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Scalable Primal Decomposition Schemes for Large-Scale Infrastructure Networks

Alexander Engelmann, Sungho Shin, François Pacaud, Victor M. Zavala

TL;DR

The paper tackles scalable optimization for large infrastructure networks by formulating problems as hierarchically structured strongly convex QPs on a star graph and introducing two primal decomposition schemes. By replacing subproblems with their optimal value functions and employing log-barrier smoothing (and slack variables) or exact ℓ1-relaxations, it achieves high feasibility with global convergence guarantees, while enabling efficient, localized computations. The authors develop sensitivity computations via the implicit-function theorem and solve the master problem with a globalized SQP framework, demonstrating global convergence under standard regularity conditions. Numerical experiments on district HVAC and optimal power flow problems show that the AL-based primal decomposition outperforms ℓ1-relaxation and ADMM in feasibility and robustness, with computation times comparable to centralized solvers for large-scale instances. The results indicate strong potential for these methods to enable fast, scalable coordination in real-world infrastructure networks, with future work focused on computational accelerations and sparsity exploitation.

Abstract

The operation of large-scale infrastructure networks requires scalable optimization schemes. To guarantee safe system operation, a high degree of feasibility in a small number of iterations is important. Decomposition schemes can help to achieve scalability. In terms of feasibility, however, classical approaches such as the alternating direction method of multipliers (ADMM) often converge slowly. In this work, we present primal decomposition schemes for hierarchically structured strongly convex QPs. These schemes offer high degrees of feasibility in a small number of iterations in combination with global convergence guarantees. We benchmark their performance against the centralized off-the-shelf interior-point solver Ipopt and ADMM on problems with up to 300,000 decision variables and constraints. We find that the proposed approaches solve problems as fast as Ipopt, but with reduced communication and without requiring a full model exchange. Moreover, the proposed schemes achieve a higher accuracy than ADMM.

Scalable Primal Decomposition Schemes for Large-Scale Infrastructure Networks

TL;DR

The paper tackles scalable optimization for large infrastructure networks by formulating problems as hierarchically structured strongly convex QPs on a star graph and introducing two primal decomposition schemes. By replacing subproblems with their optimal value functions and employing log-barrier smoothing (and slack variables) or exact ℓ1-relaxations, it achieves high feasibility with global convergence guarantees, while enabling efficient, localized computations. The authors develop sensitivity computations via the implicit-function theorem and solve the master problem with a globalized SQP framework, demonstrating global convergence under standard regularity conditions. Numerical experiments on district HVAC and optimal power flow problems show that the AL-based primal decomposition outperforms ℓ1-relaxation and ADMM in feasibility and robustness, with computation times comparable to centralized solvers for large-scale instances. The results indicate strong potential for these methods to enable fast, scalable coordination in real-world infrastructure networks, with future work focused on computational accelerations and sparsity exploitation.

Abstract

The operation of large-scale infrastructure networks requires scalable optimization schemes. To guarantee safe system operation, a high degree of feasibility in a small number of iterations is important. Decomposition schemes can help to achieve scalability. In terms of feasibility, however, classical approaches such as the alternating direction method of multipliers (ADMM) often converge slowly. In this work, we present primal decomposition schemes for hierarchically structured strongly convex QPs. These schemes offer high degrees of feasibility in a small number of iterations in combination with global convergence guarantees. We benchmark their performance against the centralized off-the-shelf interior-point solver Ipopt and ADMM on problems with up to 300,000 decision variables and constraints. We find that the proposed approaches solve problems as fast as Ipopt, but with reduced communication and without requiring a full model exchange. Moreover, the proposed schemes achieve a higher accuracy than ADMM.
Paper Structure (15 sections, 2 theorems, 48 equations, 7 figures, 4 tables, 4 algorithms)

This paper contains 15 sections, 2 theorems, 48 equations, 7 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Primal Decomposition Schemes
  4. Computing sensitivities
  5. A Method for Solving the Master Problem
  6. Implementation Aspects
  7. Numerical Case Studies
  8. District HVAC
  9. Optimal Power Flow
  10. Numerical Results
  11. Discussion of Algorithmic Properties
  12. Conclusion and Outlook
  13. Sensitivities for Augmented Lagrangians
  14. Proof of Lemma \ref{['lem:posDefHess']}
  15. Solution of \ref{['eq:sepQP']} via ADMM

Key Result

Lemma 1

Let Assumption ass:stdAss hold and assume that $(s_i,\mu_i,v_i,w_i)>0$. Then, a), the Hessian $\nabla_{yy}^2 \Phi_i^{\delta,\rho}$ is positive definite for all $\rho >0$. Moreover, b), $\nabla_{yy}^2 \Phi_i^{\delta,\bar{\lambda}}$ is positive definite if $\bar{\lambda}$ is larger than all multipli

Figures (7)

  • Figure 1: Star graph of problem \ref{['eq:sepQP']}.
  • Figure 2: Communication in Algorithms \ref{['alg:ALPrimDec']} and \ref{['alg:SL1primDec']}.
  • Figure 3: Buildings connected via network with limited capacity.
  • Figure 4: Convergence for three HVAC problems.
  • Figure 5: Convergence for two OPF problems.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Lemma 1: Positive definite Hessians
  • Theorem 1: Convergence of \ref{['alg:maProbSolve']} for fixed $\delta, \rho,\bar{\lambda}$
  • proof
  • Remark 1: Satisfying the assumptions of Theorem \ref{['thm:conv']}
  • Remark 2: Convergence rate
  • Remark 3: Scaling of the $\ell$1 formulation