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

Eugénie Marescaux 1 Nikolaus Hansen 1
1 RANDOPT - Randomized Optimisation
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : The 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 $1/(p + CONST)$.
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https://hal.archives-ouvertes.fr/hal-03205870
Contributor : Eugénie Marescaux <>
Submitted on : Thursday, April 22, 2021 - 4:30:18 PM
Last modification on : Thursday, May 20, 2021 - 9:06:02 AM

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Eugénie Marescaux, Nikolaus Hansen. Hypervolume in Biobjective Optimization Cannot Converge Faster Than Ω$(1/p)$. GECCO 2021 - The Genetic and Evolutionary Computation Conference, Jul 2021, Lille / Virtual, France. ⟨hal-03205870⟩

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