Optimal Mediated Graphs: The role of Combinatorics in Conic Optimization

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.