# 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 algorithm for linear programming which requires $O\left(\left(\left(m+n\right){n}^{2}+{\left(m+n\right)}^{1·5}n\right)L\right)$ arithmetic operations. (English) Zbl 0708.90047
The author gives an algorithm for solving a linear programming problem $max\left\{{c}^{T}x$; Ax$\ge b\right\}$ where A is an $m×n$ matrix. The algorithm employs the idea due to J. Renegar [Math. Program., Ser. A 40, No.1, 59-93 (1988; Zbl 0654.90050)] and proceeds by constructing a sequence of smaller and smaller polytopes which shrink towards the optimal vertex. At each iteration the algorithm moves from the center of the current polytope to the center of the next one by minimizing a linear function over an ellipsoid. The total number of arithmetic operations required is $O\left(\left(\left(m+n\right){n}^{2}+{\left(m+n\right)}^{1·5}n\right)L\right)$, where L is bounded by the number of bits in the input; thus the algorithm is faster than that of N. Karmarkar [Combinatorica 4, 373-395 (1984; Zbl 0557.90065)] by a factor of $\sqrt{m}$.
Reviewer: J.Rohn
##### MSC:
 90C05 Linear programming 90C60 Abstract computational complexity for mathematical programming problems 52B12 Special polytopes (linear programming, centrally symmetric, etc.)
##### Keywords:
polynomial time algorithm
##### References:
 [1] D.A. Bayer and J.C. Lagarias, ”The non-linear geometry of linear programming I. Affine and projective scaling trajectories,”Transactions of the American Mathematical Society (1989), to appear. [2] J. Edmonds, ”Systems of distinct representatives and linear algebra,”Journal of Research of the National Bureau of Standards 71B (1967) 241–245. [3] F.R. Gantmacher,Matrix Theory, Vol. 1 (Chelsea, London, 1959) Chapter 2. [4] C. Gonzaga, ”An algorithm for solving linear programming problems in O(n 3 L) operations,” Memorandum UCB/ERL M87/10, Electronics Research Laboratory, University of Berkeley (Berkeley, CA, 1987). [5] N. Karmarkar, ”A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395. · Zbl 0557.90065 · doi:10.1007/BF02579150 [6] L.G. Khachian, ”Polynomial algorithms in linear programming,”Žurnal Vyčislitel’noî Matematiki i Matematičeskoî Fiziki 20 (1980) 53–72. [7] C. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, NJ, 1982). [8] J. Renegar, ”A polynomial-time algorithm, based on Newton’s method, for linear programming,”Mathematical Programming, 40 (1988) 59–93. · Zbl 0654.90050 · doi:10.1007/BF01580724 [9] Gy. Sonnevand, ”An analytical center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming,” preprint, Department of Numerical Analysis, Institute of Mathematics, Eötvös University (Budapest, 1989) 6–8. [10] G.W. Stewart,Introduction to Matrix Computations (Academic Press, New York, 1973). [11] P.M. Vaidya, ”An algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations,”Proceedings 19th Annual ACM Symposium Theory of Computing (1987) 29–38. [12] J.H. Wilkinson,The Algebraic Eigenvalue Problem (Oxford University Press (Clarendon), London and New York, 1965). [13] G. Zoutendijk,Mathematical Programming Methods (North-Holland, New York, 1976).