Algorithmica

, Volume 6, Issue 1–6, pp 153–181

# Search directions for interior linear-programming methods

• Clovis C. Gonzaga
Article

## Abstract

Since Karmarkar published his algorithm for linear programming, several different interior directions have been proposed and much effort was spent on the problem transformations needed to apply these new techniques. This paper examines several search directions in a common framework that does not need any problem transformation. These directions prove to be combinations of two problem-dependent vectors, and can all be improved by a bidirectional search procedure.

We conclude that there are essentially two polynomial algorithms: Karmarkar's method and the algorithm that follows a central trajectory, and they differ only in a choice of parameters (respectively lower bound and penalty multiplier).

## Key words

Karmarkar's algorithm Linear programming Projective algorithm Conical projection Interior methods

## References

1. [1]
I. Adler, M. Resende, and G. Veiga. An Implementation of Karmarkar's Algorithm for Linear Programming. Report ORC86-8, Operations Research Center, University of California, Berkeley, May 1986.Google Scholar
2. [2]
K. Anstreicher. A monotonic projective algorithm for fractional linear programming.Algorithmica,1:483–498, 1986.
3. [3]
E. R. Barnes. A variation on Karmarkar's algorithm for solving linear programming problems.Mathematical Programming,36:174–182, 1986.
4. [4]
E. R. Barnes and D. L. Jensen. A Polynomial-Time Version of the Affine-Scaling Algorithm. Manuscript in preparation, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1987.Google Scholar
5. [5]
K. Bayer and J. C. Lagarias. Karmarkar's Linear Programming Algorithm and Newton's Method. Preprints, AT&T Bell Laboratories, Murray Hill, NJ, 1986.Google Scholar
6. [6]
T. Cavalier and A. Soyster. Some Computational Experience and a Modification of the Karmarkar Algorithm. Working Paper 85–105, Department of Industrial and Management System Engineering, Pennsylvania State University, February 1985.Google Scholar
7. [7]
W. C. Davidon. Conic approximations and collinear scalings for optimizers.SIAM Journal on Numerical Analysis,17:268–281, 1980.
8. [8]
A. Fiacco and G. McCormick. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York, 1955.Google Scholar
9. [9]
K. R. Frisch. The Logarithmic Potential Method of Convex Programming. Memorandum, University Institute of Economics, Oslo, May 1955.Google Scholar
10. [10]
D. Gay. A variant of Karmarkar's linear programming algorithm for problems in standard form.Mathematical Programming,37:81–89, 1987.
11. [11]
P. Gill, W. Murray, M. Saunders, J. Tomlin, and M. Wright. On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method.Mathematical Programming,36:183–209, 1986.
12. [12]
C. Gonzaga. A Conical Projection Algorithm for Linear Programming. Memorandum UCB/ERL M85/61, Electronics Research Laboratory, University of California, Berkeley, CA, July 1985.Google Scholar
13. [13]
C. Gonzaga. An Algorithm for Solving Linear Programming Problems inO(n 3 L) operations. Memorandum UCB/ERL M87/10, Electronics Research Laboratory, University of California, Berkeley, CA, March 1987.Google Scholar
14. [14]
C. Gonzaga. Conical Projection Algorithms for Linear Programming. Memorandum UCB/ERL M87/11, Electronics Research Laboratory, University of California, Berkeley, CA, March 1987. To appear inMathematical Programming.Google Scholar
15. [15]
H. Imai. On the convexity of the multiplicative version of Karmarkar's potential function.Mathematical Programming,40:29–32, 1988.
16. [16]
M. Iri and H. Imai. A multiplicative penalty function method for linear programming.Algorithmica,1:455–482, 1986.
17. [17]
N. Karmarkar. A new polynomial time algorithm for linear programming.Combinatorica,4:373–395, 1984.
18. [18]
J. C. Lagarias. Personal communication.Google Scholar
19. [19]
N. Megiddo. Pathways to the Optimal Set in Linear Programming. Research Report RJ 5295, IBM Almaden Research Center, San Jose, CA, 1986.Google Scholar
20. [20]
M. Padberg. Solution of a Nonlinear Programming Problem Arising in the Projective Method for Linear Programming. Manuscript, New York University, New York, March 1985.Google Scholar
21. [21]
J. Renegar. A polynomial-time algorithm based on Newton's method for linear programming.Mathematical Programming,40:59–94, 1988.
22. [22]
M. Todd and B. Burrell. An extension of Karmarkar's algorithm for linear programming using dual variables.Algorithmica,1:409–424, 1986.
23. [23]
P. M. Vaidya. An Algorithm for Linear Programming which RequiresO(((m+n)n 2+(m+n)1.5 n)L) Arithmetic Operations. Preprint, AT&T Bell Laboratories, Murray Hill, NJ, 1987.Google Scholar
24. [24]
R. J. Vanderbei, M. J. Meketon, and B. A. Freedman. A modification of Karmarkar's linear programming algorithm.Algorithmica,1:395–407, 1986.
25. [25]
H. Yamashita. A Polynomially and Quadratically Convergent Method for Linear Programming. Manuscript, Mathematical Systems Institute Inc., Tokyo, October 1986.Google Scholar