**Figure 1.**
**Left**: the objectives

${f}_{1}\left(x\right)={x}^{2}$ and

${f}_{2}\left(x\right)={(x-2)}^{2}$ from a multi-objective optimization problem (MOP; Equation (

1)).

**Right**: the corresponding Pareto set over the interval

$[0,2]$.

**Figure 1.**
**Left**: the objectives

${f}_{1}\left(x\right)={x}^{2}$ and

${f}_{2}\left(x\right)={(x-2)}^{2}$ from a multi-objective optimization problem (MOP; Equation (

1)).

**Right**: the corresponding Pareto set over the interval

$[0,2]$.

**Figure 2.**
According to Corollary 1, when acting on disjoint subsets, ${\Delta}_{p,q}$ behaves as a proper metric if $(p,q)$ lies in the blue sector, and according to Corollary 2, it behaves like an inframetric if $(p,q)$ lies in the orange sectors.

**Figure 2.**
According to Corollary 1, when acting on disjoint subsets, ${\Delta}_{p,q}$ behaves as a proper metric if $(p,q)$ lies in the blue sector, and according to Corollary 2, it behaves like an inframetric if $(p,q)$ lies in the orange sectors.

**Figure 3.**
Different scenarios where the ${GD}_{p}$ value of archive B is better (smaller) than the ${GD}_{p}$ value of archive A independently of the Pareto set and where the additional assumptions made in Theorem 5 are easily verifiable.

**Figure 3.**
Different scenarios where the ${GD}_{p}$ value of archive B is better (smaller) than the ${GD}_{p}$ value of archive A independently of the Pareto set and where the additional assumptions made in Theorem 5 are easily verifiable.

**Figure 4.**
Two situations where ${IGD}_{p,q}\left(B\right)$ is better (smaller) than ${IGD}_{p,q}\left(A\right)$ for sufficiently negative q: Here, the hypotheses of Proposition 3 hold true.

**Figure 4.**
Two situations where ${IGD}_{p,q}\left(B\right)$ is better (smaller) than ${IGD}_{p,q}\left(A\right)$ for sufficiently negative q: Here, the hypotheses of Proposition 3 hold true.

**Figure 5.**
Four examples where ${IGD}_{p}\left(B\right)$ is smaller (better) than ${IGD}_{p,q}\left(A\right)$ for sufficiently negative q: In each case, at least one of the requirements of Theorem 6 is satisfied.

**Figure 5.**
Four examples where ${IGD}_{p}\left(B\right)$ is smaller (better) than ${IGD}_{p,q}\left(A\right)$ for sufficiently negative q: In each case, at least one of the requirements of Theorem 6 is satisfied.

**Figure 6.**
(**Left**) A situation where the Pareto front $F\left(P\right)$ and the images $F\left(X\right)$ and $F\left(Y\right)$ of continuous archives satisfy ${\mathcal{I}}_{p,q}^{GD}\left(X\right)\u2a7d{\mathcal{I}}_{p,q}^{GD}\left(Y\right)$ and condition 1(a) of Theorem 8 holds true. (**Right**) A modification of the previous situation where conditions $\left({a}^{\prime}\right)$ and $\left({b}^{\prime}\right)$ of Remark 6 are satisfied but ${\mathcal{I}}_{p,q}^{GD}\left(X\right)\nleqq {\mathcal{I}}_{p,q}^{GD}\left(Y\right)$. Here, there are no possible partitions of the archives satisfying part 1(a) of Theorem 8.

**Figure 6.**
(**Left**) A situation where the Pareto front $F\left(P\right)$ and the images $F\left(X\right)$ and $F\left(Y\right)$ of continuous archives satisfy ${\mathcal{I}}_{p,q}^{GD}\left(X\right)\u2a7d{\mathcal{I}}_{p,q}^{GD}\left(Y\right)$ and condition 1(a) of Theorem 8 holds true. (**Right**) A modification of the previous situation where conditions $\left({a}^{\prime}\right)$ and $\left({b}^{\prime}\right)$ of Remark 6 are satisfied but ${\mathcal{I}}_{p,q}^{GD}\left(X\right)\nleqq {\mathcal{I}}_{p,q}^{GD}\left(Y\right)$. Here, there are no possible partitions of the archives satisfying part 1(a) of Theorem 8.

**Figure 7.**
A hypothetical Pareto front discretization ${P}^{\prime}$ (black circles) and two different archives: ${X}_{1}$ (blue dots) and ${X}_{2}$ (orange squares).

**Figure 7.**
A hypothetical Pareto front discretization ${P}^{\prime}$ (black circles) and two different archives: ${X}_{1}$ (blue dots) and ${X}_{2}$ (orange squares).

**Figure 8.**
Optimal ${\Delta}_{1,-1}$ archive A for the connected Pareto front ${P}_{1}$ given by Equation (12) with 10 elements (blue circles) and at the right is the respective archive coordinates and the ${\Delta}_{1,-1}$ distance.

**Figure 8.**
Optimal ${\Delta}_{1,-1}$ archive A for the connected Pareto front ${P}_{1}$ given by Equation (12) with 10 elements (blue circles) and at the right is the respective archive coordinates and the ${\Delta}_{1,-1}$ distance.

**Figure 9.**
Optimal ${\Delta}_{1,-1}$ archive A for the connected Pareto front ${P}_{2}$ given by Equation (12) with 10 elements (blue circles) and at the right is the respective archive coordinates and the ${\Delta}_{1,-1}$ distance.

**Figure 9.**
Optimal ${\Delta}_{1,-1}$ archive A for the connected Pareto front ${P}_{2}$ given by Equation (12) with 10 elements (blue circles) and at the right is the respective archive coordinates and the ${\Delta}_{1,-1}$ distance.

**Figure 10.**
Optimal ${\Delta}_{1,q}$ five-point set archives A for the connected Pareto front ${P}_{1}$ given by Equation (12) with $p=1$ and $q=\pm 1/2$.

**Figure 10.**
Optimal ${\Delta}_{1,q}$ five-point set archives A for the connected Pareto front ${P}_{1}$ given by Equation (12) with $p=1$ and $q=\pm 1/2$.

**Figure 11.**
Optimal ${\Delta}_{p,-1}$ one-point archives A for the connected Pareto front ${P}_{1}$ given by Equation (12) with $q=-1$ and different values of p: In all cases, the archives are located in the line $x=y$.

**Figure 11.**
Optimal ${\Delta}_{p,-1}$ one-point archives A for the connected Pareto front ${P}_{1}$ given by Equation (12) with $q=-1$ and different values of p: In all cases, the archives are located in the line $x=y$.

**Figure 12.**
Numerical optimal ${\Delta}_{1,-1}$ archive A for the disconnect step Pareto front ${P}_{3}^{\left(5\right)}$ given by Equation (13) with 20 elements: here, we obtain ${\Delta}_{1,-1}\left(A,{P}_{3}^{(5,\frac{1}{10})}\right)=0.111132$.

**Figure 12.**
Numerical optimal ${\Delta}_{1,-1}$ archive A for the disconnect step Pareto front ${P}_{3}^{\left(5\right)}$ given by Equation (13) with 20 elements: here, we obtain ${\Delta}_{1,-1}\left(A,{P}_{3}^{(5,\frac{1}{10})}\right)=0.111132$.

**Figure 13.**
The black horizontal segment is the set A from Equation (14), and the blue piecewise map is the respective approximation given by the set ${B}_{\delta}$ from Equation (15) for two values of $\delta $ and $\epsilon =0.10$.

**Figure 13.**
The black horizontal segment is the set A from Equation (14), and the blue piecewise map is the respective approximation given by the set ${B}_{\delta}$ from Equation (15) for two values of $\delta $ and $\epsilon =0.10$.

**Figure 14.**
(**Left**) Pareto set. (**Right**) Pareto front of MOP (Equation (16)) for $n=2$ and $\gamma =2$.

**Figure 14.**
(**Left**) Pareto set. (**Right**) Pareto front of MOP (Equation (16)) for $n=2$ and $\gamma =2$.

**Figure 15.**
The same as in

Figure 14 but for

$\gamma =1/2$.

**Figure 15.**
The same as in

Figure 14 but for

$\gamma =1/2$.

**Figure 16.**
(**Left**) The blue dots A and the blue polygonal line B are the discrete and continuous approximations, respectively, for the Pareto set which corresponds to the orange thick segment, of MOP (Equation (16)) for $n=2$. (**Right**) respective sets $F\left(A\right)$ and $F\left(B\right)$ of the Pareto front for $\gamma =2$.

**Figure 16.**
(**Left**) The blue dots A and the blue polygonal line B are the discrete and continuous approximations, respectively, for the Pareto set which corresponds to the orange thick segment, of MOP (Equation (16)) for $n=2$. (**Right**) respective sets $F\left(A\right)$ and $F\left(B\right)$ of the Pareto front for $\gamma =2$.

**Figure 17.**
The same as in

Figure 16 but for

$\gamma =1/2$.

**Figure 17.**
The same as in

Figure 16 but for

$\gamma =1/2$.

**Figure 18.**
(**Left**) the blue dots A and the blue polygonal line are the discrete and continuous approximations, respectively, for the Pareto set and the orange thick segment is for the 410th generation of the NSGA-II algorithm of MOP (Equation (16)) for $n=2$. (**Right**) corresponding sets $F\left(A\right)$ and $F\left(B\right)$ of the Pareto front for $\gamma =2$.

**Figure 18.**
(**Left**) the blue dots A and the blue polygonal line are the discrete and continuous approximations, respectively, for the Pareto set and the orange thick segment is for the 410th generation of the NSGA-II algorithm of MOP (Equation (16)) for $n=2$. (**Right**) corresponding sets $F\left(A\right)$ and $F\left(B\right)$ of the Pareto front for $\gamma =2$.

**Figure 19.**
The same as in

Figure 18 but for

$\gamma =1/2$.

**Figure 19.**
The same as in

Figure 18 but for

$\gamma =1/2$.

**Figure 20.**
(

**Left**) the blue dots

A and the blue polygonal line are the discrete and continuous approximations, respectively, for the Pareto set and the orange thick segment is for the 410th generation of the MOEA/D algorithm of MOP (Equation (

17)) for

$n=2$. (

**Right**) corresponding sets

$F\left(A\right)$ and

$F\left(B\right)$ of the Pareto front for

$\gamma =2$.

**Figure 20.**
(

**Left**) the blue dots

A and the blue polygonal line are the discrete and continuous approximations, respectively, for the Pareto set and the orange thick segment is for the 410th generation of the MOEA/D algorithm of MOP (Equation (

17)) for

$n=2$. (

**Right**) corresponding sets

$F\left(A\right)$ and

$F\left(B\right)$ of the Pareto front for

$\gamma =2$.

**Figure 21.**
The same as in

Figure 20 but for

$\gamma =1/2$.

**Figure 21.**
The same as in

Figure 20 but for

$\gamma =1/2$.

**Figure 22.**
(**Left**) Pareto set. (**Right**) Pareto front of MOP (Equation (17)).

**Figure 22.**
(**Left**) Pareto set. (**Right**) Pareto front of MOP (Equation (17)).

**Figure 23.**
The black curve is the ${\Delta}_{p,q}$ value for the discrete approximation, and the blue one is the respective curve for the continuous approximation of NSGA-II for MOP (Equation (17)).

**Figure 23.**
The black curve is the ${\Delta}_{p,q}$ value for the discrete approximation, and the blue one is the respective curve for the continuous approximation of NSGA-II for MOP (Equation (17)).

**Figure 24.**
(**Left**) The blue dots and the blue polygon line are the discrete and continuous approximation, respectively, for the Pareto set of MOP (Equation (17)) in the 300th generation. (**Right**) respective sets $F\left(A\right)$ and $F\left(B\right)$ of the Pareto front.

**Figure 24.**
(**Left**) The blue dots and the blue polygon line are the discrete and continuous approximation, respectively, for the Pareto set of MOP (Equation (17)) in the 300th generation. (**Right**) respective sets $F\left(A\right)$ and $F\left(B\right)$ of the Pareto front.

**Figure 25.**
The same as in

Figure 24 but for the 400th generation.

**Figure 25.**
The same as in

Figure 24 but for the 400th generation.

**Figure 26.**
The same as in

Figure 24 but for the 500th generation.

**Figure 26.**
The same as in

Figure 24 but for the 500th generation.

**Table 1.**
${\Delta}_{p,q}({P}^{\prime},{X}_{1})$ for several values of p and q.

**Table 1.**
${\Delta}_{p,q}({P}^{\prime},{X}_{1})$ for several values of p and q.

| p | 1 | 2 | 5 | 10 | 20 |
---|

q | |
---|

$-\infty $ | $0.9091$ | $2.7153$ | $5.5714$ | $7.0811$ | $7.9831$ |

$-100$ | $0.9272$ | $2.7701$ | $5.6839$ | $7.2241$ | $8.1443$ |

$-20$ | $0.9537$ | $2.8367$ | $5.8202$ | $7.3974$ | $8.3396$ |

$-5$ | $0.9895$ | $2.8624$ | $5.8705$ | $7.4613$ | $8.4117$ |

$-1$ | $1.1131$ | $2.8782$ | $5.8848$ | $7.4795$ | $8.4322$ |

1 | $1.3243$ | $2.9112$ | $5.8920$ | $7.4886$ | $8.4425$ |

2 | $2.9277$ | $2.9295$ | $5.8956$ | $7.4932$ | $8.4476$ |

5 | $5.8920$ | $5.8956$ | $5.9063$ | $7.5068$ | $8.4630$ |

10 | $7.4886$ | $7.4932$ | $7.5068$ | $7.5292$ | $8.4882$ |

**Table 2.**
${\Delta}_{p,q}({P}^{\prime},{X}_{2})$ for several values of p and q.

**Table 2.**
${\Delta}_{p,q}({P}^{\prime},{X}_{2})$ for several values of p and q.

| p | 1 | 2 | 5 | 10 | 20 |
---|

q | |
---|

$-\infty $ | $4.5412$ | $4.5497$ | $4.5751$ | $4.6160$ | $4.6867$ |

$-100$ | $4.6442$ | $4.6529$ | $4.6790$ | $4.7209$ | $4.7933$ |

$-20$ | $4.8425$ | $4.8518$ | $4.8795$ | $4.9239$ | $5.0003$ |

$-5$ | $4.9624$ | $4.9720$ | $5.0007$ | $5.0465$ | $5.1250$ |

$-1$ | $5.0008$ | $5.0105$ | $5.0394$ | $5.0856$ | $5.1646$ |

1 | $5.0203$ | $5.0301$ | $5.0591$ | $5.1055$ | $5.1848$ |

2 | $5.0301$ | $5.0398$ | $5.0690$ | $5.1154$ | $5.1949$ |

5 | $5.0591$ | $5.0690$ | $5.0983$ | $5.1450$ | $5.2248$ |

10 | $5.1055$ | $5.1154$ | $5.1450$ | $5.1921$ | $5.2725$ |

**Table 3.**
Triangle inequality violations, in percentage, for several values of p and q: Here, we randomly chose 80 sets, each one containing 2 points in ${[0,10]}^{2}$, and verified the triangle inequality for all possible set permutations (that is, 492,960).

**Table 3.**
Triangle inequality violations, in percentage, for several values of p and q: Here, we randomly chose 80 sets, each one containing 2 points in ${[0,10]}^{2}$, and verified the triangle inequality for all possible set permutations (that is, 492,960).

| p | 1 | 2 | 5 | 10 |
---|

q | |
---|

$-1$ | $0.05396$ | 0 | 0 | 0 |

$-2$ | $0.10265$ | $0.00041$ | 0 | 0 |

$-5$ | $0.28815$ | $0.01217$ | 0 | 0 |

$-10$ | $0.35622$ | $0.05031$ | $0.00041$ | 0 |

$-20$ | $0.43046$ | $0.08439$ | $0.00446$ | $0.00041$ |

**Table 4.**
${\Delta}_{p,q}$ results between the sets A and ${B}_{\delta}$ in Equations (14) and (15) for $\epsilon =0.10$ and some parameter values of p, q, and $\delta $.

**Table 4.**
${\Delta}_{p,q}$ results between the sets A and ${B}_{\delta}$ in Equations (14) and (15) for $\epsilon =0.10$ and some parameter values of p, q, and $\delta $.

p | q | ${\Delta}_{\mathit{pq}}(\mathit{A},{\mathit{B}}_{0.05})$ | ${\Delta}_{\mathit{pq}}(\mathit{A},{\mathit{B}}_{0.10})$ | ${\Delta}_{\mathit{pq}}(\mathit{A},{\mathit{B}}_{0.20})$ | ${\Delta}_{\mathit{pq}}(\mathit{A},{\mathit{B}}_{0.40})$ |
---|

1 | 1 | $0.7149$ | $0.7464$ | $0.8091$ | $0.9324$ |

1 | 1 | $0.4105$ | $0.4506$ | $0.5311$ | $0.6945$ |

1 | 100 | $0.1503$ | $0.1961$ | $0.2878$ | $0.4711$ |

1 | 200 | $0.1479$ | $0.1934$ | $0.2844$ | $0.4663$ |

1 | $10,000$ | $0.1451$ | $0.1901$ | $0.2802$ | $0.4602$ |

**Table 5.**
${\Delta}_{p,q}$ results for the approximations of the Pareto set and front for MOP (Equation (16)).

**Table 5.**
${\Delta}_{p,q}$ results for the approximations of the Pareto set and front for MOP (Equation (16)).

| p | q | Decision Space | Objective Space |
---|

| Finite Arch. | Cont. Arch. | Finite Arch. | Cont. Arch. |
---|

$\gamma =2$ | 1 | 1 | $0.5262$ | $0.4775$ | $0.4377$ | $0.3851$ |

1 | 1 | $0.2710$ | $0.2017$ | $0.2051$ | $0.1070$ |

1 | 100 | $0.1121$ | $0.0341$ | $0.0862$ | $0.0040$ |

1 | 200 | $0.1112$ | $0.0333$ | $0.0855$ | $0.0039$ |

1 | 10,000 | $0.1103$ | $0.0324$ | $0.0848$ | $0.0038$ |

$\gamma =\frac{1}{2}$ | 1 | 1 | $0.5262$ | $0.4775$ | $0.5520$ | $0.4965$ |

1 | 1 | $0.2710$ | $0.2017$ | $0.2587$ | $0.1120$ |

1 | 100 | $0.1121$ | $0.0341$ | $0.1079$ | $0.0012$ |

1 | 200 | $0.1112$ | $0.0333$ | $0.1071$ | $0.0012$ |

1 | 10,000 | $0.1103$ | $0.0324$ | $0.1062$ | $0.0011$ |

**Table 6.**
Parameter setting for NSGA-II and MOEA/D: Here, n denotes the dimension of the decision variable space.

**Table 6.**
Parameter setting for NSGA-II and MOEA/D: Here, n denotes the dimension of the decision variable space.

Algorithm | Parameter | Value |
---|

NSGA-II | Population size | 12 |

Number of generations | 500 |

Crossover probability | 0.8 |

Mutation probability | $1/n$ |

Distribution index for crossover | 20 |

Distribution index for mutation | 20 |

MOEA/D | Population size | 12 |

# weight vectors | 12 |

Number of generations | 500 |

Crossover probability | 1 |

Mutation probability | $1/n$ |

Distribution index for crossover | 30 |

Distribution index for mutation | 20 |

Aggregation function | Tchebycheff |

Neighborhood size | 3 |

**Table 7.**
For MOP (Equation (16)), the Table shows the ${\Delta}_{p,q}$ results for the finite and continuous Pareto front approximations. We used the NSGA-II generated archives for $p=1$ and $q=-10$.

**Table 7.**
For MOP (Equation (16)), the Table shows the ${\Delta}_{p,q}$ results for the finite and continuous Pareto front approximations. We used the NSGA-II generated archives for $p=1$ and $q=-10$.

Generation | $\mathit{\gamma}=1/2$ | $\mathit{\gamma}=2$ |
---|

Finite Arch. | Cont. Arch. | Finite Arch. | Cont. Arch. |
---|

50 | $0.0439$ | $0.0147$ | $0.0696$ | $0.0160$ |

100 | $0.0498$ | $0.0109$ | $0.0540$ | $0.0102$ |

200 | $0.0613$ | $0.0118$ | $0.0716$ | $0.0207$ |

250 | $0.0651$ | $0.0265$ | $0.0572$ | $0.0061$ |

400 | $0.0602$ | $0.0102$ | $0.0723$ | $0.0276$ |

450 | $0.0630$ | $0.0154$ | $0.0584$ | $0.0088$ |

460 | $0.0612$ | $0.0154$ | $0.0658$ | $0.0098$ |

470 | $0.0523$ | $0.0102$ | $0.0566$ | $0.0083$ |

480 | $0.0754$ | $0.0269$ | $0.0684$ | $0.0241$ |

490 | $0.0510$ | $0.0091$ | $0.0584$ | $0.0118$ |

500 | $0.0722$ | $0.0097$ | $0.0560$ | $0.0103$ |

**Table 8.**
For MOP (Equation (16)), the Table shows the ${\Delta}_{p,q}$ results for the finite and continuous Pareto front approximations. We used the MOEA/D generated archives for $p=1$ and $q=-10$.

**Table 8.**
For MOP (Equation (16)), the Table shows the ${\Delta}_{p,q}$ results for the finite and continuous Pareto front approximations. We used the MOEA/D generated archives for $p=1$ and $q=-10$.

Generation | $\mathit{\gamma}=1/2$ | $\mathit{\gamma}=2$ |
---|

Finite Arch. | Cont. Arch. | Finite Arch. | Cont. Arch. |
---|

50 | $0.0610$ | $0.0171$ | $0.0648$ | $0.0119$ |

100 | $0.0519$ | $0.0051$ | $0.1093$ | $0.0016$ |

200 | $0.0536$ | $0.0037$ | $0.0781$ | $0.0009$ |

250 | $0.0522$ | $0.0037$ | $0.0790$ | $0.0008$ |

400 | $0.0511$ | $0.0017$ | $0.0784$ | $0.0009$ |

450 | $0.0511$ | $0.0017$ | $0.0784$ | $0.0009$ |

460 | $0.0509$ | $0.0012$ | $0.0784$ | $0.0009$ |

470 | $0.0509$ | $0.0012$ | $0.0784$ | $0.0009$ |

480 | $0.0509$ | $0.0010$ | $0.0783$ | $0.0009$ |

490 | $0.0509$ | $0.0010$ | $0.0783$ | $0.0009$ |

500 | $0.0509$ | $0.0010$ | $0.0783$ | $0.0009$ |

**Table 9.**
${\Delta}_{p,q}$ results between the Pareto Front and its respective discrete and continuous approximations of NSGA-II for MOP (Equation (17)): The data shown is the averaged over the 20 independent runs above.

**Table 9.**
${\Delta}_{p,q}$ results between the Pareto Front and its respective discrete and continuous approximations of NSGA-II for MOP (Equation (17)): The data shown is the averaged over the 20 independent runs above.

Generation | Continuous Archive | Finite Archive |
---|

20 | $0.1333$ | $0.2401$ |

40 | $0.0176$ | $0.1451$ |

60 | $0.0090$ | $0.1561$ |

80 | $0.0088$ | $0.1355$ |

100 | $0.0065$ | $0.1472$ |

120 | $0.0074$ | $0.1412$ |

140 | $0.0081$ | $0.1395$ |

160 | $0.0075$ | $0.1549$ |

180 | $0.0092$ | $0.1468$ |

200 | $0.0074$ | $0.1429$ |

220 | $0.0066$ | $0.1408$ |

240 | $0.0075$ | $0.1397$ |

260 | $0.0066$ | $0.1460$ |

280 | $0.0074$ | $0.1439$ |

300 | $0.0084$ | $0.1421$ |

320 | $0.0070$ | $0.1352$ |

340 | $0.0070$ | $0.1373$ |

360 | $0.0081$ | $0.1454$ |

380 | $0.0079$ | $0.1413$ |

400 | $0.0066$ | $0.1388$ |

420 | $0.0063$ | $0.1400$ |

440 | $0.0097$ | $0.1384$ |

460 | $0.0067$ | $0.1418$ |

480 | $0.0067$ | $0.1421$ |

500 | $0.0076$ | $0.1426$ |