An Efficient Unsupervised Framework for Convex Quadratic Programs via Deep Unrolling

Linxin Yang, Bingheng Li, Tian Ding, Jianghua Wu, Akang Wang, Yuyi Wang, Jiliang Tang, Ruoyu Sun, Xiaodong Luo

Keywords: Quadratic programming, Learning to optimize, Deep unrolling, Primal-Dual Hybrid Gradient, Unsupervised learning

Abstract:

Quadratic programs (QPs) arise in various domains such as machine learning, finance, and control.
Recently, learning-enhanced primal-dual hybrid gradient (PDHG) methods have shown great potential in addressing large-scale linear programs; however, this approach has not been extended to QPs.
In this work, we focus on unrolling "PDQP", a PDHG algorithm specialized for convex QPs. Specifically, we propose a neural network model called "PDQP-net" to learn optimal QP solutions. Theoretically, we demonstrate that a PDQP-net of polynomial size can align with the PDQP algorithm, returning optimal primal-dual solution pairs.
We propose an unsupervised method that incorporates KKT conditions into the loss function. Unlike the standard learning-to-optimize framework that requires optimization solutions generated by solvers, our unsupervised method adjusts the network weights directly from the evaluation of the primal-dual gap.
This method has two benefits over supervised learning: first, it helps generate better primal-dual gap since the primal-dual gap is in the objective function; second, it does not require solvers.
We show that PDQP-net trained in this unsupervised manner can effectively approximate optimal QP solutions.
Extensive numerical experiments confirm our findings, indicating that using PDQP-net predictions to warm-start PDQP can achieve up to 45% acceleration on QP instances.
Moreover, it achieves 14% to 31% acceleration on out-of-distribution instances.

Primary Area: optimization

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Submission Number: 10659