Improved asymptotic analysis for SUMT methods

We consider the SUMT (Sequential Unconstrained Minimization Technique) method using extrapolations to link successive unconstrained sub-problems. The case when the extrapolation is obtained by a first order Taylor estimate and Newton´s method is used as a correction in this predictor-corrector scheme was analyzed in [1]. It yields a two-steps super-linear asymptotic convergence with limiting order of 4/3 for the logarithmic barrier and order two for the quadratic loss penalty. We explore both lower order variants (approximate extrapolations correction computations) as well as higher order variants (second order and further) Taylor estimate. First, we address inexact solutions of the linear systems arising within the extrapolation and the Newton´s correction steps. Depending on the inexactness allowed, asymptotic convergence order reduces, more severely so for interior variants. Second, we investigate the use of higher order path following strategies in those methods. We consider the approach based on a high order expansion of the so-called central path, somewhat reminiscent of Chebyshev´s third order method and its generalizations. The use of higher order representation of the path yields spectacular improvement in the convergence property, even more so for the interior variants.