A Tensor Optimization Algorithm for Computing Lagrangians of Hypergraphs

The Lagrangian of a hypergraph is a crucial tool for studying hypergraph extremal problems. Though Lagrangians of some special structure hypergraphs, such as complete uniform hypergraphs or three order uniform 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.