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An O(n 3 L) potential reduction algorithm for linear programming. (English) Zbl 0725.90063
Mathematical developments arising from linear programming, Proc. AMS-IMS- SIAM Jt. Summer Res. Conf., Brunswick/ME (USA) 1988, Contemp. Math. 114, 91-107 (1990).

Summary: [For the entire collection see Zbl 0722.00047.]

We describe a primal-dual potential function for linear programming

ϕ(x,s)=ρln(x T s)- j=1 n ln(x j s j ),

where ρn, x is the primal variable and s is the dual-slack variable in the standard linear programming form. As a result, we develop an interior algorithm seeking reductions in the potential function with ρ=n+n. The algorithm neither traces the central path nor uses projective transformations. It converges to the optimal solution set in O(nL) iterations and uses O(n 3 L) arithmetic operations.

MSC:
90C05Linear programming
90C60Abstract computational complexity for mathematical programming problems
65K05Mathematical programming (numerical methods)
90-08Computational methods (optimization)