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Optimal Mediated Graphs: The role of Combinatorics in Conic Optimization

Víctor Blanco, Miguel Martínez-Antón

TL;DR

The paper bridges conic optimization and graph theory by introducing A-mediated graphs, which encode midpoint relations in M-geometric digraphs and yield compact SOC/SOS representations via generalized power cones and SONC/SOS decompositions. It develops rigorous MILP/ILP formulations to compute minimal and maximal mediated graphs on continuous and discrete domains, linking these structures to convex algebraic geometry and conic representations. Extensive computational experiments on public datasets demonstrate that the optimization-based approaches outperform prior enumeration in speed and scalability, enabling effective SOS/SONC constructions and extended SOCP reformulations. This framework provides practical, scalable tools to strengthen relaxations and polynomial decompositions in conic optimization with broad implications for convex geometry and polynomial optimization.

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.

Optimal Mediated Graphs: The role of Combinatorics in Conic Optimization

TL;DR

The paper bridges conic optimization and graph theory by introducing A-mediated graphs, which encode midpoint relations in M-geometric digraphs and yield compact SOC/SOS representations via generalized power cones and SONC/SOS decompositions. It develops rigorous MILP/ILP formulations to compute minimal and maximal mediated graphs on continuous and discrete domains, linking these structures to convex algebraic geometry and conic representations. Extensive computational experiments on public datasets demonstrate that the optimization-based approaches outperform prior enumeration in speed and scalability, enabling effective SOS/SONC constructions and extended SOCP reformulations. This framework provides practical, scalable tools to strengthen relaxations and polynomial decompositions in conic optimization with broad implications for convex geometry and polynomial optimization.

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.

Paper Structure

This paper contains 11 sections, 12 theorems, 31 equations, 5 figures, 3 tables.

Table of Contents

  1. Mediated Graphs
  2. Notation
  3. Properties
  4. Optimal Mediated Graphs
  5. Minimal Mediated Graphs
  6. Maximal Mediated Graphs
  7. Computational Experiments
  8. Data
  9. Results on Maximal Mediated Graphs
  10. Results on Minimal Mediated Graphs
  11. Proof of Theorem \ref{['the:sobs']}

Key Result

Lemma 1

Let $\mathcal{A} \subset \mathbb{R}^d$ be an affine independent set and ${\bm{b}}\xspace\in \mathrm{Conv}\xspace(\mathcal{A})^\circ$. Then, there exists nonempty graph $G=(V,A)$ in $\mathcal{M}\xspace_{\mathcal{A}}({\bm{b}}\xspace)$ with where ${\bm{s}}\xspace=(s_{\bm{a}}\xspace)_{{\bm{a}}\xspace\in\mathcal{A}}\in (\mathbb{Z}_+^*)^{|\mathcal{A}|}$, $\gcd({\bm{s}}\xspace)=1$, and $\frac{1}{\|{\bm{

Figures (5)

  • Figure 1: Two versions of $\mathcal{A}$-mediated graphs but with different vertex domains. $\mathbb{R}^2$ (left) and $\mathbb{Z}^2$ (right).
  • Figure 2: The entire set $\mathrm{MinMG}\xspace_\mathcal{A}(\mathcal{B}\xspace)$ of Example \ref{['ex:listMinMG']}.
  • Figure 3: Performance profiles for our approach and the one in hartzer2022initial.
  • Figure 4: Boxplots for the CPU times (in log-scale) required to compute the maximal mediated graphs with our methodology (BM), the methodology in hartzer2022initial, and the CPU time required to enumerate the required points.
  • Figure 5: Performance profile for our experiments on the minimal mediated graphs.

Theorems & Definitions (31)

  • Definition 1.1: Mediated Graph
  • Remark 1.1
  • Example 1.1
  • proof
  • Definition 1.2: Binary Support
  • Lemma 1
  • Theorem 2
  • proof
  • Example 1.2
  • Theorem 3: powers2021note
  • ...and 21 more