Linear Programming pp 293-307 | Cite as

# Implementation Issues for Interior-Point Methods

Chapter

## Abstract

In this chapter, we discuss implementation issues that arise in connection with the path-following method. The most important issue is the efficient solution of the systems of equations discussed in the previous chapter. As we saw, there are basically three choices, involving either the reduced KKT matrix, or one of the two matrices associated with the normal equations: or (Here, \({E}^{-2} = {Y }^{-1}W\) and \({D}^{-2} = {X}^{-1}Z\).) In the previous chapter, we explained that dense columns/rows are bad for the normal equations and that therefore one might be better off solving the system involving the reduced KKT matrix. But there is also a reason one might prefer to work with one of the systems of normal equations. The reason is that these matrices are positive definite. We shall show in the first section that there are important advantages in working with positive definite matrices. In the second section, we shall consider the reduced KKT matrix and see to what extent the nice properties possessed by positive definite matrices carry over to these matrices.

$$\displaystyle{ B = \left [\begin{array}{cc} - {E}^{-2} & A \\ {A}^{T} &{D}^{-2} \end{array} \right ], }$$

(20.1)

$$\displaystyle{ A{D}^{2}{A}^{T} + {E}^{-2} }$$

(20.2)

$$\displaystyle{ {A}^{T}{E}^{2}A + {D}^{-2}. }$$

(20.3)

## References

- Fourer, R., and Mehrotra, S. (1991). Solving symmetric indefinite systems in an interior point method for linear programming.
*Mathematical Programming, 62*, 15–40.CrossRefGoogle Scholar - Gill, P., Murray, W., Ponceleón, D., and Saunders, M. (1992). Preconditioners for indefinite systems arising in optimization.
*SIAM Journal on Matrix Analysis and Applications, 13*(1), 292–311.CrossRefGoogle Scholar - Lustig, I., Marsten, R., and Shanno, D. (1994). Interior point methods for linear programming: Computational state of the art.
*ORSA Journal on Computing, 6*, 1–14.CrossRefGoogle Scholar - Turner, K. (1991). Computing projections for the Karmarkar algorithm.
*Linear Algebra and Its Applications, 152*, 141–154.CrossRefGoogle Scholar - Vanderbei, R. (1994). Interior-point methods: Algorithms and formulations.
*ORSA Journal on Computing, 6*, 32–34.CrossRefGoogle Scholar - Vanderbei, R. (1995). Symmetric quasi-definite matrices.
*SIAM Journal on Optimization, 5*(1), 100–113.CrossRefGoogle Scholar - Vanderbei, R., and Carpenter, T. (1993). Symmetric indefinite systems for interior-point methods.
*Mathematical Programming, 58*, 1–32.CrossRefGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 2014