Local and Global Convergence of Greedy Parabolic Target-Following Methods for Linear Programming

TL;DR

This paper introduces three novel greedy parabolic target-following algorithms for linear programming that achieve improved local convergence rates, including super-linear, quadratic, and cubic, while maintaining global polynomial complexity.

Contribution

It presents three new algorithms with enhanced local convergence properties and maintains global polynomial complexity, advancing interior-point methods for linear programming.

Findings

Second algorithm achieves local quadratic convergence.

Third algorithm attains local cubic convergence.

All methods maintain polynomial global complexity.

Abstract

In the first part of this paper, we prove that, under some natural non-degeneracy assumptions, the Greedy Parabolic Target-Following Method, based on {\em universal tangent direction} has a favorable local behavior. In view of its global complexity bound of the order $O(\sqrt{n} \ln {1 \over \epsilon})$, this fact proves that the functional proximity measure, used for controlling the closeness to Greedy Central Path, is large enough for ensuring a local super-linear rate of convergence, provided that the proximity to the path is gradually reduced. This requirement is eliminated in our second algorithm based on a new auto-correcting predictor direction. This method, besides the best-known polynomial-time complexity bound, ensures an automatic switching onto the local quadratic convergence in a small neighborhood of solution. Our third algorithm approximates the path by quadratic curves. On the top of the best-known global complexity bound, this method benefits from an unusual local cubic rate of convergence. This amelioration needs no serious increase in the cost of one iteration. We compare the advantages of these local accelerations with possibilities of finite termination. The conditions allowing the optimal basis detection sometimes are even weaker than those required for the local superlinear convergence. Hence, it is important to endow the practical optimization schemes with both abilities. The proposed methods have a very interesting combination of favorable properties, which can be hardly found in the most of existing Interior-Point schemes. As all other parabolic target-following schemes, the new methods can start from an arbitrary strictly feasible primal-dual pair and go directly towards the optimal solution of the problem in a single phase. The preliminary computational experiments confirm the advantage of the second-order prediction.