# A Mathematical Runtime Analysis of the Non-dominated Sorting Genetic Algorithm III (NSGA-III)

arXiv: https://arxiv.org/abs/2211.08202 · HTML: https://mlsanity.com/paper/2211.08202/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 57 references that are in the map.
- First posted on arXiv 15 Nov 2022.

## Builds on (most cited first, 7 of 7)

- [Probabilistic Tools for the Analysis of Randomized Optimization Heuristics](https://mlsanity.com/paper/1801.06733/) (2018)
- [A First Runtime Analysis of the NSGA-II on a Multimodal Problem](https://mlsanity.com/paper/2204.13750/) (2022)
- [Runtime Analysis for the NSGA-II: Provable Speed-Ups From Crossover](https://mlsanity.com/paper/2208.08759/) (2022)
- [From Understanding the Population Dynamics of the NSGA-II to the First Proven Lower Bounds](https://mlsanity.com/paper/2209.13974/) (2022)
- [Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables](https://mlsanity.com/paper/2109.05799/) (2021)
- [Mathematical Runtime Analysis for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)](https://mlsanity.com/paper/2112.08581/) (2021)
- [Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)](https://mlsanity.com/paper/2203.02693/) (2022)

## Built on by (most cited first, 11 of 11)

- [Stochastic Population Update Can Provably Be Helpful in Multi-Objective Evolutionary Algorithms](https://mlsanity.com/paper/2306.02611/) (2023)
- [Runtime Analysis for the NSGA-II: Proving, Quantifying, and Explaining the Inefficiency For Many Objectives](https://mlsanity.com/paper/2211.13084/) (2022)
- [The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem](https://mlsanity.com/paper/2305.13459/) (2023)
- [Runtime Analyses of Multi-Objective Evolutionary Algorithms in the Presence of Noise](https://mlsanity.com/paper/2305.10259/) (2023)
- [Many Objective Problems Where Crossover is Provably Essential](https://mlsanity.com/paper/2412.18375/) (2024)
- [A Crowding Distance That Provably Solves the Difficulties of the NSGA-II in Many-Objective Optimization](https://mlsanity.com/paper/2407.17687/) (2024)
- [Scalable Speed-ups for the SMS-EMOA from a Simple Aging Strategy](https://mlsanity.com/paper/2505.01647/) (2025)
- [Can Evolutionary Clustering Have Theoretical Guarantees?](https://mlsanity.com/paper/2212.01771/) (2022)
- [A Block-Coordinate Descent EMO Algorithm: Theoretical and Empirical Analysis](https://mlsanity.com/paper/2404.03838/) (2024)
- [SPEA2^+: Improved Density Estimation in SPEA2 with Provable Runtime Guarantees](https://mlsanity.com/paper/2606.12382/) (2026)
- [LLM-Driven Evolutionary Generation of Multi-Objective Bayesian Optimization Algorithms](https://mlsanity.com/paper/2607.08791/) (2026)

## Abstract (arXiv)

The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is the most prominent multi-objective evolutionary algorithm for real-world applications. While it performs evidently well on bi-objective optimization problems, empirical studies suggest that it is less effective when applied to problems with more than two objectives. A recent mathematical runtime analysis confirmed this observation by proving the NGSA-II for an exponential number of iterations misses a constant factor of the Pareto front of the simple 3-objective OneMinMax problem. In this work, we provide the first mathematical runtime analysis of the NSGA-III, a refinement of the NSGA-II aimed at better handling more than two objectives. We prove that the NSGA-III with sufficiently many reference points -- a small constant factor more than the size of the Pareto front, as suggested for this algorithm -- computes the complete Pareto front of the 3-objective OneMinMax benchmark in an expected number of O(n log n) iterations. This result holds for all population sizes (that are at least the size of the Pareto front). It shows a drastic advantage of the NSGA-III over the NSGA-II on this benchmark. The mathematical arguments used here and in previous work on the NSGA-II suggest that similar findings are likely for other benchmarks with three or more objectives.