# Extension of the LP-Newton method to SOCPs via semi-infinite representation

## TL;DR

This paper extends the LP-Newton method to solve second-order cone programming problems by reformulating them as semi-infinite programs and demonstrates its convergence and efficiency through numerical experiments.

## Contribution

The paper introduces a novel extension of the LP-Newton method to SOCPs using semi-infinite programming reformulation, with proven convergence and practical efficiency.

## Key findings

- The extended method converges globally under mild conditions.
- Numerical experiments show competitive efficiency with interior-point methods.
- The approach provides a new perspective for solving SOCPs via projection methods.

## Abstract

The LP-Newton method solves the linear programming problem (LP) by repeatedly projecting a current point onto a certain relevant polytope. In this paper, we extend the algorithmic framework of the LP-Newton method to the second-order cone programming problem (SOCP) via a linear semi-infinite programming (LSIP) reformulation of the given SOCP. In the extension, we produce a sequence by projection onto polyhedral cones constructed from LPs obtained by finitely relaxing the LSIP. We show the global convergence property of the proposed algorithm under mild assumptions, and investigate its efficiency through numerical experiments comparing the proposed approach with the primal-dual interior-point method for the SOCP.