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Learning Constrained Optimization with Deep Augmented Lagrangian Methods

James Kotary, Ferdinando Fioretto

TL;DR

This paper proposes an alternative approach, in which the ML model is trained instead to predict dual solution estimates directly, from which primal estimates are constructed to form dual-feasible solution pairs, emulating a Dual Ascent method.

Abstract

Learning to Optimize (LtO) is a problem setting in which a machine learning (ML) model is trained to emulate a constrained optimization solver. Learning to produce optimal and feasible solutions subject to complex constraints is a difficult task, but is often made possible by restricting the input space to a limited distribution of related problems. Most LtO methods focus on directly learning solutions to the primal problem, and applying correction schemes or loss function penalties to encourage feasibility. This paper proposes an alternative approach, in which the ML model is trained instead to predict dual solution estimates directly, from which primal estimates are constructed to form dual-feasible solution pairs. This enables an end-to-end training scheme is which the dual objective is maximized as a loss function, and solution estimates iterate toward primal feasibility, emulating a Dual Ascent method. First it is shown that the poor convergence properties of classical Dual Ascent are reflected in poor convergence of the proposed training scheme. Then, by incorporating techniques from practical Augmented Lagrangian methods, we show how the training scheme can be improved to learn highly accurate constrained optimization solvers, for both convex and nonconvex problems.

Learning Constrained Optimization with Deep Augmented Lagrangian Methods

TL;DR

This paper proposes an alternative approach, in which the ML model is trained instead to predict dual solution estimates directly, from which primal estimates are constructed to form dual-feasible solution pairs, emulating a Dual Ascent method.

Abstract

Learning to Optimize (LtO) is a problem setting in which a machine learning (ML) model is trained to emulate a constrained optimization solver. Learning to produce optimal and feasible solutions subject to complex constraints is a difficult task, but is often made possible by restricting the input space to a limited distribution of related problems. Most LtO methods focus on directly learning solutions to the primal problem, and applying correction schemes or loss function penalties to encourage feasibility. This paper proposes an alternative approach, in which the ML model is trained instead to predict dual solution estimates directly, from which primal estimates are constructed to form dual-feasible solution pairs. This enables an end-to-end training scheme is which the dual objective is maximized as a loss function, and solution estimates iterate toward primal feasibility, emulating a Dual Ascent method. First it is shown that the poor convergence properties of classical Dual Ascent are reflected in poor convergence of the proposed training scheme. Then, by incorporating techniques from practical Augmented Lagrangian methods, we show how the training scheme can be improved to learn highly accurate constrained optimization solvers, for both convex and nonconvex problems.
Paper Structure (22 sections, 17 equations, 2 figures, 2 algorithms)

This paper contains 22 sections, 17 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Overview
  3. Related Work
  4. End-to-End Learning with Feasibility Guarantees
  5. Learning with Soft Constraints
  6. Learning Dual Solutions
  7. Preliminaries: Lagrangian Duality
  8. Deep Dual Ascent
  9. Training Algorithm
  10. Dual Ascent Interpretation
  11. Deep Augmented Lagrangian Method
  12. Box-Constrained Reformulation
  13. Box-Constrained Optimization Step
  14. Training Optimization Proxy Models with Deep ALM
  15. Experiments
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Training with Deep Dual Ascent (left) and Deep ALM (right) to solve the convex QP problem \ref{['eq:QP']}. Mean and standard deviation over the test set shown by bold curves and shaded regions.
  • Figure 2: Training with Deep ALM to solve the nonconvex program \ref{['eq:nonconvex_variant']}.