Pareto Set Characterization in Constrained Multiobjective Optimization and the COBI Problem Generator *

TL;DR

This paper introduces a new class of analytically tractable constrained multiobjective optimization problems with characterized Pareto sets, and presents COBI, a scalable generator for benchmarking optimization algorithms.

Contribution

It provides a formal characterization of Pareto sets in a new problem class and introduces COBI, a generator for constrained bi-objective test problems.

Findings

Problems can be analytically characterized with convex-quadratic functions.

COBI enables scalable generation of constrained bi-objective problems.

The approach supports multimodality, ill-conditioning, and non-separability.

Abstract

Benchmark problems play a central role in assessing the performance of numerical optimization algorithms. However, many existing constrained multiobjective optimization benchmark problems rely on overly restricted constructions or lack formal analysis of their optimal solution sets, limiting their relevance for systematic algorithm evaluation. In this work, we introduce a class of analytically tractable constrained multiobjective optimization problems whose Pareto sets can be formally characterized. The construction is based on convex-quadratic functions with positive definite Hessians, combined through multipeak formulations in which each objective is defined as the minimum over several convex-quadratic components. This approach preserves analytical structure while enabling multimodality (non-convexity), ill-conditioning and non-separability. The constraints are built as sublevel sets of multipeak functions giving rise to problems with potentially disconnected feasible regions. Building on these results, we propose COBI, a scalable generator of constrained bi-objective test problems designed for benchmarking derivative-free optimization algorithms. We provide a reference Python implementation that enables straightforward integration of COBI instances into benchmarking workflows.