zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
A polynomial-time algorithm, based on Newton’s method, for linear programming. (English) Zbl 0654.90050
The paper includes a new interior method for linear optimization problems based on Newton’s method. A polynomial time bound is proven for this algorithm. The algorithm is compared with the ellipsoid algorithm and with Karmarkar’s algorithm. The proposed algorithm is conceptually simpler than either of those algorithms. Furthermore, the bounds are compared. The most important result is the O $(m+n)L)$ bound on the number of iterations where n is the number of variables and m the number of constraints.
Reviewer: J.Guddat

90C05Linear programming
68Q25Analysis of algorithms and problem complexity
65K05Mathematical programming (numerical methods)
Full Text: DOI
[1] A. Aho, J. Hopcroft and J. Ullman,The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, MA, 1974). · Zbl 0326.68005
[2] D.A. Bayer and J.C. Lagarias, ”The non-linear geometry of linear programming, I. Affine and projective scaling trajectories, II. Legendre transform coordinates, III. Central trajectories,” preprints, AT&T Bell Laboratories (Murray Hill, NJ, 1986).
[3] L. Blum, talk at Workshop on Problems Relating Numerical Analysis to Computer Science, Mathematical Sciences Research Institute, Berkeley, California (January 1986).
[4] L. Blum, ”Towards an asymptotic analysis of Karmarkar’s algorithm,”Information Processing Letters 23 (1986) 189--194. · Zbl 0625.90050 · doi:10.1016/0020-0190(86)90134-1
[5] P. Huard, ”Resolution of mathematical programming with non-linear constraints by the method of centers,” in: J. Abadie, ed.,Non-Linear Programming (North-Holland, Amsterdam, 1967) pp. 207--219. · Zbl 0157.49701
[6] N. Karmarkar, ”A new polynomial-time algorithm for linear programming,” in:Proceedings of the 16th Annual ACM Symposium on Theory of Computing (1984), ACM, New York, 1984, pp. 302--311; revised version:Combinatorica 4 (1984) pp. 373--395. · Zbl 0557.90065
[7] L.G. Khachiyan, ”A polynomial algorithm in linear programming,”Soviet Mathematics Doklady 20 (1979) pp. 191--194. · Zbl 0414.90086
[8] L.G. Khachiyan, ”Polynomial algorithms in linear programming,”USSR Computational Mathematics and Mathematical Physics 20 (1980) pp. 53--72. · Zbl 0459.90047 · doi:10.1016/0041-5553(80)90061-0
[9] J. Lagarias, talk at Mathematical Sciences Research Institute (Berkeley, California, December, 1985).
[10] N. Megiddo and M. Shub, ”Boundary behavior of interior point algorithms in linear programming,” Research Report RJ5319, IBM Thomas J. Watson Research Center (1986). · Zbl 0675.90050
[11] S. Smale, ”On the efficiency of algorithms of analysis,”Bulletin of the American Mathematical Society 13 (1985) pp. 87--121. · Zbl 0592.65032 · doi:10.1090/S0273-0979-1985-15391-1
[12] S. Smale, ”Algorithms for solving equations,” to appear in:Proceedings, International Congress of Mathematicians (Berkeley, 1986). · Zbl 0595.65048
[13] Gy. Sonnevend, ”A new method for solving a set of linear (convex) inequalities and its applications for identification and optimization,” preprint, Department of Numerical Analysis, Institute of Mathematics, Eötvös University, 1088, Budapest, Muzeum Körut 6--8. · Zbl 0638.90085
[14] Gy. Sonnevend, ”An analytical centre’ for polyhedrons and new classes of global algorithms for linear (smooth convex) programming,” preprint, Department of Numerical Analysis, Institute of Mathematics, Eötvös University, 1088, Budapest, Muzeum Körut 6--8. · Zbl 0602.90106
[15] P. Vaidya, ”An algorithm for linear programming which requires O((m+n)n 2+(m+n)1.5 n)L) arithmetic operations,” AT&T Bell Laboratories, Murray Hill, NJ (1987). · Zbl 0708.90047
[16] J.H. Wilkinson,The algebraic Eigenvalue Problem (Oxford University Press, Oxford, 1965). · Zbl 0258.65037