# The Hypervolume Newton Method for Constrained Multi-Objective Optimization Problems

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Background and Related Work

#### 2.1. Notations

#### 2.2. Multi-Objective Optimization

#### 2.3. Hypervolume Indicator and Its First-Order Derivatives

#### 2.4. Hypervolume Hessian and Hypervolume Newton Method

## 3. Hypervolume Newton Method for Constrained MOPs

#### 3.1. Handling Equalities

#### 3.2. Handling Inequalities

#### 3.3. Handling Dominated Points

#### 3.4. The HVN Method for Constrained MOPs

Algorithm 1: Standalone hypervolume Newton algorithm for equality-constrained
MOPs |

#### 3.5. Computational Cost

## 4. Numerical Results

#### 4.1. HVN as Standalone Algorithm

#### 4.2. HVN within NSGA-III

Algorithm 2: Hybridization of HVN and MOEA |

`Pymoo`library: https://pymoo.org/constraints/eps.html (accessed on 1 November 2022). The method considers a solution feasible subject to a small $\epsilon $ threshold, which decreases linearly to zero. The initial value of $\epsilon $ is set to the average constraint value of the initial population. In our experiment, we control the $\epsilon $ decrease to zero after $50\%$ of the iterations of NSGA-III. In addition, we use Das and Dennis’s approach [28] to generate well-spaced reference directions (18 partitions which lead to 190 directions) for NSGA-III. As for its hyperparameters, we use the default setting: $\eta =30$ and $p=1$ for simulated binary crossover and $\eta =20$ for polynomial mutation. Furthermore, the hybrid algorithm first executes NSGA-III with $\mu =200$ for 1000 iterations and then runs the HVN method for 10 iterations. In HVN, the total function evaluations and AD operations take ca. 270 s CPU time on an Intel(R) Core(TM) i5-8257U CPU. Considering the CPU time of a single function evaluation, which is on average ca. 5.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ s measured on the same hardware, the total function evaluations plus the AD operations are equivalent to roughly $270/5.6\phantom{\rule{4.pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}\approx 4.8\phantom{\rule{4.pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$ FEs. Therefore, the total budget of the hybrid algorithm is roughly $4.8\phantom{\rule{4.pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}/200+1000\approx 3400$ iterations. We will execute the standalone NSGA-III algorithm for the same iterations to keep the fairness of comparisons.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of a hypervolume indicator Hessian computation in three-dimensional objective space with a collection of points $\{{\mathbf{y}}^{\left(1\right)},{\mathbf{y}}^{\left(2\right)},{\mathbf{y}}^{\left(3\right)}\}$ and reference point $\mathbf{r}$.

**Figure 2.**On problem P1, the convergence of the HVN method is shown for three different initializations of the starting approximation set ($\mu =50$)—linear (

**top row**), logistic (

**middle**), and logit spacing (

**bottom**). We depict the final approximation set (

**left column**; green stars), the corresponding objective points (

**middle column**; green stars), and the evolution of the HV value and $\u2225G(\mathbf{X},\lambda )\u2225$ (

**right column**).

**Figure 3.**On problem P2 with a spherical constraint, we depict for three sizes of the approximation set ($\mu \in \{20,40,60\}$; from

**top**to

**bottom**), the final approximation set (

**left column**; green stars), the corresponding objective points (

**middle column**; green stars), and the evolution of the HV value and $\u2225G(\mathbf{X},\lambda )\u2225$ (

**right column**). The initial points are sampled uniformly at random in the convex hull of three points ${(1,1,0)}^{\top},{(1,-1,0)}^{\top}$, and ${(-1,0,0)}^{\top}$.

**Figure 4.**On problem P3 with a spherical constraint, we depict for three sizes of the initial approximation set ($\mu \in \{20,40,60\}$; from

**top**to

**bottom**), the final approximation set (

**left column**; green stars), the corresponding objective points (

**middle column**; green stars), and the evolution of the HV value and $\u2225G\left(\mathbf{X}\right)\u2225$ (

**right column**). The initial decision points are sampled uniformly at random in the feasible space of $[0,4]\times {[-4,4]}^{2}$.

**Figure 5.**On Eq-DTLZ1-3 problems, the HVN method starts from a small local perturbation (black crosses) of the Pareto set (sphere in the decision space), i.e., ${\mathbf{X}}^{*}+0.02\mathcal{U}(0,1)$, where ${\mathbf{X}}^{*}$ (of size 200) is sampled uniformly at random on the Pareto set. The final approximation set of the HVN method is depicted as green points. Only the first three search dimensions are shown for the decision space.

**Figure 6.**On the Eq-DTLZ2 (

**a**) and the Eq-IDTLZ1 (

**b**) problem, we compare the hybridization of HVN and NSGA-III to NSGA-III with roughly the same budget: for the former, the hybrid algorithm first executes NSGA-III with $\mu =200$ for 1000 iterations and then runs the HVN method for 10 iterations. In HVN, the total function evaluations and AD takes ca. 270 s CPU time on an Intel(R) Core(TM) i5-8257U CPU, which corresponds to ca. $4.8\times {10}^{5}$ FEs. Hence, for the latter, we set 3400 ($=4.8\times {10}^{5}/200+1000$) iterations in total for $\mu =200$. We use the same hyperparameter setting for the standalone NSGA-III and the one used in the hybridization. The decision space is ${[0,1]}^{11}$, and the reference point is ${(1,1,1)}^{\top}$ for HVN.

**Table 1.**The evolution of $\u2225G(\mathbf{X},\lambda )\u2225$ on problems P1 with three different initialization strategies.

Linear | Logistic | Logit | |
---|---|---|---|

1 | 4.23 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 4.55 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 4.20 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ |

2 | 2.33 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 2.54 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 2.27 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ |

3 | 8.81 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 1.01 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 8.52 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ |

4 | 8.19 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 7.82 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 8.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ |

5 | 2.29 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 2.17 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 2.29 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ |

6 | 1.06 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 8.77 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.11 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

7 | 1.91 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.48 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.93 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

8 | 7.38 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 7.05 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 1.03 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

9 | 1.76 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | 1.06 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 1.55 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ |

10 | 1.62 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | 1.79 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | 2.33 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

Problem P2 | Problem P3 | |||||
---|---|---|---|---|---|---|

$\mathbf{\mu}=\mathbf{20}$ | $\mathbf{\mu}=\mathbf{40}$ | $\mathbf{\mu}=\mathbf{60}$ | $\mathbf{\mu}=\mathbf{20}$ | $\mathbf{\mu}=\mathbf{40}$ | $\mathbf{\mu}=\mathbf{60}$ | |

1 | 1.365 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 1.055 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 18.036836 | 1.433 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | 569.121097 | 438.983791 |

2 | 9.454 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 9.259 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 16.083916 | 9.368 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ | 541.523806 | 434.703270 |

3 | 1.247 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 8.977 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 16.403643 | 1.197 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | 444.774066 | 365.952443 |

4 | 1.589 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 8.628 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 18.052126 | 9.522 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ | 261.636562 | 362.014326 |

5 | 9.791 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 6.888 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 12.364802 | 6.194 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ | 212.841570 | 341.897644 |

6 | 8.618 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 1.123 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 3.899254 | 5.232 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ | 145.076665 | 254.253017 |

7 | 8.024 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 8.779 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 11.323440 | 3.557 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ | 103.986300 | 240.719767 |

8 | 4.737 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 7.632 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 13.320606 | 2.419 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ | 57.592159 | 165.603954 |

9 | 9.037 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 7.090 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 2.543622 | 1.511 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ | 12.628821 | 109.411195 |

10 | 7.393 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.816 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 5.984437 | 8.527 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 0.104307 | 70.516402 |

11 | 1.182 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.660 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 5.749496 | 3.732 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{1}$ | 0.097777 | 41.699152 |

12 | 4.399 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 2.877 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.702964 | 3.248 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 0.097013 | 19.525977 |

13 | 1.535 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 3.232 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 2.240449 | 2.008 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 0.096634 | 0.447690 |

14 | 2.299 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 3.694 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 13.274468 | 1.829 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | 0.096256 | 0.257345 |

15 | 7.425 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 7.159 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 15.201915 | 5.425 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.005277 | 2.066379 |

16 | 1.572 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.378 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 11.273571 | 1.735 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.002934 | 2.016149 |

17 | 4.216 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.231 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 3.318978 | 6.372 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.001602 | 1.019636 |

18 | 1.630 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 5.630 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.818340 | 9.702 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 0.001552 | 0.944297 |

19 | 1.674 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 5.454 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 0.400360 | 9.373 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.001528 | 4.904926 |

20 | 1.733 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 5.411 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 0.335107 | 2.546 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 0.001522 | 2.937413 |

21 | 1.803 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 4.697 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 0.074058 | 7.243 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.001516 | 3.118031 |

22 | 1.761 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 5.901 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 0.081798 | 8.897 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.001139 | 0.336917 |

23 | 1.384 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 6.140 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.057776 | 6.654 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.001072 | 0.004270 |

24 | 1.020 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 4.508 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.029809 | 7.210 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.001010 | 0.000866 |

25 | 9.765 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | 2.759 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 0.001956 | 5.851 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.000994 | 0.000489 |

26 | 9.788 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | 7.794 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 0.081949 | 5.851 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.000990 | 0.000477 |

27 | 1.177 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 7.767 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 0.031275 | 5.851 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.000954 | 0.000459 |

28 | 1.176 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 7.688 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 0.000492 | 5.851 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.128140 | 0.000460 |

29 | 1.052 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 5.918 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 0.000053 | 5.851 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.252721 | 0.000460 |

30 | 1.314 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 6.821 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 0.000002 | 5.851 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 0.022233 | 0.000460 |

**Table 3.**On Eq-DTLZ1-4 and Eq-IDTLZ1-4 problems, the sample mean and standard error of the hypervolume (HV) value and the number of final non-dominated (ND) points over 15 independent runs for each algorithm. The hypervolume values are computed with reference point ${(1,1,1)}^{\top}$ for all problems except Eq-DTLZ4, Eq-IDTLZ3, and Eq-IDTLZ4, which we use ${(1.2,5\phantom{\rule{4.pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3},5\phantom{\rule{4.pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4})}^{\top}$, ${(800,800,700)}^{\top}$, and ${(-0.4,0.6,0.6)}^{\top}$, respectively. The initial population is $\mu =200$ for all algorithms. Hybridization = NSGA-III (iter $=1000$) + HVN (iter $=10$), which consumes roughly the same CPU time on function evaluations with NSGA-III for 3400 iterations (see caption of Figure 6 for the detail).

Eq-DTLZ1 | Eq-DTLZ2 | Eq-DTLZ3 | Eq-DTLZ4 | |||||
---|---|---|---|---|---|---|---|---|

Algorithm | HV | #ND | HV | #ND | HV | #ND | HV | #ND |

NSGA-III (1000) | 0.867 ± 1.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 28.4 ± 0.7 | 0.297 ± 1.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 32.7 ± 0.9 | 0.292 ± 1.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 26.0 ± 1.0 | 8.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ ± 7.0 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 12.3 ± 0.8 |

Hybridization | 0.876 ± 2.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 80.9 ± 2.0 | 0.324 ± 3.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 95.3 ± 1.9 | 0.321 ± 6.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 75.2 ± 2.4 | 1.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ ± 5.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 200.0 ± 0.0 |

NSGA-III (3400) | 0.873 ± 4.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 38.5 ± 1.3 | 0.304 ± 9.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 32.6 ± 0.9 | 0.301 ± 1.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 30.1 ± 0.7 | 9.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ ± 5.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 14.5 ± 0.6 |

Eq-IDTLZ1 | Eq-IDTLZ2 | Eq-IDTLZ3 | Eq-IDTLZ4 | |||||

Algorithm | HV | #ND | HV | #ND | HV | #ND | HV | #ND |

NSGA-III (1000) | 0.517 ± 1.8 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 23.2 ± 0.5 | 3.224 ± 2.0 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 74.1 ± 1.2 | 1.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$ ± 8.0 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 81.7 ± 1.6 | 8.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ ± 7.0 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 12.3 ± 0.8 |

Hybridization | 0.534 ± 1.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 112.1 ± 2.1 | 3.388 ± 1.7 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 198.2 ± 0.4 | 1.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$ ± 5.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 197.1 ± 0.4 | 1.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ ± 5.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 200.0 ± 0.0 |

NSGA-III (3400) | 0.529 ± 2.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 33.4 ± 0.4 | 3.359 ± 4.7 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 88.3 ± 0.4 | 1.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$ ± 2.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 92.1 ± 0.8 | 9.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ ± 5.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 14.5 ± 0.6 |

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**MDPI and ACS Style**

Wang, H.; Emmerich, M.; Deutz, A.; Hernández, V.A.S.; Schütze, O.
The Hypervolume Newton Method for Constrained Multi-Objective Optimization Problems. *Math. Comput. Appl.* **2023**, *28*, 10.
https://doi.org/10.3390/mca28010010

**AMA Style**

Wang H, Emmerich M, Deutz A, Hernández VAS, Schütze O.
The Hypervolume Newton Method for Constrained Multi-Objective Optimization Problems. *Mathematical and Computational Applications*. 2023; 28(1):10.
https://doi.org/10.3390/mca28010010

**Chicago/Turabian Style**

Wang, Hao, Michael Emmerich, André Deutz, Víctor Adrián Sosa Hernández, and Oliver Schütze.
2023. "The Hypervolume Newton Method for Constrained Multi-Objective Optimization Problems" *Mathematical and Computational Applications* 28, no. 1: 10.
https://doi.org/10.3390/mca28010010