Local Superlinear Convergence of Polynomial-Time Interior-Point Methods for Hyperbolic Cone Optimization Problems

TL;DR

This paper proves local superlinear convergence of certain polynomial-time interior-point methods for hyperbolic cone optimization, leveraging the negative curvature of self-concordant barriers and introducing a dual-space predictor-corrector scheme.

Contribution

It introduces a new dual-space predictor-corrector method with an automatic transition from linear to superlinear convergence under mild conditions.

Findings

Proved local superlinear convergence for specific interior-point methods.

Developed a gradient proximity measure for automatic convergence rate enhancement.

Designed a step-size rule that maintains feasibility and tightens the neighborhood near the solution.

Abstract

In this paper, we establish the local superlinear convergence property of some polynomial-time interior-point methods for an important family of conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier function, which must have {\em negative curvature}. We propose a new path-following predictor-corrector scheme, which work only in the dual space. It is based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the local superlinear one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size maintaining feasibility. As the optimal solution set is approached, our algorithm automatically tightens the neighborhood of the central path proportionally to the current duality gap.