Abstract
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.
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Notes
- 1.
In that scenario, we set
, 
(\(f_{1}\) and \(f_{2}\) are seen as squares of the Mahalanobis distance to the optima, with respect to the Hessian matrices), \(g_{1}(u) = 1-\frac{h_{1}}{ 1+\frac{\sqrt{u}}{r_{1}} }\), \(g_{2}(u) = 1-\frac{h_{2}}{ 1+\frac{\sqrt{u}}{r_{2}} }\).
, 
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Acknowledgments
The Ph.D. of Cheikh Touré is funded by Inria and Storengy. We particularly thank F. Huguet and A. Lange from Storengy for their strong support, practical ideas and expertise.
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Toure, C., Auger, A., Brockhoff, D., Hansen, N. (2019). On Bi-objective Convex-Quadratic Problems. In: Deb, K., et al. Evolutionary Multi-Criterion Optimization. EMO 2019. Lecture Notes in Computer Science(), vol 11411. Springer, Cham. https://doi.org/10.1007/978-3-030-12598-1_1
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