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Automating Reformulation for Parallel ADMM

Kaizhao Sun, Baihao Wu, Kun Yuan, Wotao Yin

Abstract

Many real-world optimization models contain exploitable sparsity and block structure, but this structure is often obscured in algebraic form, limiting the effectiveness of modern parallel algorithms. We propose an automatic pipeline that converts a generic multiblock problem into a canonical two-block formulation suitable for parallel Alternating Direction Method of Multipliers (ADMM). The method constructs a coupling graph, applies an edge-subdivision-based bipartization to obtain a bipartite representation, and produces an ADMM-ready decomposition with independent subproblems. Fast graph-traversal heuristics, a new mixed-integer linear program (MILP), and a learning-based graph neural network (GNN) surrogate model are developed to guide edge subdivision. Numerical experiments demonstrate that the resulting reformulations yield strong parallel ADMM performance. The entire pipeline is implemented in the open-source Julia package PDMO.jl.

Automating Reformulation for Parallel ADMM

Abstract

Many real-world optimization models contain exploitable sparsity and block structure, but this structure is often obscured in algebraic form, limiting the effectiveness of modern parallel algorithms. We propose an automatic pipeline that converts a generic multiblock problem into a canonical two-block formulation suitable for parallel Alternating Direction Method of Multipliers (ADMM). The method constructs a coupling graph, applies an edge-subdivision-based bipartization to obtain a bipartite representation, and produces an ADMM-ready decomposition with independent subproblems. Fast graph-traversal heuristics, a new mixed-integer linear program (MILP), and a learning-based graph neural network (GNN) surrogate model are developed to guide edge subdivision. Numerical experiments demonstrate that the resulting reformulations yield strong parallel ADMM performance. The entire pipeline is implemented in the open-source Julia package PDMO.jl.
Paper Structure (45 sections, 1 theorem, 26 equations, 18 figures, 2 tables, 3 algorithms)

This paper contains 45 sections, 1 theorem, 26 equations, 18 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The obstacle: structure for splitting methods is hard to craft by hand.
  3. A small example: equivalent splittings, different behavior.
  4. Contributions preview.
  5. Background on ADMM and FLiP-ADMM
  6. Related Work on Problem Reformulation and Decomposition
  7. Contributions
  8. Notations
  9. Automatic Reformulation Pipeline
  10. Multiblock Problem Formulation
  11. Graph-based Transformation
  12. Bipartization via Edge Subdivision
  13. BFS-based Bipartization Algorithm
  14. An MILP Formulation for Optimal Bipartization
  15. Variables and constraints.
  16. ...and 30 more sections

Key Result

proposition 1

Consider the graph-based formulation eq: graph_problem_formulation defined on a graph $G(V,E)$. There exists a partition of variable blocks into two disjoint groups $V = \texttt{L}\sqcup \texttt{R}$ such that every coupling block constraint eq: graph_problem_formulation_constr involves exactly one v

Figures (18)

  • Figure 1: A triangle resistive network and power dissipation minimization.
  • Figure 2: Residual trajectories of ADMM with $(R_{1}, R_{2}, R_{3})$ = (1.0e-6, 1.0e2, 1.0e8) and $(J_1, J_2, J_3) = (-50,100,-50)$.
  • Figure 3: Proposed bipartization pipeline.
  • Figure 4: A 3-block example and its graph representation.
  • Figure 5: Bipartite graph obtained by subdividing $(\mathbf{x}_2, C_1)$ in Fig. \ref{['fig: graph']}.
  • ...and 13 more figures

Theorems & Definitions (4)

  • remark 1
  • proposition 1
  • proof
  • remark 2