Transformation-based Hypervolume Indicator: A Framework for Designing Hypervolume Variants

2020 ◽  
Author(s):  
Ke Shang ◽  
Hisao Ishibuchi ◽  
Yang Nan ◽  
Weiyu Chen
Keyword(s):  
Hypervolume Indicator

A Test Case Prioritization Genetic Algorithm Guided by the Hypervolume Indicator

2020 ◽  
Vol 46 (6) ◽  
pp. 674-696 ◽  
Author(s):  
Dario Di Nucci ◽  
Annibale Panichella ◽  
Andy Zaidman ◽  
Andrea De Lucia
Keyword(s):  
Genetic Algorithm ◽  
Test Case ◽  
Test Case Prioritization ◽  
Hypervolume Indicator

An Efficient Algorithm for Computing Hypervolume Contributions

2010 ◽  
Vol 18 (3) ◽  
pp. 383-402 ◽  
Author(s):  
Karl Bringmann ◽  
Tobias Friedrich
Keyword(s):  
Evolutionary Algorithms ◽  
Efficient Algorithm ◽  
Optimization Algorithms ◽  
Multi Objective ◽  
Multiplicative Factor ◽  
Optimal Set ◽  
Hypervolume Indicator ◽  
Additive Term

The hypervolume indicator serves as a sorting criterion in many recent multi-objective evolutionary algorithms (MOEAs). Typical algorithms remove the solution with the smallest loss with respect to the dominated hypervolume from the population. We present a new algorithm which determines for a population of size n with d objectives, a solution with minimal hypervolume contribution in time [Formula: see text](nd/2 log n) for d > 2. This improves all previously published algorithms by a factor of n for all d > 3 and by a factor of [Formula: see text] for d = 3. We also analyze hypervolume indicator based optimization algorithms which remove λ > 1 solutions from a population of size n = μ + λ. We show that there are populations such that the hypervolume contribution of iteratively chosen λ solutions is much larger than the hypervolume contribution of an optimal set of λ solutions. Selecting the optimal set of λ solutions implies calculating [Formula: see text] conventional hypervolume contributions, which is considered to be computationally too expensive. We present the first hypervolume algorithm which directly calculates the contribution of every set of λ solutions. This gives an additive term of [Formula: see text] in the runtime of the calculation instead of a multiplicative factor of [Formula: see text]. More precisely, for a population of size n with d objectives, our algorithm can calculate a set of λ solutions with minimal hypervolume contribution in time [Formula: see text](nd/2 log n + nλ) for d > 2. This improves all previously published algorithms by a factor of nmin{λ,d/2} for d > 3 and by a factor of n for d = 3.

scholarly journals Population size matters: Rigorous runtime results for maximizing the hypervolume indicator

2015 ◽  
Vol 561 ◽  
pp. 24-36 ◽  
Author(s):  
Anh Quang Nguyen ◽  
Andrew M. Sutton ◽  
Frank Neumann
Keyword(s):  
Population Size ◽  
Hypervolume Indicator

Greedy Hypervolume Subset Selection in Low Dimensions

2016 ◽  
Vol 24 (3) ◽  
pp. 521-544 ◽  
Author(s):  
Andreia P. Guerreiro ◽  
Carlos M. Fonseca ◽  
Luís Paquete
Keyword(s):  
Pareto Front ◽  
Subset Selection ◽  
Approximation Problem ◽  
Exact Algorithms ◽  
Submodular Function ◽  
Dimensional Case ◽  
Low Dimensions ◽  
Point Set ◽  
Suitable Reference ◽  
Hypervolume Indicator

Given a nondominated point set [Formula: see text] of size [Formula: see text] and a suitable reference point [Formula: see text], the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size [Formula: see text] that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of [Formula: see text]. In contrast, the best upper bound available for [Formula: see text] is [Formula: see text]. Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of [Formula: see text] using a greedy strategy. In this article, greedy [Formula: see text]-time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem.

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