On the Convergence of the Newton/Log-Barrier Method

Stephen J. Wright

In the Newton/log-barrier method, Newton steps are taken for the log barrier function for a fixed value of the barrier parameter until a certain convergence criterion is satisfied. The barrier parameter is then decreased and the Newton process is repeated. A naive analysis indicates that Newton's method does not exhibit superlinear convergence to the minimizer of each instance of the log-barrier function until it reaches a very small neighborhood of the minimizer. By partitioning according to the subspace of active constraint gradients, however, we show that this neighborhood is actually quite large, thus explaining why reasonably fast local convergence can be attained in practice. Finally, we show that the overall convergence rate of the Newton/log-barrier algorithm % to the solution of the nonlinear programming problem is superlinear in the number of function/derivative evaluations, provided that the nonlinear program is formulated with a linear objective and the schedule for decreasing the barrier parameter is related in a certain way to the convergence criterion for each Newton process.

Preprint ANL/MCS-P681-0897, Mathematics and Computer Science Division, Argonne National Laboratory, August, 1997. Revised: January, 1999.

Contact: wright@mcs.anl.gov