A tensor optimization algorithm for computing Lagrangians of hypergraphs

TL;DR

This paper introduces a fast tensor-based algorithm combined with a gradient projection method to efficiently compute the Lagrangian of large hypergraphs, addressing a challenging problem in hypergraph extremal theory.

Contribution

It proposes a novel computational scheme using adjacency tensors and gradient projection for large-scale hypergraph Lagrangian calculation, with convergence analysis.

Findings

The method efficiently computes Lagrangians of large hypergraphs.

Numerical experiments demonstrate the algorithm's effectiveness.

Convergence is guaranteed under the Lojasiewicz gradient inequality.

Abstract

The Lagrangian of a hypergraph is a crucial tool for studying hypergraph extremal problems. Though Lagrangians of some special structure hypergraphs have closed-form solutions, it is a challenging problem to compute the Lagrangian of a general large scale hypergraph. In this paper, we exploit a fast computational scheme involving the adjacency tensor of a hypergraph. Furthermore, we propose to utilize the gradient projection method on a simplex from nonlinear optimization for solving the Lagrangian of a large scale hypergraph iteratively. Using the Lojasiewicz gradient inequality, we analyze the global and local convergence of the gradient projection method. Numerical experiments illustrate that the proposed numerical method could compute Lagrangians of large scale hypergraphs efficiently.