Iteration Complexity of an Infeasible Interior Point Methods for Seconder-order Cone Programming and its Warmstarting

TL;DR

This paper analyzes the worst-case iteration complexity of an infeasible interior point method for second-order cone programming, showing it matches the best known bounds for feasible methods and exploring warmstarting benefits.

Contribution

It establishes the iteration complexity of an infeasible IPM for SOCP based on the homogeneous self-dual model and examines warmstarting conditions.

Findings

Worst-case complexity is $O(k^{1/2} ext{log}(rac{1}{ extepsilon}))$

Warmstarting can improve complexity bounds under certain conditions

Method is based on Monteiro-Zhang search directions

Abstract

This paper studies the worst case iteration complexity of an infeasible interior point method (IPM) for seconder order cone programming (SOCP), which is more convenient for warmstarting compared with feasible IPMs. The method studied bases on the homogeneous and self-dual model and the Monteiro-Zhang family of searching directions. Its worst case iteration complexity is $O\left(k^{1/2}\log\left(\epsilon^{-1}\right)\right)$, to reduce the primal residual, dual residual, and complementarity gap by a factor of $\epsilon$, where $k$ is the number of cone constraints. The result is the same as the best known result for feasible IPMs. The condition under which warmstarting improves the complexity bound is also studied.