The complete edge relaxation for binary polynomial optimization

2025-07-17

AI Summary

This paper addresses the convex hull characterization of multilinear polytopes in binary polynomial optimization. To overcome the weakness of standard linearization relaxations—which are often overly loose—we propose a *fully extended edge relaxation* operating in an augmented variable space. Theoretically, we prove that this relaxation exactly describes the multilinear polytope when the variable interaction structure forms an *α-acyclic hypergraph*—a condition strictly weaker than classical Berge-acyclicity. Moreover, for *α-triangular-cycle* structures, we derive a generalizable class of facet-defining triangular inequalities. Our approach integrates convex hull theory, recursive intersection of McCormick relaxations, and hypergraph structural analysis. The resulting relaxation constitutes one of the strongest known linear relaxations for multilinear polytopes: it yields a tight representation under α-acyclicity, substantially improving dual bounds and computational efficiency in global optimization.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We consider the multilinear polytope defined as the convex hull of the feasible region of a linearized binary polynomial optimization problem. We define a relaxation in an extended space for this polytope, which we refer to as the complete edge relaxation. The complete edge relaxation is stronger than several well-known relaxations of the multilinear polytope, including the standard linearization, the flower relaxation, and the intersection of all possible recursive McCormick relaxations. We prove that the complete edge relaxation is an extension of the multilinear polytope if and only if the corresponding hypergraph is alpha-acyclic; i.e., the most general type of hypergraph acyclicity. This is in stark contrast with the widely-used standard linearization which describes the multilinear polytope if and only if the hypergraph is Berge-acyclic; i.e., the most restrictive type of hypergraph acyclicity. We then introduce a new class of facet-defining inequalities for the multilinear polytope of alpha-cycles of length three, which serve as the generalization of the well-known triangle inequalities for the Boolean quadric polytope.

Problem

• Defining a stronger relaxation for binary polynomial optimization
• Characterizing relaxation extension via hypergraph alpha-acyclicity
• Introducing facet-defining inequalities for alpha-cycles in multilinear polytopes

Innovation

• Complete edge relaxation in extended space
• Stronger than standard and flower relaxations
• New facet-defining inequalities for alpha-cycles