PREPRINT MCS-P485-1294, MCS DIVISION, ARGONNE NATIONAL LABORATORY
To simplify the analysis of interior-point methods, one commonly formulates the problem so that the objective function is linear, by introducing a single extra variable if necessary. Here we show that a linear objective function makes the Newton direction for a barrier function a useful search direction if the current iterate is sufficiently close to the central path. Hence, there are two advantages to using a linear objective and staying close to the central path. First, the Newton direction (which coincides with the affine scaling direction on the central path) gives a very accurate approximation to the direction to the minimum. Second, a long step along the Newton direction is possible without violating the inequality constraints.