# A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems

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## Abstract

**:**

## 1. Introduction

## 2. Background and Related Work

## 3. GD_{p} Newton Method

#### 3.1. Derivatives of $G{D}_{2}^{2}$

#### 3.1.1. Gradient of $G{D}_{2}^{2}$

- (i)
- $F\left({a}_{i}\right)={z}_{{j}_{i}},$ that is, if the image of ${a}_{i}$ is equal to one of the elements of the reference set. This is for instance never the case if Z is chosen utopian.
- (ii)
- If $F\left({a}_{i}\right)\ne {z}_{{j}_{i}}$, we have$$J{\left({a}_{i}\right)}^{T}(F\left({a}_{i}\right)-{z}_{{j}_{i}})=\sum _{l=1}^{k}\nabla {f}_{l}\left({a}_{i}\right)\underset{=:{\alpha}_{l}^{\left(i\right)}}{\underset{\u23df}{({f}_{l}\left({a}_{i}\right)-{\left({z}_{{j}_{i}}\right)}_{l})}}=\sum _{l=1}^{k}{\alpha}_{l}^{\left(i\right)}\nabla {f}_{l}\left({a}_{i}\right)=0$$

#### 3.1.2. Hessian of $G{D}_{2}^{2}$

#### 3.2. Gradient and Hessian for General $p>1$

#### 3.3. $G{D}_{2}^{2}$-Newton Method

**Proposition**

**1.**

**Proof.**

#### 3.4. Example

## 4. IGD_{p} Newton Method

#### 4.1. Gradient of $IG{D}_{p}$

- $rank\left(J\left({a}_{l}\right)\right)=k,$ then $F\left({a}_{l}\right)=\frac{{\sum}_{i\in {I}_{l}}{z}_{i}}{{m}_{l}}={C}_{l}.$
- $rank\left(J\left({a}_{l}\right)\right)=k-1,$ then $F\left({a}_{l}\right)-{C}_{l}$ is orthogonal to the linearized image of F at $F\left({a}_{l}\right)$, and orthogonal to the linearized Pareto front at $F\left({a}_{l}\right)$ in case $F\left({a}_{l}\right)-{C}_{l}{\ge}_{p}0$ and $F\left({a}_{l}\right)-{C}_{l}\ne 0$ (see Figure 5 for such a scenario).

#### 4.2. Hessian Matrix of $IG{D}_{p}$

#### 4.3. Gradient and Hessian for General $p>1$

#### 4.4. $IG{D}_{2}^{2}$ Newton Method

- if ${m}_{l}=0$ for a $l\in \{1,\dots ,N\}$ (since then $\mathcal{D}g\left({a}_{l}\right)=0$) (see also the discussion above), and
- if one element ${z}_{l}$ of Z is feasible (since then $\mathcal{D}g\left({a}_{l}\right)=J{\left({a}_{l}\right)}^{T}J\left({a}_{l}\right)$ which has a rank $\le k$, and under the assumption that $k<n$).

**Proposition**

**2.**

**Proof.**

#### 4.5. Examples

## 5. ${\mathsf{\Delta}}_{\mathbf{2}}$-Newton Method

#### 5.1. ${\mathsf{\Delta}}_{2}$-Newton Method

#### 5.2. Examples

#### 5.3. A Bootstrap Method for the Computation of the Pareto Front

- Compute the minima ${x}_{i}^{*}$ of the individual objectives ${f}_{i}$, $i=1,\dots ,k$. Let ${y}_{i}^{*}=F\left({x}_{i}^{*}\right)$, and let ${\tilde{Z}}_{0}$ be the convex hull of the ${y}_{i}^{*}$’s (also called convex hull of individual minima (CHIM) [23]). Let ${\delta}_{0}>0$ and set$${Z}_{0}={\tilde{Z}}_{0}-{\delta}_{0},$$
- In step l of the iteration, use the set ${A}^{(l-1)}$ computed in the previous iteration to compute a set ${\tilde{Z}}_{l}$. This can be done via interpolation of the elements of ${A}^{(l-1)}$ so that ${\tilde{Z}}_{l}$ only contains mutually non-dominated elements. As new reference set use$${Z}_{l}={\tilde{Z}}_{l}-{\delta}_{l},$$

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Pareto fronts with different shapes together with their best approximations in the sense of (5) for $p=2$ and $N=20$, and where Z is an approximation of the Pareto front.

**Figure 2.**Geometrical interpretation of the optimality condition (ii) for $G{D}_{2}^{2}$. Note that $\alpha $ is orthogonal to the linearized Pareto front. (

**a**) shows this behavior on a concave Pareto front, (

**b**) on a convex Pareto front, and (

**c**) on a concave/convex Pareto front.

**Figure 3.**(

**Left**) application of the $G{D}_{2}^{2}$ Newton method on bi-objective oriented problem (BOP) (27). (

**Right**) only the final archive is shown.

**Figure 4.**Example of a relation between the reference set Z and the approximation set $F\left(A\right)$.

**Figure 5.**Geometric interpretation when $F\left({a}_{l}\right)-C$ is orthogonal to the linearized Pareto front.

**Figure 6.**Potential problem of the Inverted Generational Distance (IGD) Newton method: if ${m}_{l}=0$ (here it is ${m}_{2}=0$) then the l-th sub-gradient is equal to zero, and ${a}_{l}$ will stay fixed under the Newton iteration.

**Figure 7.**Dominance and distance are different concepts. (

**Left**) an example where ${a}_{1}$ and ${a}_{2}$ are mutually non-dominated, but where $\mid {I}_{2}\mid =0.$ (

**Right**) an example where ${a}_{1}\prec {a}_{2}$, but $\mid {I}_{l}\mid \ne 0$ for $l\in \{1,2\}.$

**Figure 8.**(

**Left**) application of the $IG{D}_{2}^{2}$-Newton method on BOP (27). (

**Right**) image of the final archive (green) together with the images for those ${m}_{l}=0$ (red).

**Figure 9.**(

**Left**) application of the $IG{D}_{2}^{2}$-Newton method on BOP (50). (

**Right**) the image of the final archive (green) together with the images for which ${m}_{l}=0$.

**Figure 10.**(

**Left**) application of the $IG{D}_{2}^{2}$-Newton method on BOP (51). (

**Right**) only the final archive is shown.

**Figure 11.**(

**Left**) application of the ${\mathsf{\Delta}}_{2}$-Newton method on BOP (27). (

**Right**) the final archive.

**Figure 12.**(

**Left**) application of the ${\mathsf{\Delta}}_{2}^{2}$-Newton method on BOP (50). (

**Right**) the final archive.

**Figure 13.**(

**Left**) application of the ${\mathsf{\Delta}}_{2}$-Newton method on BOP (51). (

**Right**) the final archive.

**Figure 15.**Result of the ${\mathsf{\Delta}}_{2}^{2}$-Newton method for MOP (53), where Z is a triangle (two different views of the same result).

**Figure 16.**Different iterations of the ${\mathsf{\Delta}}_{2}^{2}$-Newton method to obtain the Pareto front of MOP (27) via the bootstrapping method.

**Figure 17.**Different iterations of the ${\mathsf{\Delta}}_{2}^{2}$-Newton method to obtain the Pareto front of MOP (50) via the bootstrapping method.

**Figure 18.**Different iterations of the ${\mathsf{\Delta}}_{2}^{2}$-Newton method to obtain the Pareto front of MOP (51) via the bootstrapping method.

**Figure 19.**Initial candidate set (

**left**) and numerical result of the ${\mathsf{\Delta}}_{2}$-Newton method on minus DTLZ2 (

**right**).

Iter. | $\parallel \mathit{\nabla}{\mathit{GD}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathit{GD}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{F}\left({\mathit{P}}_{\mathit{Q}}\right))$ | ${\mathit{GD}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ |
---|---|---|---|

0 | - | 12.000000000000000 | 2.102524077758237 |

1 | 24.575789798441914 | 2.443855313744088 | 1.364160070353236 |

2 | 10.174108923911083 | 0.155893831541973 | 1.099322040004995 |

3 | 5.003263195893473 | 0.002209872937986 | 1.014751905633911 |

4 | 3.169714351377499 | 0.000015254816873 | 0.976329099745630 |

5 | 1.947602617177173 | 0.000000021343544 | 0.957865673825140 |

6 | 1.758375206901766 | 0.000000000020256 | 0.945790145414235 |

7 | 1.433193382521511 | 0.000000000000013 | 0.939274242767051 |

8 | 1.012249366157551 | 0.000000000000000 | 0.936469149315602 |

9 | 0.006408088020990 | 0.000000000000000 | 0.936469035893491 |

10 | 0.000000182419413 | 0 | 0.936469035893491 |

11 | 0.000000000000002 | 0 | 0.936469035893491 |

Iter. | $\parallel \mathit{\nabla}{\mathit{IGD}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathit{GD}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{F}\left({\mathit{P}}_{\mathit{Q}}\right))$ | ${\mathit{IGD}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ |
---|---|---|---|

0 | - | 12.000000000000000 | 0.280604068205798 |

1 | 18.378420484981000 | 1.930506027522264 | 0.206869895378755 |

2 | 5.432039146770605 | 0.043471951768718 | 0.192756253890335 |

3 | 0.817043487084936 | 0.000003701993633 | 0.192391225092683 |

4 | 0.706420510642436 | 0.000000000171966 | 0.192326368208389 |

5 | 0.006184251273371 | 0.000000000000000 | 0.192326331507963 |

6 | 0.000000481423311 | 0 | 0.192326331505474 |

7 | 0.000000000000007 | 0 | 0.192326331505474 |

Iter. | $\parallel \mathit{\nabla}{\mathit{IGD}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathit{IGD}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{F}\left({\mathit{P}}_{\mathit{Q}}\right))$ | ${\mathit{IGD}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ |
---|---|---|---|

0 | - | 0.278373584606464 | 0.023220380628487 |

1 | 0.336586538953659 | 0.201027590101253 | 0.008847986218368 |

2 | 0.037592704195443 | 0.208694136517504 | 0.008453782829010 |

3 | 0.020266018162657 | 0.205037496184976 | 0.008317268653462 |

4 | 0.004947265003093 | 0.197436504168332 | 0.008331864711939 |

5 | 0.012675095498115 | 0.196899482529974 | 0.008335126593931 |

6 | 0.011342458560546 | 0.195934897889421 | 0.008256278024484 |

7 | 0.001644428661330 | 0.194957155951972 | 0.008252791569886 |

8 | 0.001355427544721 | 0.194529344670235 | 0.008248151529685 |

9 | 0.000531420743367 | 0.194275287179155 | 0.008247144931520 |

10 | 0.000462304446354 | 0.194197084382389 | 0.008244551897282 |

11 | 0.000160142304221 | 0.194173320804311 | 0.008243923138606 |

12 | 0.000083987735463 | 0.194171755397215 | 0.008243322467470 |

13 | 0.000007989306735 | 0.194171718496160 | 0.008243262849450 |

14 | 0.000000313502013 | 0.194171718485903 | 0.008243260421944 |

15 | 0.000000004246450 | 0.194171718485903 | 0.008243260388981 |

16 | 0.000000000077470 | 0.194171718485903 | 0.008243260388530 |

17 | 0.000000000010284 | 0.194171718485903 | 0.008243260388522 |

18 | 0.000000000001988 | 0.194171718485903 | 0.008243260388521 |

19 | 0.000000000000385 | 0.194171718485903 | 0.008243260388521 |

20 | 0.000000000000075 | 0.194171718485903 | 0.008243260388521 |

Iter. | $\parallel \mathit{\nabla}{\mathit{GD}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathit{IGD}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{F}\left({\mathit{P}}_{\mathit{Q}}\right))$ | ${\mathit{IGD}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ |
---|---|---|---|

0 | - | 0.743479945976417 | 0.089706026039859 |

1 | 1.774223579432539 | 0.598423777888877 | 0.069797721774084 |

2 | 1.059917072891813 | 0.580388865733755 | 0.065730197299576 |

3 | 0.436102700877237 | 0.575606525856331 | 0.065186338106827 |

4 | 0.044524401746943 | 0.575571396044490 | 0.065172539475108 |

5 | 0.000368536791948 | 0.575571392056566 | 0.065172518364193 |

6 | 0.000000023167599 | 0.575571392056566 | 0.065172518358461 |

7 | 0.000000000000001 | 0.575571392056566 | 0.065172518358461 |

Iter. | $\parallel \mathit{\nabla}{\mathsf{\Delta}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{F}\left({\mathit{P}}_{\mathit{Q}}\right))$ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ | Indicator |
---|---|---|---|---|

0 | - | 2.000000000000000 | 2.102524077758237 | GD |

1 | 24.575789798441914 | 0.410124138028425 | 1.364160070353236 | GD |

2 | 10.174108923911083 | 0.026608923543674 | 1.099322040004995 | IGD |

3 | 3.542645526228592 | 0.000015520657566 | 1.104000035194028 | GD |

4 | 5.213757247004612 | 0.000000441436988 | 1.020226507943846 | IGD |

5 | 9.260923965020773 | 0.000000016030350 | 1.036249331407054 | IGD |

6 | 2.625118989394418 | 0.000000384599061 | 1.047878336296791 | IGD |

7 | 0.042216669617238 | 0.000000384618980 | 1.047678945935868 | IGD |

8 | 0.000010909557366 | 0.000000384618980 | 1.047678891593882 | IGD |

9 | 0.000000000000756 | 0.000000384618980 | 1.047678891593878 | IGD |

10 | 0.000000000000006 | 0.000000384618980 | 1.047678891593878 | IGD |

Iter. | $\parallel \mathit{\nabla}{\mathsf{\Delta}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{F}\left({\mathit{P}}_{\mathit{Q}}\right))$ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ | Indicator |
---|---|---|---|---|

0 | - | 0.278373584606464 | 0.139932582443422 | GD |

1 | 0.056371237267200 | 0.066618294837097 | 0.056728504338161 | GD |

2 | 0.057045938719184 | 0.039369161609912 | 0.041433057966044 | GD |

3 | 0.037484475625202 | 0.024109339347752 | 0.031977133783097 | IGD |

4 | 0.050812222533911 | 0.023513431743364 | 0.033996922698945 | GD |

5 | 0.031160653990564 | 0.014395303481718 | 0.024321095970292 | IGD |

6 | 0.037264116905168 | 0.016443622791045 | 0.025809212757710 | IGD |

7 | 0.018144934519792 | 0.024703492528406 | 0.029965847813991 | IGD |

8 | 0.019468781951843 | 0.028619439695385 | 0.030778919683651 | GD |

9 | 0.035410330655855 | 0.017309625336888 | 0.019853297293327 | IGD |

10 | 0.043091325647137 | 0.021752471483838 | 0.024124052991424 | IGD |

11 | 0.008311561314162 | 0.017118271463333 | 0.026490329733338 | GD |

12 | 0.029726132461309 | 0.011015521887052 | 0.017647857545199 | IGD |

13 | 0.049385612870240 | 0.013547698417436 | 0.021426583957166 | IGD |

14 | 0.014525769797194 | 0.020436747509906 | 0.034124722919781 | IGD |

15 | 0.030582462613590 | 0.012480668370491 | 0.021750946945761 | IGD |

16 | 0.030419387651439 | 0.006090064700256 | 0.023544581385908 | IGD |

17 | 0.012758353366778 | 0.000030594174296 | 0.025131789220814 | IGD |

18 | 0.009657374280631 | 0.001454430574110 | 0.025813967350062 | IGD |

19 | 0.005296333332650 | 0.001374135866037 | 0.026375698139894 | IGD |

20 | 0.005548518084090 | 0.002112406054454 | 0.027521017269386 | IGD |

21 | 0.005856819919213 | 0.002804811375528 | 0.029968509026162 | IGD |

22 | 0.012701286040104 | 0.001922080339960 | 0.030132357483057 | IGD |

23 | 0.003183819547848 | 0.001504297326063 | 0.030456038027207 | IGD |

24 | 0.003253860331803 | 0.002240659587601 | 0.031310708007687 | IGD |

25 | 0.003580104890061 | 0.001602870721889 | 0.031465842685442 | IGD |

26 | 0.002074689422294 | 0.001367127795787 | 0.031805380040383 | IGD |

27 | 0.001414150903872 | 0.000126099902661 | 0.031769100819775 | IGD |

28 | 0.001111604812819 | 0.001688362662578 | 0.031742469387990 | IGD |

29 | 0.000901680741441 | 0.003036794425943 | 0.031689031421120 | IGD |

30 | 0.000257772611116 | 0.003797901156449 | 0.031672123060034 | IGD |

31 | 0.000101230696412 | 0.003991409932376 | 0.031663374641811 | IGD |

32 | 0.000007716198343 | 0.004008504531546 | 0.031662626381853 | IGD |

33 | 0.000000360445308 | 0.004008654325951 | 0.031662593054177 | IGD |

34 | 0.000000015185568 | 0.004008654388153 | 0.031662591709428 | IGD |

35 | 0.000000000643412 | 0.004008654388154 | 0.031662591654952 | IGD |

36 | 0.000000000027461 | 0.004008654388154 | 0.031662591652896 | IGD |

37 | 0.000000000001332 | 0.004008654388154 | 0.031662591652858 | IGD |

38 | 0.000000000000154 | 0.004008654388154 | 0.031662591652867 | IGD |

39 | 0.000000000000033 | 0.004008654388154 | 0.031662591652869 | IGD |

Iter. | $\parallel \mathit{\nabla}{\mathsf{\Delta}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{F}\left({\mathit{P}}_{\mathit{Q}}\right))$ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ | Indicator |
---|---|---|---|---|

0 | - | 0.706541653137130 | 0.794786137846191 | GD |

1 | 2.052766083969590 | 0.169306894160018 | 0.477606371056880 | GD |

2 | 1.043651307474400 | 0.001325637478010 | 0.312621943719186 | IGD |

3 | 0.335281361638164 | 0.000782828935146 | 0.318925114182142 | IGD |

4 | 0.091979632627269 | 0.000782533520334 | 0.322156690814970 | IGD |

5 | 0.070361802550103 | 0.000782533238333 | 0.325270306125291 | IGD |

6 | 0.001111027357469 | 0.000782533238004 | 0.325312843569155 | IGD |

7 | 0.000000574058899 | 0.000782533238004 | 0.325312858713113 | IGD |

8 | 0.000000000000187 | 0.000782533238004 | 0.325312858713118 | IGD |

9 | 0.000000000000000 | 0.000782533238004 | 0.325312858713118 | IGD |

Iter. | $\parallel \mathit{\nabla}{\mathsf{\Delta}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ | |
---|---|---|---|

0 | - | 1.378676581248260 | 5.918796450955248 |

1 | 7.515915778732357 | 1.244858568391063 | 5.821725966611490 |

2 | 1.853655193889793 | 1.249239765031169 | 5.811343998211261 |

3 | 0.145669456936361 | 1.250137056379269 | 5.810973507376134 |

4 | 0.000671971342724 | 1.250139614137738 | 5.810971792145200 |

5 | 0.000000033865300 | 1.250139614289546 | 5.810971792043552 |

6 | 0.000000000000003 | 1.250139614289546 | 5.810971792043552 |

Iter. | $\parallel \mathit{\nabla}{\mathsf{\Delta}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ | |
---|---|---|---|

0 | 1.000000000000000 | 1.378676581248260 | 5.136345894189382 |

1 | 9.078968824204878 | 1.190673858342912 | 5.014877171961635 |

2 | 12.361924917381627 | 1.250986213036476 | 3.359444827553141 |

3 | 4.188649668005252 | 1.076592897207513 | 3.232994469439562 |

4 | 0.664877643630374 | 1.053683832191072 | 3.217085086974496 |

5 | 0.686656379329441 | 1.036451498535567 | 3.213854172896627 |

6 | 0.005761784413677 | 1.036607563930994 | 3.213850773658971 |

7 | 0.000001010224622 | 1.036607573211355 | 3.213850772877354 |

8 | 0.000000000000157 | 1.036607573211355 | 3.213850772877354 |

9 | 0.000000000000002 | 1.036607573211354 | 3.213850772877354 |

**Table 10.**Numerical results of the ${\mathsf{\Delta}}_{2}^{2}$-Newton method to obtain the Pareto front of MOP (27) via the bootstrapping method.

Iter. | $\parallel \mathit{\nabla}{\mathsf{\Delta}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ | Indicator | |
---|---|---|---|---|

0 | - | 2.565838356405802 | 5.454852860388515 | GD |

1 | 14.958401000284267 | 1.230817708101819 | 1.024752009175881 | IGD |

2 | 6.553591835159349 | 0.754749784421640 | 0.553744685923295 | IGD |

3 | 1.930428808338138 | 0.613768117290251 | 0.490492908923838 | IGD |

4 | 0.937132679630156 | 0.537139549603782 | 0.481517242208329 | IGD |

5 | 0.540200357139832 | 0.476989307368074 | 0.471988242018486 | IGD |

6 | 0.394304493982539 | 0.438480476946482 | 0.467546882767516 | IGD |

7 | 0.153675036875927 | 0.419941366901776 | 0.468192970378975 | IGD |

8 | 0.059462724070887 | 0.413040100805383 | 0.468716613758660 | IGD |

9 | 0.039636672484473 | 0.412237873646849 | 0.468845106760589 | IGD |

10 | 0.016104664984103 | 0.412336536272929 | 0.468844846612736 | IGD |

11 | 0.001970967225026 | 0.412348016205662 | 0.468845003745375 | IGD |

12 | 0.000010540592599 | 0.412348100926435 | 0.468845005883349 | IGD |

13 | 0.000000006447819 | 0.412348100981951 | 0.468845005884675 | IGD |

14 | 0.000000000000000 | 0.412348100981951 | 0.468845005884675 | IGD |

**Table 11.**Numerical results of the ${\mathsf{\Delta}}_{2}^{2}$-Newton method to obtain the Pareto front of MOP (50) via the bootstrapping method.

Iter. | $\parallel \mathit{\nabla}{\mathsf{\Delta}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ | Indicator | |
---|---|---|---|---|

0 | - | 0.455981539616886 | 0.695079920183452 | GD |

1 | 0.335395116261223 | 0.073901038363798 | 0.755243658407092 | GD |

2 | 0.091052248527535 | 0.037763808963224 | 0.074896196233522 | IGD |

3 | 0.014233219389476 | 0.037763808963224 | 0.037618719614753 | IGD |

4 | 0.012918924846453 | 0.037763808963224 | 0.037607435705178 | IGD |

5 | 0.012918504651879 | 0.037763808963224 | 0.037607435705178 | IGD |

**Table 12.**Numerical results of the ${\mathsf{\Delta}}_{2}^{2}$-Newton method to obtain the Pareto front of MOP (51) via the bootstrapping method.

Iter. | $\parallel \mathit{\nabla}{\mathsf{\Delta}}_{2}^{2}\left({\mathit{A}}^{\mathit{i}}\right)\parallel $ | ${\mathsf{\Delta}}_{2}^{2}(\mathit{F}\left({\mathit{A}}^{\mathit{i}}\right),\mathit{Z})$ | Indicator | |
---|---|---|---|---|

0 | - | 0.702540625580303 | 2.214433876989687 | GD |

1 | 1.437150990002929 | 0.389087697290865 | 0.477018278137838 | IGD |

2 | 0.581565628190262 | 0.342604825739356 | 0.357844471083072 | IGD |

3 | 0.461927350893728 | 0.164460636656196 | 0.182910255512032 | IGD |

4 | 0.082455998873464 | 0.158481979725082 | 0.175440371075156 | IGD |

5 | 0.074464658261760 | 0.191541386471772 | 0.208964468823487 | IGD |

6 | 0.131398860002112 | 0.153900963788386 | 0.171965063484733 | IGD |

7 | 0.040291187202238 | 0.152033891842905 | 0.166135305826676 | IGD |

8 | 0.005689443259245 | 0.151817516262598 | 0.165264799265542 | IGD |

9 | 0.002103237914802 | 0.151819267034376 | 0.165273277734145 | IGD |

10 | 0.000466641874904 | 0.151819600203537 | 0.165273337507943 | IGD |

11 | 0.000013655772453 | 0.151819670338732 | 0.165273320812711 | IGD |

12 | 0.000000209461574 | 0.151819687437130 | 0.165273316697773 | IGD |

13 | 0.000000050653504 | 0.151819691559064 | 0.165273315706448 | IGD |

14 | 0.000000012209033 | 0.151819692552588 | 0.165273315467506 | IGD |

15 | 0.000000002942920 | 0.151819692792067 | 0.165273315409911 | IGD |

16 | 0.000000000709377 | 0.151819692849792 | 0.165273315396027 | IGD |

17 | 0.000000000167849 | 0.151819692849792 | 0.165273315396027 | IGD |

**Table 13.**Number of function ($\#F$), Jacobian ($\#J$), and Hessian ($\#H$) calls used by the ${\mathsf{\Delta}}_{2}^{2}$-Newton method using bootstrapping for the three test problems.

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## Share and Cite

**MDPI and ACS Style**

Uribe, L.; Bogoya, J.M.; Vargas, A.; Lara, A.; Rudolph, G.; Schütze, O.
A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems. *Mathematics* **2020**, *8*, 1822.
https://doi.org/10.3390/math8101822

**AMA Style**

Uribe L, Bogoya JM, Vargas A, Lara A, Rudolph G, Schütze O.
A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems. *Mathematics*. 2020; 8(10):1822.
https://doi.org/10.3390/math8101822

**Chicago/Turabian Style**

Uribe, Lourdes, Johan M Bogoya, Andrés Vargas, Adriana Lara, Günter Rudolph, and Oliver Schütze.
2020. "A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems" *Mathematics* 8, no. 10: 1822.
https://doi.org/10.3390/math8101822