Convex Analysis and Optimization with Submodular Functions: a Tutorial

Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of submodular functions is presented, in a self-contained way, with all results shown from first principles. A good knowledge of convex analysis is assumed.

💡 Research Summary

The tutorial “Convex Analysis and Optimization with Submodular Functions” offers a self‑contained exposition that bridges the theory of submodular set functions with the well‑established machinery of convex analysis. It begins by recalling the definition of a submodular function f on a finite ground set V, emphasizing the diminishing‑returns property: for any A ⊆ B ⊆ V and any element s ∉ B, the marginal gain f(A∪{s}) − f(A) ≥ f(B∪{s}) − f(B). The authors then present several equivalent characterizations—pairwise inequality, lattice‑theoretic formulation, and a Lagrangian duality view—showing that submodularity is the combinatorial analogue of convexity.

A central construct is the Lovász extension (\hat f), which lifts the discrete function to a continuous, piecewise‑linear convex function on the hypercube