A polynomial-time algorithm, based on Newton’s method, for linear programming. (English) Zbl 0654.90050

Renegar, James

Math. Program., Ser. A 40, No. 1, 59-93 (1988).

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

MSC:

90C05	Linear programming
68Q25	Analysis of algorithms and problem complexity
65K05	Numerical mathematical programming methods

Keywords:

interior method; linear optimization; Newton’s method; polynomial time bound; ellipsoid algorithm; Karmarkar’s algorithm

References:

[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