On the Impact of Crossover in Many-Objective Optimization: A Runtime Analysis of NSGA-III
TLDR
This paper theoretically analyzes NSGA-III, showing crossover significantly speeds up optimization on many-objective problems like m-OJZJ.
Key contributions
- Analyzes NSGA-III's runtime with crossover on the $m$-objective $m$-OneJumpZeroJump function.
- Shows NSGA-III with crossover is asymptotically faster than without on $m$-OJZJ for any $m$.
- Establishes a lower runtime bound for NSGA-III without crossover on the 4-OJZJ problem.
Why it matters
This paper provides crucial theoretical insights into the benefits of crossover in many-objective optimization, a topic where understanding has lagged behind practical application. It validates the use of crossover in NSGA-III, demonstrating its significant performance advantage. This helps bridge the gap between theory and practice in evolutionary algorithms.
Original Abstract
In recent years, a theoretical understanding has rapidly advanced regarding how popular multi-objective evolutionary algorithms (MOEAs) can optimize many-objective problems. However, the benefits of using crossover in many-objective optimization are theoretically not understood, except for specifically designed benchmark functions tuned to particular crossover operators, and still lag significantly behind its practical use. In this paper, we build upon this line of research and present a theoretical runtime analysis of the widely used NSGA-III algorithm on the classical $m$-objective $m$-OneJumpZeroJump function ($m$-OJZJ for short). Our results demonstrate that NSGA-III with crossover optimizes $m$-OJZJ asymptotically faster than NSGA-III without crossover for any number $m$ of objectives for huge parameter regimes. We complement our analysis by providing a lower runtime bound on $4$-OJZJ when crossover is turned off.
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