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An algorithm for solving linear programming problems in $O\left({n}^{3}L\right)$ operations. (English) Zbl 0691.90053
Progress in mathematical programming. Interior-point and related methods, Proc. Conf., Pacific Grove/Calif. 1987, 1-28 (1989).

[For the entire collection see Zbl 0669.00026.]

This paper describes a short-step penalty function algorithm that solves linear programming problems in no more than $O\left({n}^{0·5}L\right)$ iterations. The total number of arithmetic operations is bounded by $O\left({n}^{3}L\right)$, carried on with the same precision as that in Karmarkar’s algorithm. Each iteration updates a penalty multiplier and solves a Newton-Raphson iteration on the traditional logarithmic barrier function using approximated Hessian matrices. The resulting sequence follows the path of optimal solutions for the penalized functions as in a predictor-corrector homotopy algorithm.

Reviewer: J.Abrham

##### MSC:
 90C05 Linear programming 49M15 Newton-type methods in calculus of variations 68Q25 Analysis of algorithms and problem complexity 65K05 Mathematical programming (numerical methods) 49M30 Other numerical methods in calculus of variations