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. To address the feasibility issues, existing methods have explored the imposing of constraints on the output space of deep learning models. Section 2 provides an overview of the existing techniques. The imposing of constraints has inspired the use of end-to-end learning approaches that directly consider the original objectives and constraints in the NN training process without the need of expert labeled data. However, even the state-of-the-art methods to impose constraints can hardly guarantee a reliable convergence with perfect feasibility on unseen data for large problems. Penalty method which treats the constraints as a form of regularization requires careful selection of penalty weights and such soft-constraint treatment cannot guarantee satisfying constraints to machine precision. Primal-dual Lagrangian-based formulation theoretically provides a hard constraint methodology, whereas empirical evidence indicate it can perform worse than penalty method (the reason remains unclear, see (Márquez-Neila et al., 2017) ). Another strategy (Donti et al., 2021) adds completion layer after the NN model to reconstruct the complete solution from an incomplete one given by the NN, using the equality constraints. This enables a hard constraint method for equality constraints, whereas when facing nonlinear constraints, the completion layer, as an iterative solver, adds to the computation complexity, and can potentially diverge when a bad incomplete output from NN causes a non-existence of feasible solution to be reconstructed. Due to the lack of consensus in the community, these new approaches are often called by different names such as constrained deep learning, end-to-end neural networks, differentiable optimization layers, or deep declarative networks. In this paper we contribute to this diversity by referring to the proposed method as differentiable parametric programming (DPP) to emphasize the connection with sensitivity analysis developed in the context of operations research (Gal & Nedoma, 1972; Gal & Greenberg) and later adopted in control theory applications (Bemporad et al., 2000; Herceg et al., 2013) . As a main contribution, we present a novel method that combines homotopy, deep learning and parametric programming formulations into one coherent algorithmic framework. The aim is to obtain a more reliable convergence of constrained deep learning models whose solution feasibility can be further improved. Homotopy based meta-optimization heuristics are developed to create a continuous transformation of objective and constraint sets, making a homotopy path that drives the training of NN to gradually learn from easy problems to harder problems. Our contribution includes 2 types of homotopy heuristics which are different ways of utlizing the basin of attraction: 1) homotopy heuristics based on relaxation of objective and constraints to manipulate the basin of attraction, 2) domain-aware homotopy heuristics based on physics-informed transformation of the problem to make it available trivial starting points within the basin of attraction 2 RELATED WORK

2.1. CONSTRAINED NEURAL NETWORKS

Imposing constraints onto the output space of NNs can be done via supervised learning (where labels are used to write the constraints) or unsupervised learning; using either soft constraint (which usually treats the constraints as a regularization) or hard constraint method (which usually means enforcing satisfaction of constraints to machine precision, i.e., perfect satisfaction). We briefly describe the different categories of existing methods, according to the type of constraints to be imposed: General equality and inequality constraints can be imposed by augmented objective functions, reprojection (as hard constraints), completion layer (as hard constraints), etc. Among augmented objective function methods, penalty method (Yang et al., 2019; Hu et al., 2020; Donti et al., 2021; Pan et al., 2019) augments the objective function by additional terms that penalize the violation of constraints, treating the constraints as a regularization with pre-defined weights to control the regularization strength, whereas the primal-dual based formulation (or lagrangian formulation) (Nandwani et al., 2019; Fioretto et al., 2020; Márquez-Neila et al., 2017) exploits Lagrangian duality and iteratively updates both primal and dual variables to minimize a Lagrangian loss function. Penalty method, as a soft constraint method, has some theoretical deficits of requiring extra weight tuning for the multi-objective loss function, and no guarantee of satisfying constraints. However, evidence (Márquez-Neila et al., 2017) has shown that Lagriangian formulation, as a hard constraint method, is empirically worse. Reprojection method makes corrections on out-of-constraint-set outputs by projecting them onto the feasible region, either during the training cycle using different variants of projected gradient descent methods (Donti et al., 2021; Márquez-Neila et al., 2017) , or during the test as a post-processing step (e.g., (Pan et al., 2019) passed outputs to a physical equation solver). A completion layer method (e.g., DC3(Donti et al., 2021 ), ACnet(Beucler et al., 2021) ) developed NN to only produce a subset of the target output variables, and then an extra constraint layer attached



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

