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Conference Papers Year : 2021

## Hypervolume in Biobjective Optimization Cannot Converge Faster Than $\Omega(1/p)$

Eugénie Marescaux
Nikolaus Hansen
• Function : Author

#### 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 $1/(p + CONST)$.

### Dates and versions

hal-03205870 , version 1 (22-04-2021)

### Identifiers

• HAL Id : hal-03205870 , version 1
• DOI :

### Cite

Eugénie Marescaux, Nikolaus Hansen. Hypervolume in Biobjective Optimization Cannot Converge Faster Than $\Omega(1/p)$. GECCO 2021 - The Genetic and Evolutionary Computation Conference, Jul 2021, Lille / Virtual, France. ⟨10.1145/3449639.3459371⟩. ⟨hal-03205870⟩

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