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The search direction

For later discussion, let us first introduce the operator defined by

The adjoint of with respect to the standard inner products in and is the operator

The main step at each iteration of our algorithms is the computation of the search direction from the symmetrized Newton equation (with respect to an invertible matrix P which is usually chosen as a function of the current iterate X,Z) given below.

where and is the centering parameter. Here is the symmetrization operator defined by

and and are the linear operators

where denotes the linear operator defined by

Assuming that , we compute the search direction via a Schur complement equation as follows (the reader is referred to [2] and [8] for details). First compute from the Schur complement equation

where

Then compute and from the equations

If , solving (9) by a direct method is overwhelmingly expensive; in this case, the user should consider using an implementation that solves (9) by an iterative method such as CG or QMR. In our package, we assume that and (9) is solved by a direct method such as LU or Cholesky decomposition.