Challenges of Convex Quadratic Bi-objective Benchmark Problems

TL;DR

This paper investigates the challenges of convex quadratic bi-objective benchmark problems, highlighting issues like non-separability and ill-conditioning, and analyzing algorithm performance across diverse problem classes.

Contribution

It introduces a comprehensive problem class design for bi-objective quadratic benchmarks and provides experimental insights into algorithm performance related to problem structure.

Findings

Performance varies significantly across problem classes

Non-separability impacts algorithm effectiveness

Alignment with coordinate axes influences difficulty

Abstract

Convex quadratic objective functions are an important base case in state-of-the-art benchmark collections for single-objective optimization on continuous domains. Although often considered rather simple, they represent the highly relevant challenges of non-separability and ill-conditioning. In the multi-objective case, quadratic benchmark problems are under-represented. In this paper we analyze the specific challenges that can be posed by quadratic functions in the bi-objective case. Our construction yields a full factorial design of 54 different problem classes. We perform experiments with well-established algorithms to demonstrate the insights that can be supported by this function class. We find huge performance differences, which can be clearly attributed to two root causes: non-separability and alignment of the Pareto set with the coordinate system.