# Polyhedral approximations of the semidefinite cone and their application

## TL;DR

This paper introduces a new sparse polyhedral approximation of the semidefinite cone using expanded SD bases, improving efficiency in solving semidefinite relaxations of combinatorial problems.

## Contribution

It proposes an expanded SD basis for polyhedral approximation that maintains sparsity and enhances computational efficiency in semidefinite programming.

## Key findings

- Approximation contains diagonally dominant matrices
- Approximation is contained in scaled diagonally dominant matrices
- Methods outperform existing approaches in efficiency

## Abstract

We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.