Optimal Mediated Graphs: The role of Combinatorics in Conic Optimization

TL;DR

This paper introduces mediated graphs, explores their properties, and demonstrates their significance in conic optimization, providing formulations and experimental validation for their computation and applications.

Contribution

It offers a unified definition of mediated graphs, analyzes their properties, and develops mixed integer linear formulations for their computation in optimization contexts.

Findings

Mediated graphs are crucial in sum of squares decomposition.

They enable second-order cone representations of convex cones.

Experimental results validate the proposed formulations.

Abstract

In this paper, we provide a unified definition of mediated graph, a combinatorial structure with multiple applications in mathematical optimization. We study some geometric and algebraic properties of this family of graphs and analyze extremal mediated graphs under the partial order induced by the cardinalty of their vertex sets. We derive mixed integer linear formulations to compute these challenging graphs and show that these structures are crucial in different fields, such as sum of squares decomposition of polynomials and second-order cone representations of convex cones, with a direct impact on conic optimization. We report the results of an extensive battery of experiments to show the validity of our approaches.