Existence, uniqueness, and convergence of the regularized primal–dual central path

In a recent work [J. Castro, J. Cuesta, Quadratic regularizations in an interior-point method for primal block-angular problems, Mathematical Programming, in press (doi:10.1007/s10107-010-0341-2)] the authors improved one of the most efficient interior-point approaches for some classes of block-angular problems. This was achieved by adding a quadratic regularization to the logarithmic barrier. This regularized barrier was shown to be self-concordant, thus fitting the general structural optimization interior-point framework. In practice, however, most codes implement primal–dual path-following algorithms. This short paper shows that the primal–dual regularized central path is well defined, i.e., it exists, it is unique, and it converges to a strictly complementary primal–dual solution.