Learning based convex approximation for constrained parametric optimization

TL;DR

The paper addresses constrained parametric optimization at scale by blending an input convex neural network (ICNN) with an augmented Lagrangian framework and a gradient-based constraint-correction mechanism. The proposed Augmented Lagrangian Network (ALN) uses a convex ICNN inner solver to enable parallel, GPU-accelerated optimization while preserving convergence guarantees, proving that the overall ALM can converge to a KKT point of the original problem even with inexact NN solutions. It introduces two feasible-correction schemes and provides rigorous convergence and stability analyses, including inner-approximation accuracy and Berge-based continuity results for the solution mappings. Empirically, the approach (AIC) demonstrates strong performance on quadratic programs, nonconvex problems, and large-scale ACOPF, achieving near-optimal objectives, tight feasibility, and fast runtimes compared to classical solvers (OSQP, IPOPT) and learning-based baselines (DC3, PDL). The work offers a scalable, theoretically grounded pathway to integrating neural surrogates into constrained optimization with practical impact on engineering and operations research tasks such as power systems optimization.

Abstract

We propose an input convex neural network (ICNN)-based self-supervised learning framework to solve continuous constrained optimization problems. By integrating the augmented Lagrangian method (ALM) with the constraint correction mechanism, our framework ensures \emph{non-strict constraint feasibility}, \emph{better optimality gap}, and \emph{best convergence rate} with respect to the state-of-the-art learning-based methods. We provide a rigorous convergence analysis, showing that the algorithm converges to a Karush-Kuhn-Tucker (KKT) point of the original problem even when the internal solver is a neural network, and the approximation error is bounded. We test our approach on a range of benchmark tasks including quadratic programming (QP), nonconvex programming, and large-scale AC optimal power flow problems. The results demonstrate that compared to existing solvers (e.g., \texttt{OSQP}, \texttt{IPOPT}) and the latest learning-based methods (e.g., DC3, PDL), our approach achieves a superior balance among accuracy, feasibility, and computational efficiency.

Figures (1)

• Figure 1: Illustration of ICNN architecture.

Theorems & Definitions (17)

• Lemma 4.1
• Theorem 5.4: Convergence to KKT point
• proof
• Lemma A.4: Uniform Approximation Implies Objective Value Approximation
• proof
• Theorem A.5: ICNN Inner Approximation Accuracy
• proof
• Theorem A.6: Inexact ALM Convergence
• proof
• Lemma A.7: Properties of ReLU Function