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Decentralized non-convex optimization via bi-level SQP and ADMM

Gösta Stomberg, Alexander Engelmann, Timm Faulwasser

TL;DR

This work addresses decentralized non-convex optimization with coupled subsystems by formulating a bi-level Sequential Quadratic Programming (SQP) method in which the outer loop solves SQP steps and the inner loop uses a decentralized ADMM to solve the resulting Quadratic Programs. The authors establish local convergence guarantees for non-convex objectives and constraints, even when the inner ADMM solves are inexact, by introducing a decentralized inexact Newton stopping criterion and a related modified stopping test. They prove that, near a KKT point, the active sets align and the overall method converges locally at q-linear, q-superlinear, or q-quadratic rates depending on the inner tolerance. Numerical results on a distributed optimal power flow problem demonstrate competitive performance against alternative decentralized methods while maintaining fully decentralized communication and the desirable property of solving only QPs in the inner loop. Overall, the approach offers provable convergence for decentralized non-convex NLPs with reduced subproblem complexity and practical applicability to power systems and distributed MPC.

Abstract

Decentralized non-convex optimization is important in many problems of practical relevance. Existing decentralized methods, however, typically either lack convergence guarantees for general non-convex problems, or they suffer from a high subproblem complexity. We present a novel bi-level SQP method, where the inner quadratic problems are solved via ADMM. A decentralized stopping criterion from inexact Newton methods allows the early termination of ADMM as an inner algorithm to improve computational efficiency. The method has local convergence guarantees for non-convex problems. Moreover, it only solves sequences of Quadratic Programs, whereas many existing algorithms solve sequences of Nonlinear Programs. The method shows competitive numerical performance for an optimal power flow problem.

Decentralized non-convex optimization via bi-level SQP and ADMM

TL;DR

This work addresses decentralized non-convex optimization with coupled subsystems by formulating a bi-level Sequential Quadratic Programming (SQP) method in which the outer loop solves SQP steps and the inner loop uses a decentralized ADMM to solve the resulting Quadratic Programs. The authors establish local convergence guarantees for non-convex objectives and constraints, even when the inner ADMM solves are inexact, by introducing a decentralized inexact Newton stopping criterion and a related modified stopping test. They prove that, near a KKT point, the active sets align and the overall method converges locally at q-linear, q-superlinear, or q-quadratic rates depending on the inner tolerance. Numerical results on a distributed optimal power flow problem demonstrate competitive performance against alternative decentralized methods while maintaining fully decentralized communication and the desirable property of solving only QPs in the inner loop. Overall, the approach offers provable convergence for decentralized non-convex NLPs with reduced subproblem complexity and practical applicability to power systems and distributed MPC.

Abstract

Decentralized non-convex optimization is important in many problems of practical relevance. Existing decentralized methods, however, typically either lack convergence guarantees for general non-convex problems, or they suffer from a high subproblem complexity. We present a novel bi-level SQP method, where the inner quadratic problems are solved via ADMM. A decentralized stopping criterion from inexact Newton methods allows the early termination of ADMM as an inner algorithm to improve computational efficiency. The method has local convergence guarantees for non-convex problems. Moreover, it only solves sequences of Quadratic Programs, whereas many existing algorithms solve sequences of Nonlinear Programs. The method shows competitive numerical performance for an optimal power flow problem.
Paper Structure (11 sections, 6 theorems, 35 equations, 1 figure, 3 algorithms)

This paper contains 11 sections, 6 theorems, 35 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. Sequential quadratic programming
  4. Alternating direction method of multipliers
  5. Decentralized sequential quadratic programming
  6. Outer convergence
  7. Inner convergence
  8. Local convergence of decentralized SQP
  9. Communication requirements and discussion
  10. Numerical Results
  11. Conclusion and Outlook

Key Result

Theorem 1

Let Assumption ass:kkt hold. Then, there exists a constant $\varepsilon_1 > 0$ such that, for all $p^0 \in \mathcal{B}_{\varepsilon_1} ( p^\star)$, the sequence $\{p^k\}$ generated by Algorithm alg:basicSQP converges q-quadratically to $p^\star$. $\square$

Figures (1)

  • Figure 1: d-SQP, ADMM, bi-level ALADIN, and d-IP for 118-bus OPF.

Theorems & Definitions (19)

  • Definition 1: Convergence rates
  • Theorem 1: Local convergence of SQP
  • proof
  • Remark 1: Decentralized ADMM
  • Lemma 1: Local convergence with inexact SQP steps
  • proof
  • Lemma 2: Modified stopping criterion
  • proof
  • Lemma 3: Active set of the ADMM subsystem QPs
  • proof
  • ...and 9 more