Some disadvantages of a Mehrotra-type primal-dual corrector interior point algorithm for linear programming

Employing a new primal-dual corrector algorithm, we investigate the impact that corrector directions may have on the convergence behaviour of predictor–corrector methods. The Primal-Dual Corrector (pdc) algorithm that we propose computes on each iteration a corrector direction in addition to the direction of the standard primal-dual path-following interior point method [M. Kojima, S. Mizuno, A. Yoshise, A primal-dual interior point algorithm for linear programming, in: Progress in Mathematical Programming, Pacific Grove, CA, 1987, Springer, New York, 1989, pp. 29–47] for Linear Programming (lp), in an attempt to improve performance. The new iterate is chosen by moving along the sum of these directions, from the current iterate. This technique is similar to the construction of Mehrotra's highly popular predictor–corrector algorithm [S. Mehrotra, On finding a vertex solution using interior point methods, Linear Algebra Appl. 152 (1991) 233–253]. We present examples, however, that show that the pdc algorithm may fail to converge to a solution of the lp problem, in both exact and finite arithmetic, regardless of the choice of stepsize that is employed. The cause of this bad behaviour is that the correctors exert too much influence on the direction in which the iterates move.