A Globally Convergent Distributed Jacobi Scheme for Block-Structured Nonconvex Constrained Optimization Problems

TL;DR

This paper introduces a distributed Jacobi-based algorithm for large-scale nonconvex constrained optimization, offering convergence guarantees and practical parameter tuning, demonstrated on power system optimization problems.

Contribution

It presents a novel parallel Jacobi scheme with convergence analysis and automatic parameter tuning for nonconvex problems, outperforming existing methods.

Findings

Proven global and local convergence of the algorithm.

Effective for large-scale power system optimization.

Achieved significant speedup on supercomputers.

Abstract

Motivated by the increasing availability of high-performance parallel computing, we design a distributed parallel algorithm for linearly-coupled block-structured nonconvex constrained optimization problems. Our algorithm performs Jacobi-type proximal updates of the augmented Lagrangian function, requiring only local solutions of separable block nonlinear programming (NLP) problems. We provide a cheap and explicitly computable Lyapunov function that allows us to establish global and local sublinear convergence of our algorithm, its iteration complexity, as well as simple, practical and theoretically convergent rules for automatically tuning its parameters. This in contrast to existing algorithms for nonconvex constrained optimization based on the alternating direction method of multipliers that rely on at least one of the following: Gauss-Seidel or sequential updates, global solutions of NLP problems, non-computable Lyapunov functions, and hand-tuning of parameters. Numerical experiments showcase its advantages for large-scale problems, including the multi-period optimization of a 9000-bus AC optimal power flow test case over 168 time periods, solved on the Summit supercomputer using an open-source Julia code.