Linear Programming pp 289-301 | Cite as

# The Central Path

## Abstract

In this chapter, we begin our study of an alternative to the simplex method for solving linear programming problems. The algorithm we are going to introduce is called a *path-following method*. It belongs to a class of methods called *interior-point methods*. The path-following method seems to be the simplest and most natural of all the methods in this class, so in this book we focus primarily on it. Before we can introduce this method, we must define the path that appears in the name of the method. This path is called the *central path* and is the subject of this chapter. Before discussing the central path, we must lay some groundwork by analyzing a nonlinear problem, called the *barrier problem*, associated with the linear programming problem that we wish to solve.

## Keywords

Lagrange Multiplier Barrier Function Linear Programming Problem Central Path Nonempty Interior## Preview

Unable to display preview. Download preview PDF.

## References

- Fiacco, A. & McCormick, G. (1968),
*Nonlinear Programming: Sequential Unconstrainted Minimization Techniques*, Research Analysis Corporation, McLean Virginia. Republished in 1990 by SIAM, Philadelphia.Google Scholar - Karmarkar, N. (1984), ‘A new polynomial time algorithm for linear programming’,
*Combinatorica***4**, 373–395.MATHCrossRefMathSciNetGoogle Scholar - Megiddo, N. (1989), Pathways to the optimal set in linear programming,
*in*N. Megiddo, ed., ‘Progress in Mathematical Programming’, Springer-Verlag, New York, pp. 131–158.Google Scholar - Huard, P. (1967), Resolution of mathematical programming with nonlinear constraints by the method of centers,
*in*J. Abadie, ed., ‘Nonlinear Programming’, North-Holland, Amsterdam, pp. 209–219.Google Scholar - Bayer, D. & Lagarias, J. (1989a), ‘The nonlinear geometry of linear programming. I. affine and projective scaling trajectories’,
*Transactions of the AMS***314**, 499–525.MATHMathSciNetGoogle Scholar - Bayer, D. & Lagarias, J. (1989b), ‘The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories’,
*Transactions of the AMS***314**, 527–581.MATHMathSciNetGoogle Scholar - Luenberger, D. (1984),
*Introduction to Linear and Nonlinear Programming*, Addison-Wesley, Reading MA.Google Scholar - Bertsekas, D. (1995),
*Nonlinear Programming*, Athena Scientific, Belmont MA.MATHGoogle Scholar - Nash, S. & Sofer, A. (1996),
*Linear and Nonlinear Programming*, McGraw-Hill, New York.Google Scholar