity and practical performance of simplex-type algorithms. Computational tests have been carried out on randomly generated sparse linear problems and on a small set of benchmark problems. Specifically, 6780 linear problems were randomly generated, in order to formulate a respectable amount of experiments. Our study consists of the measurement of the number of iterations that the exterior point simplex algorithm needs for the solution of the above mentioned problems and benchmark dataset. Our purpose is to formulate representative regression models for these measurements, which would play a significant role for the evaluation of an algorithm’s efficiency. For this examination, specific characteristics, such as the number of constraints and variables, the sparsity and bit length, and the condition of matrix A, of each linear problem, were taken into account. What drew our attention was that the formulated model for the randomly generated problems reveal a linear relation among these characteristics.

Efficient GPU-based implementations of simplex type algorithms

Recent hardware advances have made it possible to solve large scale Linear Programming problems in a short amount of time. Graphical Processing Units (GPUs) have gained a lot of popularity and have been applied to linear programming algorithms. In this paper, we propose two efficient GPU-based implementations of the Revised Simplex Algorithm and a Primal–Dual Exterior Point Simplex Algorithm. Both parallel algorithms have been implemented in MATLAB using MATLAB’s Parallel Computing Toolbox. Computational results on randomly generated optimal sparse and dense linear programming problems and on a set of benchmark problems (netlib, kennington, Mészáros) are also presented. The results show that the proposed GPU implementations outperform MATLAB’s interior point method.

On Fuzzy Primal Exterior Point Simplex Algorithm for Solving Linear Programming Problems with Fuzzy Numbers

On Fuzzy Primal Exterior Point Simplex Algorithm for Solving Linear Programming Problems with Fuzzy Numbers

An experimental investigation of a primal–dual exterior point simplexalgorithm

The aim of this paper is to present an experimental investigation of a Primal–Dual Exterior Point Simplex Algorithm (PDEPSA) for Linear Programming problems (LPs). There was a huge gap between the theoretical worst-case complexity and practical performance of simplex type algorithms. PDEPSA traverses across the interior of the feasible region in an attempt to avoid combinatorial complexities of vertex following algorithms. In order to gain an insight into the practical behaviour of the proposed algorithm, we have performed some computational experiments. Our computational results demonstrate that PDEPSA is faster comparable with simplex algorithm and the primal exterior point algorithm. A clear advantage is obtained by PDEPSA in number of iterations and CPU time.

Dynamic trees in exterior-point Simplex-type algorithms for network flow problems

Abstract Recently, a new Dual Network Exterior-Point Simplex Algorithm (DNEPSA) for the Minimum Cost Network Flow Problem (MCNFP) has been developed. In extensive computational studies, DNEPSA performed better than the classical Dual Network Simplex Algorithm (DNSA). In this paper, we present for the first time how to utilize the dynamic trees data structure in the DNEPSA algorithm, in order to achieve an improvement of the amortized complexity per pivot. Our work constitutes a first step towards the development of an efficient implementation of DNEPSA.

On a dual network exterior point simplex type algorithm and its computational behavior

The minimum cost network flow problem, (MCNFP) constitutes a wide category of network flow problems. Recently a new dual network exterior point simplex algorithm (DNEPSA) for the MCNFP has been developed. This algorithm belongs to a special “exterior point simplex type” category. Similar to the classical dual network simplex algorithm (DNSA), this algorithm starts with a dual feasible tree-solution and after a number of iterations, it produces a solution that is both primal and dual feasible, i.e. it is optimal. However, contrary to the DNSA, the new algorithm does not always maintain a dual feasible solution. Instead, it produces tree-solutions that can be infeasible for the dual problem and at the same time infeasible for the primal problem. In this paper, we present for the first time, the mathematical proof of correctness of DNEPSA, a detailed comparative computational study of DNEPSA and DNSA on sparse and dense random problem instances, a statistical analysis of the experimental results, and finally some new results on the empirical complexity of DNEPSA. The analysis proves the superiority of DNEPSA compared to DNSA in terms of cpu time and iterations.

On the initialization methods of an exterior point algorithm for the assignment problem

In this paper, we present a theoretical investigation and an extensive computational study of exterior point simplex algorithm (EPSA) initialization methods for the assignment problem (AP). We describe the exterior point algorithm using three different initialization methods. Effective implementations are explored for each initialization method. Then we perform an experimental evaluation on a large set of benchmark problems from the TSPLib 95 and OR Library collections. The results obtained demonstrate the advantages of the three initialization methods. Finally, we give a theoretical justification of the initialization methods efficiency. We explain theoretically the computational ranking for these methods.

An exterior simplex type algorithm for the Minimum Cost Network Flow Problem

In this paper a new Network Exterior Point Simplex Algorithm (NEPSA) for the Minimum Cost Network Flow Problem (MCNFP) is analytically presented. NEPSA belongs to a special simplex type category and is a modification of the classical network simplex algorithm. The main idea of the algorithm is to compute two flows. One flow is basic but not always feasible and the other is feasible but not always basic. A complete proof of correctness for the proposed algorithm is also presented. Moreover, the computational behavior of NEPSA is shown by an empirical study carried out for randomly generated sparse instances created by the well-known GRIDGEN network problem generator.

An efficient simplex type algorithm for sparse and dense linear programs

We present a big- M method for solving general linear programming problems with a recently developed exterior point simplex algorithm (EPSA). We also provide intuitive motivations to EPSA. We describe major steps in implementing EPSA for large-scale linear programming. Our implementation was carried out under the MATLAB environment. This algorithm seems to be more efficient than the classical primal simplex algorithm (PSA), employing Dantzig’s rule. Preliminary computational studies on randomly generated sparse and dense small and medium linear programs support this belief. In particular, EPSA is up to 10 times faster than PSA on medium size linear programs. This translates into corresponding savings in CPU time. Although the computational effort required in each step of EPSA requires more time compared to an iteration step of PSA, the improvement of EPSA comes from the fact that it requires adequately less iterations than PSA. Moreover, as the problem size increases and the problem density decreases, EPSA gets relatively faster.

An exterior point simplex algorithm for (general) linear programming problems

We present an exterior point simplex type algorithm that possesses a new monotonic property. A dual feasible basic solution is required to start with. Intermediate solutions are neither primal nor dual feasible. Cycling-free pivoting rules and an exponentional example are presented.