A Primal-Dual Interior Point Method for a Novel Type-2 Second Order Cone Optimization Problem

TL;DR

This paper introduces a primal-dual interior point method tailored for a new type-2 second order cone optimization problem, extending traditional SOCO techniques with specialized kernel functions and iteration bounds.

Contribution

It develops the first primal-dual interior point algorithm specifically for type-2 second order cone problems, incorporating new kernel functions and theoretical iteration bounds.

Findings

Derived an iteration bound for the proposed algorithm

Extended SOCO theory to a new type-2 cone with additional variables

Provided a framework for solving the novel cone optimization problem

Abstract

In this paper, we define a new, special second order cone as a type-$k$ second order cone. We focus on the case of $k=2$, which can be viewed as SOCO with an additional {\em complicating variable}. For this new problem, we develop the necessary prerequisites, based on previous work for traditional SOCO. We then develop a primal-dual interior point algorithm for solving a type-2 second order conic optimization (SOCO) problem, based on a family of kernel functions suitable for this type-2 SOCO. We finally derive the following iteration bound for our framework: \[\frac{L^\gamma}{\theta \kappa \gamma} \left[2N \psi\left( \frac{\varrho \left(\tau /4N\right)}{\sqrt{1-\theta}}\right)\right]^\gamma\log \frac{3N}{\epsilon}.\]