scholarly journals A primal-dual exterior point algorithm for linear programming problems

2009 ◽  
Vol 19 (1) ◽  
pp. 123-132 ◽  
Author(s):  
Nikolaos Samaras ◽  
Angelo Sifelaras ◽  
Charalampos Triantafyllidis
Keyword(s):  
Linear Programming ◽  
Computational Study ◽  
Optimal Solution ◽  
Simplex Algorithm ◽  
Dual Algorithm ◽  
Optimal Linear ◽  
Primal Dual ◽  
Primal And Dual ◽  
Two Phases ◽  
Feasible Solutions

The aim of this paper is to present a new simplex type algorithm for the Linear Programming Problem. The Primal - Dual method is a Simplex - type pivoting algorithm that generates two paths in order to converge to the optimal solution. The first path is primal feasible while the second one is dual feasible for the original problem. Specifically, we use a three-phase-implementation. The first two phases construct the required primal and dual feasible solutions, using the Primal Simplex algorithm. Finally, in the third phase the Primal - Dual algorithm is applied. Moreover, a computational study has been carried out, using randomly generated sparse optimal linear problems, to compare its computational efficiency with the Primal Simplex algorithm and also with MATLAB's Interior Point Method implementation. The algorithm appears to be very promising since it clearly shows its superiority to the Primal Simplex algorithm as well as its robustness over the IPM algorithm.

scholarly journals Convergence of a short-step primal-dual algorithm based on the Gauss-Newton direction

2003 ◽  
Vol 2003 (10) ◽  
pp. 517-534 ◽  
Author(s):  
Serge Kruk ◽  
Henry Wolkowicz
Keyword(s):  
Optimal Solution ◽  
Path Following ◽  
Singular Value ◽  
Dual Algorithm ◽  
Strict Complementarity ◽  
Semidefinite Programs ◽  
Primal Dual ◽  
Newton Direction ◽  
Proof Of Convergence ◽  
Smallest Singular Value

We prove the theoretical convergence of a short-step, approximate path-following, interior-point primal-dual algorithm for semidefinite programs based on the Gauss-Newton direction obtained from minimizing the norm of the perturbed optimality conditions. This is the first proof of convergence for the Gauss-Newton direction in this context. It assumes strict complementarity and uniqueness of the optimal solution as well as an estimate of the smallest singular value of the Jacobian.

scholarly journals Automatic limit and shakedown analysis of 3-D steel frames

2007 ◽  
Vol 29 (3) ◽  
pp. 337-351
Author(s):  
Nguyen Dang Hung ◽  
Hoang Van Long
Keyword(s):  
Linear Programming ◽  
Three Dimensional ◽  
Steel Frames ◽  
Direct Calculation ◽  
Simplex Algorithm ◽  
Rigid Plastic ◽  
Programming Technique ◽  
Shakedown Analysis ◽  
Primal Dual ◽  
Automatic Choice

This paper presents an efficient algorithm for both limit and shakedown analysis of 3-D steel frames by kinematical method using linear programming technique. Several features in the application of linear programming for rigid-plastic analysis of three-dimensional steel frames are discussed, as: change of the variables, automatic choice of the initial basic matrix for the simplex algorithm, direct calculation of the dual variables by primal-dual technique. Some numerical examples are presented to demonstrate the robustness, efficiency of the proposed technique and computer program.

A least-squares primal-dual algorithm for solving linear programming problems

2002 ◽  
Vol 30 (5) ◽  
pp. 289-294 ◽  
Author(s):  
Earl Barnes ◽  
Victoria Chen ◽  
Balaji Gopalakrishnan ◽  
Ellis.L. Johnson
Keyword(s):  
Linear Programming ◽  
Least Squares ◽  
Dual Algorithm ◽  
Primal Dual ◽  
Linear Programming Problems

ON SOLVING SHORTEST PATHS WITH A LEAST-SQUARES PRIMAL-DUAL ALGORITHM

2008 ◽  
Vol 25 (02) ◽  
pp. 135-150
Author(s):  
I.-LIN WANG
Keyword(s):  
Linear Programming ◽  
Least Squares ◽  
Shortest Path ◽  
Shortest Paths ◽  
Arc Length ◽  
Dual Algorithm ◽  
Shortest Path Problems ◽  
Dijkstra's Algorithm ◽  
Primal Dual ◽  
Linear Programming Problems

Recently a new least-squares primal-dual (LSPD) algorithm, that is impervious to degeneracy, has effectively been applied to solving linear programming problems by Barnes et al., 2002. In this paper, we show an application of LSPD to shortest path problems with nonnegative arc length is equivalent to the Dijkstra's algorithm. We also compare the LSPD algorithm with the conventional primal-dual algorithm in solving shortest path problems and show their difference due to degeneracy in solving the 1-1 shortest path problems.

scholarly journals Considerations on Cycling in the Case of Linear Programming Problem (Lpp)

2018 ◽  
Vol 24 (3) ◽  
pp. 14-19
Author(s):  
Vasile Carutasu
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Optimal Solution ◽  
Simplex Algorithm ◽  
Fundamental Question ◽  
Cyclic Phenomenon ◽  
General Aspects ◽  
Do So

Abstract Ever since the onset of algorithms for determining the optimal solution or solutions for a linear programming problem (LPP), the question of the possibility of occurrence of cycling when one or other of these algorithms are applied was born. Thus, the fundamental question regarding this issue is under what conditions the cyclic phenomenon appears for a problem of linear programming and how to construct examples in which to do so, and as a continuation of it, which methods can be developed to avoid this phenomenon. In this study we will present some aspects regarding this issue starting from the primal simplex algorithm, by highlighting some general aspects that occur when this phenomenon happens

scholarly journals A Proposed Technique for Solving Linear Fractional Bounded Variable Problems

2012 ◽  
Vol 60 (2) ◽  
pp. 223-230
Author(s):  
H.K. Das ◽  
M. Babul Hasan
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Language ◽  
Simplex Algorithm ◽  
Computational Effort ◽  
New Method ◽  
Bounded Variables ◽  
Primal Dual ◽  
Dual Simplex ◽  
Dual Simplex Algorithm

In this paper, a new method is proposed for solving the problem in which the objective function is a linear fractional Bounded Variable (LFBV) function, where the constraints functions are in the form of linear inequalities and the variables are bounded. The proposed method mainly based upon the primal dual simplex algorithm. The Linear Programming Bounded Variables (LPBV) algorithm is extended to solve Linear Fractional Bounded Variables (LFBV).The advantages of LFBV algorithm are simplicity of implementation and less computational effort. We also compare our result with programming language MATHEMATICA.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11522 Dhaka Univ. J. Sci. 60(2): 223-230, 2012 (July) 

Aspects of the Cycling Phenomenon in the Linear Programming Problem (Lpp) Through the Example of Marshall and Suurballe

2018 ◽  
Vol 24 (3) ◽  
pp. 20-25
Author(s):  
Vasile Carutasu
Keyword(s):  
Complex System ◽  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Optimal Solution ◽  
Simplex Algorithm ◽  
Fundamental Question ◽  
Complete Analysis ◽  
System Of Inequalities ◽  
The Given

Abstract A complete analysis of the cycling phenomenon in the case of the linear programming problem (LPP) is far from being achieved. Even if [5] states that the answer to the fundamental question of this problem is found, the proposed solution is very difficult to apply, being necessary to find a solution of a complex system of inequalities. Additionally, it is difficult to recognize a problem that, by applying the primal simplex algorithm, leads us to the occurrence of this phenomenon. The example given by Marshall and Suurballe, but also the example given by Danzig, lead us to draw some useful conclusions about this phenomenon, whether the given problem admits the optimal solution or has an infinite optimal solution

scholarly journals Computing on Vertices in Data Mining

2021 ◽  
Author(s):  
Leon Bobrowski
Keyword(s):  
Data Mining ◽  
Linear Programming ◽  
Big Data ◽  
Problem Solving ◽  
Parameter Space ◽  
Predictive Models ◽  
Optimal Solution ◽  
Simplex Algorithm ◽  
Data Sets ◽  
Exchange Algorithms

The main challenges in data mining are related to large, multi-dimensional data sets. There is a need to develop algorithms that are precise and efficient enough to deal with big data problems. The Simplex algorithm from linear programming can be seen as an example of a successful big data problem solving tool. According to the fundamental theorem of linear programming the solution of the optimization problem can found in one of the vertices in the parameter space. The basis exchange algorithms also search for the optimal solution among finite number of the vertices in the parameter space. Basis exchange algorithms enable the design of complex layers of classifiers or predictive models based on a small number of multivariate data vectors.

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