Implementation Issues for Interior-Point Methods

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, 20.1 $$\displaystyle{ B = \left [\begin{array}{cc} - {E}^{-2} & A \\ {A}^{T} &{D}^{-2} \end{array} \right ], }$$ or one of the two matrices associated with the normal equations: 20.2 A D 2 A T + E − 2 $$\displaystyle{ A{D}^{2}{A}^{T} + {E}^{-2} }$$ or 20.3 A T E 2 A + D − 2 . $$\displaystyle{ {A}^{T}{E}^{2}A + {D}^{-2}. }$$ (Here, E − 2 = Y − 1 W $${E}^{-2} = {Y }^{-1}W$$ and D − 2 = X − 1 Z $${D}^{-2} = {X}^{-1}Z$$ .) 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.