# A Bundle Approach for SDPs with Exact Subgraph Constraints

## TL;DR

This paper introduces a bundle method-based computational framework for solving large-scale semidefinite relaxations with exact subgraph constraints, improving efficiency in tackling NP-hard graph problems.

## Contribution

It proposes a novel partial Lagrangian dual approach that decomposes the problem, enabling the use of bundle methods to efficiently solve SDPs with many subgraph constraints.

## Key findings

- Efficiently solves SDPs with many subgraph constraints.
- Demonstrates improved performance on Max-Cut, stable set, and coloring problems.
- Shows the approach's scalability and effectiveness.

## Abstract

The 'exact subgraph' approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into two independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Computational experiments on the Max-Cut, stable set and coloring problem show the efficiency of this approach.