A Spatial Branch-and-Bound Approach for Maximization of Non-Factorable Monotone Continuous Submodular Functions

TL;DR

This paper introduces a novel spatial branch-and-bound method for maximizing non-factorable, monotone continuous submodular functions, demonstrating that submodularity can be more advantageous than factorability in certain optimization problems.

Contribution

The paper develops a cutting plane and branch-and-bound algorithm tailored for non-decreasing continuous submodular functions, addressing a gap in global optimization methods.

Findings

The proposed method effectively finds approximate solutions and bounds.

Submodularity can outperform factorability in global optimization.

Comparison shows advantages over existing commercial solvers.

Abstract

In contrast to the many continuous global optimization methods that assume the objective function and constraints are factorable, we study how to find globally maximal solutions to problems that are not factorable, focusing on a particular class of problems. Specifically, we develop a method for non-decreasing continuous submodular functions subject to constraints. We characterize the hypograph of such functions and develop a cutting plane algorithm that finds approximate solutions and bounds using an approximation of the convex hull of the hypograph. We also test a spatial branch-and-bound approach that utilizes the approximate cutting plane algorithm to form an outer approximation and obtain upper bounds for a sub-rectangle and compare our method with a state-of-the-art commercial solver. The main result is that for some problems the property of submodularity is more useful than factorability.