A polynomial-time solvable class of sparse box-constrained polynomial optimization problems

TL;DR

This paper introduces a polynomial-time method for solving a class of sparse box-constrained polynomial optimization problems by exploiting hypergraph structures and variable binary properties.

Contribution

It extends classical binary optimization tractability results to box-constrained problems using hypergraph sparsity and variable elimination techniques.

Findings

Identifies a polynomial-time solvable class of sparse box-constrained polynomial problems.

Shows how to reduce the problem to a structured binary optimization problem.

Extends tractability results from binary to box-constrained polynomial optimization.

Abstract

We study the problem of minimizing a multivariate polynomial function over the unit hypercube. By representing the polynomial through a hypergraph and exploiting its sparsity structure, we establish a new sufficient condition under which the problem can be solved in time polynomial in the encoding length of the input. Our approach identifies a subset of variables that attain binary values at optimality and shows how the remaining continuous variables can be eliminated locally when they appear in small, weakly coupled blocks, yielding a reduction to a structured binary optimization problem that can be solved efficiently. Our result extends the classical tractability result for binary polynomial optimization, namely, that problems with bounded treewidth are solvable in polynomial time, to box-constrained polynomial optimization.