LEARNING A LATENT SEARCH SPACE FOR ROUTING PROBLEMS USING VARIATIONAL AUTOENCODERS

Abstract

Methods for automatically learning to solve routing problems are rapidly improving in performance. While most of these methods excel at generating solutions quickly, they are unable to effectively utilize longer run times because they lack a sophisticated search component. We present a learning-based optimization approach that allows a guided search in the distribution of high-quality solutions for a problem instance. More precisely, our method uses a conditional variational autoencoder that learns to map points in a continuous (latent) search space to highquality, instance-specific routing problem solutions. The learned space can then be searched by any unconstrained continuous optimization method. We show that even using a standard differential evolution search strategy our approach is able to outperform existing purely machine learning based approaches.

1. INTRODUCTION

Significant progress has been made in learning to solve optimization problems via machine learning (ML). Especially for practical applications, learning-based approaches are of great interest because of the high labor costs associated with the development of completely hand-crafted solution approaches. For routing problems such as the traveling salesperson problem (TSP) and the capacitated vehicle routing problem (CVRP), recent ML-based approaches are able to generate good solutions for small problem instances in a fraction of a second (e.g., Kool et al. (2019) ). However, in many real-world applications of these problems users gladly accept more computation time for solutions of even higher quality. Recently proposed approaches (e.g., Hottung & Tierney (2020)) address this demand and integrate learning-based components with high-level search procedures. While these approaches offer improved performance over non-search-based methods, they rely on domain knowledge encapsulated in the high-level search procedures. In this work, we present a learning-based optimization approach for routing problems that is able to perform an extensive search for high-quality solutions. In contrast to other approaches, our method does not rely on domain-specific high-level search procedures. Our approach learns an instancespecific mapping of solutions to a continuous search space that can then be searched via any existing continuous optimization method. We use a conditional variational autoencoder (CVAE) that learns to encode a solution to a given instance as a numerical vector and vice versa. Some genetic algorithm variants (e.g., Gonc ¸alves & Resende (2012)) use numerical vectors to represent solutions to combinatorial optimization problems. However, these approaches rely on decoding schemes that are carefully handcrafted by domain experts. In contrast, our approach learns the problem-specific decoding schema on its own, requiring no domain or optimization knowledge on the side of the user. The performance of an optimization algorithm heavily depends on the structure of the fitness landscape of the search space, such as its smoothness. If solutions close to each other in the search space are semantically similar, resulting in a smooth landscape, the employed search algorithm can

