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A variation on Karmarkar’s algorithm for solving linear programming problems. (English) Zbl 0626.90052

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 $x\ge 0$ is replaced by

$\sum _{i=1}^{n}\frac{{\left({x}_{i}-{y}_{i}\right)}^{2}}{{y}_{i}^{2}}<{R}^{2},$

where $0. 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

MSC:
 90C05 Linear programming
Keywords:
nondegeneracy; Karmarkar algorithm
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. [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. [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.