Inverse barrier methods for linear programming

Abstract

In the recent interior point methods for linear programming much attention has been given to the logarithmic barrier method In this paper we will analyse the class of inverse barrier methods for linear programming, in which the barrier is SIGMAx(i)-r, where r > 0 is the rank of the barrier. There are many similarities with the logarithmic barrier method. The minima of an inverse barrier function for different values of the barrier parameter define a `'central path'' dependent on r, called the r-path of the problem. For r down 0 this path coincides with the central path determined by the logarithmic barrier function. We introduce a metric to measure the distance of a feasible point to a point on the path. We prove that in a certain region around a point on the path the Newton process converges quadratically. Moreover, outside this region, taking a step into the Newton direction decreases the barrier function value at least with a constant. We will derive upper bounds for the total number of iterations needed to obtain an epsilon-optimal solution. Unfortunately, these bounds are not polynomial in the input length. Only if the rank r goes to zero we get a polynomiality result, but then we are actually working with the logarithmic barrier method.