Solving linear program as linear system in polynomial time

A physically concise polynomial-time iterative-cum-non-iterative algorithm is presented to solve the linear program (LP) Minctxsubject toAx=b,x≥0. The iterative part–a variation of Karmarkar projective transformation algorithm–is essentially due to Barnes only to the extent of detection of basic var...

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Bibliographic Details
Published in:Mathematical and computer modelling Vol. 53; no. 5-6; pp. 1056 - 1073
Main Authors: Sen, Syamal K., Ramakrishnan, Suja, Agarwal, Ravi P.
Format: Journal Article
Language:English
Published: Kidlington Elsevier Ltd 01.03.2011
Elsevier
Subjects:
Algebra
Algebraic geometry
algorithms
Barnes algorithm
Calculus of variations and optimal control
Error-free computation
Exact sciences and technology
Linear program
linear programming
Linear system
Mathematical analysis
Mathematics
Matlab program
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Polynomial-time iterative-cum-non-iterative algorithm
Sciences and techniques of general use
Barnes algorithm
Error-free computation
Matlab program
Linear system
Polynomial-time iterative-cum-non-iterative algorithm
Linear program
Solution uniqueness
Error estimation
Iterative method
Linear time
Iteration
Polynomial-time
Implementation
Convergence
Direct method
Mathematical model
Computer aided analysis
Numerical linear algebra
iterative-cum-non-iterative algorithm
Algorithm
Computational complexity
Polynomial time
Numerical analysis
Matrix inversion
Applied mathematics
Optimal solution
Problem solving
ISSN:0895-7177, 1872-9479
Online Access:Get full text
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Summary:A physically concise polynomial-time iterative-cum-non-iterative algorithm is presented to solve the linear program (LP) Minctxsubject toAx=b,x≥0. The iterative part–a variation of Karmarkar projective transformation algorithm–is essentially due to Barnes only to the extent of detection of basic variables of the LP taking advantage of monotonic convergence. It involves much less number of iterations than those in the Karmarkar projective transformation algorithm. The shrunk linear system containing only the basic variables of the solution vector x resulting from Ax=b is then solved in the mathematically non-iterative part. The solution is then tested for optimality and is usually more accurate because of reduced computation and has less computational and storage complexity due to smaller order of the system. The computational complexity of the combination of these two parts of the algorithm is polynomial-time O(n3). The boundedness of the solution, multiple solutions, and no-solution (inconsistency) cases are discussed. The effect of degeneracy of the primal linear program and/or its dual on the uniqueness of the optimal solution is mentioned. The algorithm including optimality test is implemented in Matlab which is found to be useful for solving many real-world problems.
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ISSN:0895-7177
1872-9479
DOI:10.1016/j.mcm.2010.11.065