# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
An $O\left({n}^{3}L\right)$ 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

$\phi \left(x,s\right)=\rho ln\left({x}^{T}s\right)-\sum _{j=1}^{n}ln\left({x}_{j}{s}_{j}\right),$

where $\rho \ge n$, 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 $\rho =n+\sqrt{n}$. The algorithm neither traces the central path nor uses projective transformations. It converges to the optimal solution set in O($\sqrt{n}L\right)$ iterations and uses $O\left({n}^{3}L\right)$ arithmetic operations.

##### MSC:
 90C05 Linear programming 90C60 Abstract computational complexity for mathematical programming problems 65K05 Mathematical programming (numerical methods) 90-08 Computational methods (optimization)
##### Keywords:
primal-dual potential function; interior algorithm