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A Learning-Based Inexact ADMM for Solving Quadratic Programs

Xi Gao, Jinxin Xiong, Linxin Yang, Akang Wang, Weiwei Xu, Jiang Xue

TL;DR

This work targets convex quadratic programs (QPs) and the challenge of solving them efficiently at scale. It integrates a neural accelerator, I‑ADMM‑LSTM, into an inexact ADMM framework by replacing exact subproblem solves with a learned LSTM predictor, while preserving convergence through residual‑based criteria. The authors establish convergence guarantees within the inexact ADMM formalism and introduce a two‑stage feasibility restoration (FR) step to eliminate mild feasibility violations. Extensive experiments show substantial speedups—up to 7×, 28×, and 22× over Gurobi, SCS, and OSQP, respectively—while maintaining high solution quality, with the FR variant further tightening feasibility and objective gaps. The approach demonstrates strong practical potential for high‑throughput and GPU‑friendly optimization tasks, though future work aims to extend beyond convex QPs to general NLPs and nonlinear programs.

Abstract

Convex quadratic programs (QPs) constitute a fundamental computational primitive across diverse domains including financial optimization, control systems, and machine learning. The alternating direction method of multipliers (ADMM) has emerged as a preferred first-order approach due to its iteration efficiency - exemplified by the state-of-the-art OSQP solver. Machine learning-enhanced optimization algorithms have recently demonstrated significant success in speeding up the solving process. This work introduces a neural-accelerated ADMM variant that replaces exact subproblem solutions with learned approximations through a parameter-efficient Long Short-Term Memory (LSTM) network. We derive convergence guarantees within the inexact ADMM formalism, establishing that our learning-augmented method maintains primal-dual convergence while satisfying residual thresholds. Extensive experimental results demonstrate that our approach achieves superior solution accuracy compared to existing learning-based methods while delivering significant computational speedups of up to $7\times$, $28\times$, and $22\times$ over Gurobi, SCS, and OSQP, respectively. Furthermore, the proposed method outperforms other learning-to-optimize methods in terms of solution quality. Detailed performance analysis confirms near-perfect compliance with the theoretical assumptions, consequently ensuring algorithm convergence.

A Learning-Based Inexact ADMM for Solving Quadratic Programs

TL;DR

This work targets convex quadratic programs (QPs) and the challenge of solving them efficiently at scale. It integrates a neural accelerator, I‑ADMM‑LSTM, into an inexact ADMM framework by replacing exact subproblem solves with a learned LSTM predictor, while preserving convergence through residual‑based criteria. The authors establish convergence guarantees within the inexact ADMM formalism and introduce a two‑stage feasibility restoration (FR) step to eliminate mild feasibility violations. Extensive experiments show substantial speedups—up to 7×, 28×, and 22× over Gurobi, SCS, and OSQP, respectively—while maintaining high solution quality, with the FR variant further tightening feasibility and objective gaps. The approach demonstrates strong practical potential for high‑throughput and GPU‑friendly optimization tasks, though future work aims to extend beyond convex QPs to general NLPs and nonlinear programs.

Abstract

Convex quadratic programs (QPs) constitute a fundamental computational primitive across diverse domains including financial optimization, control systems, and machine learning. The alternating direction method of multipliers (ADMM) has emerged as a preferred first-order approach due to its iteration efficiency - exemplified by the state-of-the-art OSQP solver. Machine learning-enhanced optimization algorithms have recently demonstrated significant success in speeding up the solving process. This work introduces a neural-accelerated ADMM variant that replaces exact subproblem solutions with learned approximations through a parameter-efficient Long Short-Term Memory (LSTM) network. We derive convergence guarantees within the inexact ADMM formalism, establishing that our learning-augmented method maintains primal-dual convergence while satisfying residual thresholds. Extensive experimental results demonstrate that our approach achieves superior solution accuracy compared to existing learning-based methods while delivering significant computational speedups of up to , , and over Gurobi, SCS, and OSQP, respectively. Furthermore, the proposed method outperforms other learning-to-optimize methods in terms of solution quality. Detailed performance analysis confirms near-perfect compliance with the theoretical assumptions, consequently ensuring algorithm convergence.
Paper Structure (29 sections, 3 theorems, 63 equations, 6 figures, 3 tables, 3 algorithms)

This paper contains 29 sections, 3 theorems, 63 equations, 6 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. ADMM for Convex QPs
  3. Problem Formulation
  4. OSQP: An ADMM-Based Solver
  5. Inexact ADMM Framework
  6. Learning-based Inexact ADMM
  7. I-ADMM-LSTM
  8. Adaptive Parameters
  9. The Two-Stage Framework
  10. Convergence Analysis
  11. Experiments
  12. Experimental Settings
  13. Datasets.
  14. Baseline Algorithms.
  15. Model Settings.
  16. ...and 14 more sections

Key Result

Proposition 1

Suppose Assumption assump:bound holds, and Conditions cond:x_subp_cond_1, cond:x_subp_cond_2, cond:z_subp_cond_2 are satisfied. If the following conditions are satisfied: then for all $k\geq 1$, we have

Figures (6)

  • Figure 1: An illustration of the I-ADMM-LSTM approach.
  • Figure 2: Coordinate-wise LSTM architecture for solving \ref{['prob:ls']}, where "L" represents the affine transformation layer.
  • Figure 3: An illustration of the I-ADMM-LSTM-FR approach.
  • Figure 4: Convergence characteristics of I-ADMM-LSTM (-FR) across representative instances, illustrating objective value trajectories, linear system residuals, and primal/dual residual dynamics.
  • Figure 5: The satisfaction of conditions \ref{['cond:x_subp_cond_1']} -- \ref{['cond:alpha_subp_cond']} across I-ADMM-LSTM iterations.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Remark 1
  • Proposition 1
  • Theorem 1
  • Lemma 1