Hypervolume in Biobjective Optimization Cannot Converge Faster Than Ω(1/p)(1/p)

International audienceThe hypervolume indicator is widely used by multi-objective optimization algorithms and for assessing their performance. We investigate a set of $p$ vectors in the biobjective space that maximizes the hypervolume indicator with respect to some reference point, referred to as $p$-optimal distribution. We prove explicit lower and upper bounds on the gap between the hypervolumes of the $p$-optimal distribution and the $\infty$-optimal distribution (the Pareto front) as a function of $p$, of the reference point, and of some Lipschitz constants. On a wide class of functions, this optimality gap can not be smaller than $\Omega(1/p)$, thereby establishing a bound on the optimal convergence speed of any algorithm. For functions with either bilipschitz or convex Pareto fronts, we also establish an upper bound and the gap is hence $\Theta(1/p)$. The presented bounds are not only asymptotic. In particular, functions with a linear Pareto front have the normalized exact gap of $1/(p + 1)$ for any reference point dominating the nadir point. We empirically investigate on a small set of Pareto fronts the exact optimality gap for values of $p$ up to 1000 and find in all cases a dependency resembling $1/(p + CONST)$

On Bi-Objective convex-quadratic problems

International audienceIn 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