On Bi-Objective convex-quadratic problems
This work addresses theoretical and benchmarking challenges in multi-objective optimization for researchers and practitioners, but it is incremental as it builds on existing convex-quadratic frameworks.
The paper analyzes bi-objective convex-quadratic problems, providing a complete description of their Pareto set and proving the convexity of their Pareto front, with specific results like the Pareto set being a line segment under proportional Hessian matrices. It also proposes a novel set of convex-quadratic test problems to evaluate algorithm abilities in areas such as separability and rotational invariance.
In this paper we analyze theoretical properties of bi-objective convex-quadratic problems. We give a complete description of their Pareto set and prove the convexity of their Pareto front. We show that the Pareto set is a line segment when both Hessian matrices are proportional. We then propose a novel set of convex-quadratic test problems, describe their theoretical properties and the algorithm abilities required by those test problems. This includes in particular testing the sensitivity with respect to separability, ill-conditioned problems, rotational invariance, and whether the Pareto set is aligned with the coordinate axis.