Barnes, Earl R. A variation on Karmarkar’s algorithm for solving linear programming problems. (English) Zbl 0626.90052 Math. Program. 36, 174-182 (1986). The algorithm described in the paper uses the ideas of the original Karmarkar algorithm, but differs in some respects. At first the minimum value of the objective function has not to be known in advance. The algorithm solves the standard form of a linear programming problem with the requirement that the primal and dual problems have no degenerate basic feasible solutions. The algorithm starts from the feasible point \(y\in R^ n\) such that \(y_ i>0\), \(i=1\), 2,..., n and produces a monotone decreasing sequence of values of the goal function. The main difference between this algorithm and that given by R. J. Vanderbei, M. S. Meketon and B. A. Freedman [Algorithmica 1, 395-407 (1986; Zbl 0626.90056)] is that the constraint \(x\geq 0\) is replaced by \[ \sum^{n}_{i=1}\frac{(x_ i-y_ i)^ 2}{y^ 2_ i}<R^ 2, \] where \(0<R<1\). The convergence of the algorithm is proved and a numerical method for finding a starting point is shown. At last in the case of absence of degeneracy it is proved that the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero. Reviewer: M.Todorov Cited in 6 ReviewsCited in 115 Documents MSC: 90C05 Linear programming Keywords:nondegeneracy; Karmarkar algorithm Citations:Zbl 0626.90056 PDFBibTeX XMLCite \textit{E. R. Barnes}, Math. Program. 36, 174--182 (1986; Zbl 0626.90052) Full Text: DOI References: [1] T.M. Cavalier and A.L. Soyster, ”Some computational experience and a modification of the Karmarkar algorithm,” The Pennsylvania State University, ISME Working Paper 85-105, 1985. [2] P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin and M.H. Wright, ”On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method,” Manuscript, Stanford University, 1985. · Zbl 0624.90062 [3] N. Karmarkar, ”A new polynomial-time algorithm for linear programming,” Proceedings of the 16th Annual ACM Symposium on Theory of Computing, 1984, pp. 302–311. · Zbl 0557.90065 [4] R.J. Vanderbei, M.S. Meketon and B.A. Freedman, ”A modification of Karmarkar’s linear programming algorithm,” Manuscript, AT & T Bell Laboratories, Holmdel, New Jersey, June 1985. · Zbl 0626.90056 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.