Biobjective Hypervolume Based HV-ISOOMOO Algorithms Converge with At Least Sublinear Speed to the Entire Pareto Front - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... Year : 2022

## Biobjective Hypervolume Based HV-ISOOMOO Algorithms Converge with At Least Sublinear Speed to the Entire Pareto Front

Eugénie Marescaux
Anne Auger
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• PersonId : 751513
• IdHAL : anne-auger

#### Abstract

In multiobjective optimization, one is interested in finding a good approximation of the Pareto set and the Pareto front, i.e the sets of best compromises in the decision and objective spaces, respectively. In this context, we introduce a new algorithm framework based on the hypervolume and called HyperVolume based Incremental Single-Objective Optimization for MultiObjective Optimization (HV-ISOOMOO) for approximating the Pareto front with an increasing number of points. The hypervolume is a set-quality indicator which is widely used for algorithms design and performance assessment. The class of HV-ISOOMOO algorithms approximate the Pareto front by greedily maximizing this indicator. At each meta-iteration of HV-ISOOMOO algorithms, a single-objective subproblem is solved. We study the convergence to the entire Pareto front of HV-ISOOMOO under the assumption that these subproblems are solved perfectly. The convergence is defined as the convergence of the hypervolume of the sets of all metaiterations incumbents towards the hypervolume of the Pareto front. We prove tight lower bounds on the convergence-speed for convex and bilipschitz Pareto fronts in O(1/n c) with n being the number of metaiterations and c = 1 and c ≤ 1, respectively. For convex Pareto fronts, the convergence speed is in Θ(1/n), namely the highest convergence rate achievable by a biobjective optimization algorithm. These are the first results on speed of convergence of multiobjective optimization algorithms towards the entire Pareto front. We also analyze theoretically the asymptotic convergence behavior.

#### Domains

Optimization and Control [math.OC]

### Dates and versions

hal-03198414 , version 1 (13-07-2021)
hal-03198414 , version 2 (16-09-2021)
hal-03198414 , version 3 (22-04-2022)

### Identifiers

• HAL Id : hal-03198414 , version 3

### Cite

Eugénie Marescaux, Anne Auger. Biobjective Hypervolume Based HV-ISOOMOO Algorithms Converge with At Least Sublinear Speed to the Entire Pareto Front. 2022. ⟨hal-03198414v3⟩

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