Theoretical Perspectives on Jabr-Type Convex Relaxations for AC Optimal Power Flow

TL;DR

This paper reviews and unifies various strengthening techniques for Jabr-type convex relaxations of the AC optimal power flow problem, highlighting their geometric and graph-theoretic foundations.

Contribution

It provides a structured, unifying perspective on multilinear convexification and cycle constraints, connecting graph structure with conic relaxations.

Findings

Cycle constraints are multilinear consistency conditions.

Convex hull theory explains relaxation strengthening.

Structural conditions unify primal and dual relaxations.

Abstract

The alternating current optimal power flow problem is a fundamental yet highly nonconvex optimization problem whose structure reflects both nonlinear power flow physics and the topology of the underlying network. Among convex relaxations, the second-order cone relaxation introduced by Jabr has proven particularly influential, serving as a computationally efficient alternative to semidefinite relaxations and a foundation for numerous strengthening techniques. In recent years, a variety of approaches have been proposed to tighten Jabr-type relaxations, including cycle-based constraints, convex envelopes of multilinear terms, and dual reformulations. However, these developments are often presented independently, concealing their common geometric and graph-theoretic foundations. This paper provides a structured review of strengthening techniques for the Jabr relaxation and develops a unifying perspective based on multilinear equalities. We reinterpret cycle constraints as multilinear consistency conditions, analyze their convexification through classical convex hull theory, and investigate the relationship between primal McCormick relaxations and dual extended formulations. In particular, we identify structural conditions under which these relaxations coincide and clarify the distinction between convexifying the interaction graph and convexifying the feasible set of the ACOPF. The resulting framework connects graph structure, multilinear convexification, and conic relaxations in a unified manner, offering both a conceptual synthesis of existing results and new insights for the design of stronger relaxations.