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Self-supervised Equality Embedded Deep Lagrange Dual for Approximate Constrained Optimization

Minsoo Kim, Hongseok Kim

TL;DR

This work tackles the challenge of fast, feasible solutions to constrained optimization by introducing DeepLDE, a self-supervised framework that embeds equality constraints directly into neural network outputs and employs a primal-dual training loop to enforce inequality constraints. By predicting a reduced-dimension variable x and recovering the full solution via an equality-embedding function Ψ_h, DeepLDE enables backpropagation through implicit layers, while the outer primal-dual updates ensure feasibility and convergence. Theoretical results show convergence under mild conditions and demonstrate that without equality embedding, equality constraints cannot be guaranteed. Empirically, DeepLDE achieves the smallest NN-based optimality gaps across convex, non-convex, and AC-OPF problems, with substantial speedups (up to 5–250× faster than DC3 and traditional solvers) and robust feasibility on large-scale power-system benchmarks.

Abstract

Conventional solvers are often computationally expensive for constrained optimization, particularly in large-scale and time-critical problems. While this leads to a growing interest in using neural networks (NNs) as fast optimal solution approximators, incorporating the constraints with NNs is challenging. In this regard, we propose deep Lagrange dual with equality embedding (DeepLDE), a framework that learns to find an optimal solution without using labels. To ensure feasible solutions, we embed equality constraints into the NNs and train the NNs using the primal-dual method to impose inequality constraints. Furthermore, we prove the convergence of DeepLDE and show that the primal-dual learning method alone cannot ensure equality constraints without the help of equality embedding. Simulation results on convex, non-convex, and AC optimal power flow (AC-OPF) problems show that the proposed DeepLDE achieves the smallest optimality gap among all the NN-based approaches while always ensuring feasible solutions. Furthermore, the computation time of the proposed method is about 5 to 250 times faster than DC3 and the conventional solvers in solving constrained convex, non-convex optimization, and/or AC-OPF.

Self-supervised Equality Embedded Deep Lagrange Dual for Approximate Constrained Optimization

TL;DR

This work tackles the challenge of fast, feasible solutions to constrained optimization by introducing DeepLDE, a self-supervised framework that embeds equality constraints directly into neural network outputs and employs a primal-dual training loop to enforce inequality constraints. By predicting a reduced-dimension variable x and recovering the full solution via an equality-embedding function Ψ_h, DeepLDE enables backpropagation through implicit layers, while the outer primal-dual updates ensure feasibility and convergence. Theoretical results show convergence under mild conditions and demonstrate that without equality embedding, equality constraints cannot be guaranteed. Empirically, DeepLDE achieves the smallest NN-based optimality gaps across convex, non-convex, and AC-OPF problems, with substantial speedups (up to 5–250× faster than DC3 and traditional solvers) and robust feasibility on large-scale power-system benchmarks.

Abstract

Conventional solvers are often computationally expensive for constrained optimization, particularly in large-scale and time-critical problems. While this leads to a growing interest in using neural networks (NNs) as fast optimal solution approximators, incorporating the constraints with NNs is challenging. In this regard, we propose deep Lagrange dual with equality embedding (DeepLDE), a framework that learns to find an optimal solution without using labels. To ensure feasible solutions, we embed equality constraints into the NNs and train the NNs using the primal-dual method to impose inequality constraints. Furthermore, we prove the convergence of DeepLDE and show that the primal-dual learning method alone cannot ensure equality constraints without the help of equality embedding. Simulation results on convex, non-convex, and AC optimal power flow (AC-OPF) problems show that the proposed DeepLDE achieves the smallest optimality gap among all the NN-based approaches while always ensuring feasible solutions. Furthermore, the computation time of the proposed method is about 5 to 250 times faster than DC3 and the conventional solvers in solving constrained convex, non-convex optimization, and/or AC-OPF.
Paper Structure (25 sections, 3 theorems, 31 equations, 5 figures, 6 tables, 2 algorithms)

This paper contains 25 sections, 3 theorems, 31 equations, 5 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Methodology
  5. Embedding Equality Constraints
  6. Learning Constrained Optimization with NNs
  7. Analysis about Equality Constraint Embedding
  8. Experiments
  9. Simulation Setup
  10. Linearly Constrained Nonlinear Programming
  11. Feasibility
  12. Optimality
  13. Computation time
  14. AC Optimal Power Flow
  15. Feasibility
  16. ...and 10 more sections

Key Result

Proposition 4.1

Let $\beta\in\mathbb{Z}$ and $\gamma\in\mathbb{R}$ be nonnegative values. Then, Algorithm alg:DeepLDE_noW converges if $\beta > 0$ or $\gamma > 0$.

Figures (5)

  • Figure 1: Overall framework of DeepLDE.
  • Figure 2: Comparsion of NN-based baselines and DeepLDE.
  • Figure 3: The number of optimization variables ($n$) and constraints ($n_{\text{eq}}$ and $n_{\text{ineq}}$) in Case 57 (black) and Case 118 (dotted).
  • Figure 4: Tradeoff between computation time (x-axis, log scale) and maximum violation (y-axis) in Case 118. The size of the circle is proportional to the optimality gap.
  • Figure 5: Optimality and feasibility of DC3, LDF, and DeepLDE. For LDF and DeepLDE, we use the sum of every inner and warm-up iteration as an epoch as described in Table \ref{['table:iteration']}.

Theorems & Definitions (7)

  • Proposition 4.1
  • Definition 4.2: $\varepsilon$-inexact solution
  • Proposition 4.4
  • Corollary 4.5
  • proof
  • proof
  • proof