HOMOTOPY LEARNING OF PARAMETRIC SOLUTIONS TO CONSTRAINED OPTIMIZATION PROBLEMS

Abstract

Building deep learning (DL) alternatives to constrained optimization problems has been proposed as a cheaper solution approach than classical constrained optimization solvers. However, these approximate learning-based solutions still suffer from constraint violations. From this perspective, reaching a reliable convergence remains an open challenge to DL models even with state-of-the-art methods to impose constraints, especially when facing a large set of nonlinear constraints forming a non-convex feasible set. In this paper, we propose the use of homotopy meta-optimization heuristics which creates a continuous transformation of the objective and constraints during training, to promote a more reliable convergence where the solution feasibility can be further improved. The method developed in this work includes 1) general-purpose homotopy heuristics based on the relaxation of objectives and constraint bounds to enlarge the basin of attraction and 2) physics-informed transformation of domain problem leading to trivial starting points lying within the basin of attraction. Experimentally, we demonstrate the efficacy of the proposed method on a set of abstract constrained optimization problems and real-world power grid optimal power flow problems with increasing complexity. Results show that constrained deep learning models with homotopy heuristics can improve the feasibility of the resulting solutions while achieving near-optimal objective values when compared with non-homotopy counterparts.

1. INTRODUCTION

Recent years have seen a rich literature of deep learning (DL) models for solving constrained optimization problems on real-world tasks such as power grid, traffic, or wireless system optimization. These applications can largely benefit from data-driven alternatives enabling fast real-time inference. The problems remain that these problems commonly include a large set of nonlinear system constraints that lead to non-convex parametric nonlinear programming (pNLP) problems which are NP-hard. Earlier attempts simply adopt imitation learning (i.e., supervised learning) to train function approximators via a minimization of the prediction error using labeled data of pre-computed solutions. Unfortunately, these models can hardly perform well on unseen data as the outputs are not trained to satisfy physical constraints, leading ifeasible solutions. 



Figure 1: Imitation learning VS end-to-end learning using Differentiable Parametric Programming

