[Paper Review] Heuristic Rectangle Splitting: Leveraging Single-Objective Heuristics to Efficiently Solve Multi-Objective Problems

Piotr Matl, Richard F. Hartl+1 more|arXiv (Cornell University)|May 29, 2017

Vehicle Routing Optimization Methods23 references3 citations

TL;DR

This paper proposes Heuristic Rectangle Splitting (HRS), a novel adaptation of the $ε$-constraint method that leverages state-of-the-art single-objective metaheuristics to solve multi-objective problems efficiently. By integrating a tailored CVRP heuristic within an $ε$-constraint framework using penalty functions and search memory, HRS significantly outperforms existing multi-objective metaheuristics on the VRP with Route Balancing (VRPRB), achieving superior solution quality and convergence speed with minimal implementation overhead.

ABSTRACT

Real-life problems are often characterized by conflicting optimization objectives. Consequently, there has been a growing interest not only in multi-objective models, but also in specialized multi-objective metaheuristics for solving those models. A wide variety of methods, e.g. NSGA-II, SPEA, IBEA, scatter search, Pareto local search, and many others, have thus been proposed over the years. Yet in principle, multi-objective problems can be efficiently solved with existing tailored single-objective solvers -- this is the central idea behind the well-known epsilon-constraint method (ECM). Despite its theoretical properties and conceptual simplicity, the epsilon-constraint method has been largely ignored in the domain of heuristics and remains associated mostly with exact algorithms. In this article we dispel these preconceptions and demonstrate that the epsilon-constraint framework can be a highly effective way to directly leverage the existing research on single-objective optimization for solving multi-objective problems. We propose an improved version of the classical ECM adapted to the challenges and requirements specific to heuristic search. The resulting framework is implemented with an existing state-of-the-art single-objective solver for the Capacitated Vehicle Routing Problem (CVRP) and tested on the VRP with Route Balancing (VRPRB). Based on an extensive computational study, we show the added value of our adaptations compared to the classical ECM, and demonstrate that our simple epsilon-constraint algorithm significantly outperforms the current state-of-the-art multi-objective metaheuristics with respect to multiple quality metrics. We conclude with a discussion of relevant success factors and promising directions for further research.

Motivation & Objective

• To address the underutilization of the $ε$-constraint method in heuristic optimization despite its theoretical advantages.
• To bridge the gap between decades of progress in single-objective metaheuristics and the current reliance on complex, problem-specific multi-objective algorithms.
• To develop a generic, efficient, and easy-to-implement framework that enables existing single-objective solvers to solve multi-objective problems effectively.
• To evaluate the performance of the proposed HRS framework against state-of-the-art multi-objective metaheuristics on the VRPRB benchmark.
• To identify key success factors enabling high performance in the $ε$-constraint framework when applied to heuristic search.

Proposed method

• Adapt the classical $ε$-constraint method for heuristic search by reformulating constraints as penalty terms in the objective function.
• Integrate a state-of-the-art single-objective CVRP heuristic (Hybrid Genetic Search) as the underlying solver within the HRS framework.
• Use a CPU budget parameter to control computational effort and enable fair comparison with other metaheuristics.
• Employ search memory to store and reuse previous search states, enabling warm-starts and faster convergence to new non-dominated solutions.
• Apply a box-splitting strategy to systematically explore the objective space and refine the Pareto set approximation.
• Utilize penalty functions to allow exploration of infeasible solutions, enhancing heuristic flexibility and performance.

Experimental results

Research questions

• RQ1 Can the $ε$-constraint method be effectively repurposed for heuristic-based multi-objective optimization, despite its historical association with exact methods?
• RQ2 How does the performance of the proposed HRS framework compare to established multi-objective metaheuristics such as GRASP-ASP and MSSPR on the VRPRB?
• RQ3 What are the key design choices—such as penalty functions, search memory, and solution harvesting—that significantly impact HRS performance?
• RQ4 To what extent does HRS benefit from future advances in single-objective optimization, given its generic and modular design?
• RQ5 Can the $ε$-constraint framework be used to efficiently target minimal representative Pareto sets in multi-objective problems?

Key findings

• HRS converges significantly faster than the classical $ε$-constraint method while maintaining or improving solution quality on the VRPRB benchmark.
• HRS outperforms both GRASP-ASP and MSSPR in terms of hypervolume, unary epsilon, and cardinality metrics, demonstrating superior convergence and diversity.
• The use of penalty functions in the objective function enhances heuristic performance and enables effective exploration of infeasible regions.
• Search memory plays a critical role in computational efficiency by allowing warm-starts from previous search states, accelerating convergence to new non-dominated solutions.
• Solution harvesting has only a marginal impact on approximation set quality, suggesting that single-solution-based heuristics can be as effective as population-based ones within the HRS framework.
• The HRS framework is problem-agnostic, requires minimal implementation effort (only a CPU budget parameter), and directly benefits from future advances in single-objective optimization.