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An algorithm for solving linear programming problems in $O(n\sp 3L)$ 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 $O(n\sp{0.5}L)$ iterations. The total number of arithmetic operations is bounded by $O(n\sp 3L)$, 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

90C05Linear programming
49M15Newton-type methods in calculus of variations
68Q25Analysis of algorithms and problem complexity
65K05Mathematical programming (numerical methods)
49M30Other numerical methods in calculus of variations