New adaptive interior point algorithms for linear optimization.

摘要

Interior Point Methods (IPMs) have shown their power in solving large scale optimization problems. In this thesis, using the notion of self-regularity, various classes of algorithms are proposed. First, we present an adaptive technique to compute the target barrier parameter. Using this adaptive technique a single step large-update algorithm is proposed that enjoys better worst case iteration complexity than the classical large-update IPMs. A new variant of infeasible IPMs, using the same adaptive technique, is proposed that enjoys better worst case iteration complexity than the classical infeasible IPMs. By further use of the adaptive technique a new family of predictor-corrector IPMs is proposed that are different both in the predictor and the corrector steps from their classical counter parts. Worst case iteration complexity analysis reveals significant improvement for some special cases over the classical analogues.;In the second part of thesis we analyze the theoretical behavior of the most widely used algorithm in IPM based software packages, the so called Mehrotra-type predictor-corrector algorithm. This analysis reveals some drawbacks of the algorithm that motivated us to modify it. We combine it with a safeguard that prevents the drawback we observed and enables us to prove strong theoretical results about the new algorithm. Motivated by this drawback, we analyze it from different perspective. In the new approach, contrary to the existing variants in the literature, first we fix the step size and then aim to compute the smallest value of the barrier parameter that ensures the prescribed step size is feasible. After the barrier parameter is computed, the algorithm computes the next iterate by a line-search as usual. The worst case behavior of the proposed algorithms is discussed. Finally, some limited encouraging computational results are reported.