Using Well-Understood Single-Objective Functions in Multiobjective Black-Box Optimization Test Suites

TL;DR

This paper introduces a new approach for constructing multiobjective optimization test suites by combining existing single-objective functions, aiming to better reflect real-world problem characteristics and improve benchmarking practices.

Contribution

It proposes a method to create multiobjective test functions from single-objective problems, including the bbob-biobj and bbob-biobj-ext suites, with implementation in the COCO platform.

Findings

New test suites with 55 and 92 bi-objective functions

Formal function definitions and property analysis

Enhanced benchmarking capabilities for deterministic and stochastic solvers

Abstract

Several test function suites are being used for numerical benchmarking of multiobjective optimization algorithms. While they have some desirable properties, like well-understood Pareto sets and Pareto fronts of various shapes, most of the currently used functions possess characteristics that are arguably under-represented in real-world problems. They mainly stem from the easier construction of such functions and result in improbable properties such as separability, optima located exactly at the boundary constraints, and the existence of variables that solely control the distance between a solution and the Pareto front. Here, we propose an alternative way to constructing multiobjective problems-by combining existing single-objective problems from the literature. We describe in particular the bbob-biobj test suite with 55 bi-objective functions in continuous domain, and its extended version with 92 bi-objective functions (bbob-biobj-ext). Both test suites have been implemented in the COCO platform for black-box optimization benchmarking. Finally, we recommend a general procedure for creating test suites for an arbitrary number of objectives. Besides providing the formal function definitions and presenting their (known) properties, this paper also aims at giving the rationale behind our approach in terms of groups of functions with similar properties, objective space normalization, and problem instances. The latter allows us to easily compare the performance of deterministic and stochastic solvers, which is an often overlooked issue in benchmarking.