# On Generalized Dominance Structures for Multi-Objective Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Optimality Principles for Single-Objective Optimization

**Definition**

**1**

**Definition**

**2**

**Irreflexive property:**A solution $\mathbf{x}$ is not inferior to itself, that is, $\mathbf{x}\nprec \mathbf{x}$.**Asymmetric property:**If $\mathbf{x}\prec \mathbf{y}$, then $\mathbf{y}\nprec \mathbf{x}$.**Transitive property:**If $\mathbf{x}\prec \mathbf{y}$ and $\mathbf{y}\prec \mathbf{z}$, then $\mathbf{x}\prec \mathbf{z}$.

**Definition**

**3**

**Definition**

**4**

**Theorem**

**1**

**Theorem**

**2**

**Definition**

**5**

**Semi-transitive Property:**If $\mathbf{x}\prec \mathbf{y}$ and $\mathbf{y}\prec \mathbf{z}$, then $\mathbf{z}\nprec \mathbf{x}$.

**Theorem**

**3**

**Proof.**

#### 2.1. Epsilon-Inferiority Conditions

## 3. Optimality Principles for Multi-Objective Optimization

**Definition**

**6**

**Definition**

**7**

**Definition**

**8**

**Theorem**

**4**

#### 3.1. Defining a Generalized Dominance Structure for Multi-Objective Optimization

#### 3.2. Relationship between Dominance and Anti-Dominance Structures

**Theorem**

**5**

**Proof.**

**Corollary**

**1.**

**Irreflexive property:**${\Omega}_{0}$ (and its boundary ${B}_{\Omega}$) must exclude the $\mathbf{0}$-vector (origin) from its set.**Asymmetric property:**${\Omega}_{0}\cap {{\Omega}^{\prime}}_{0}=\varnothing $ (recommended, as discussed in the paragraph before Theorem 2).**Transitive property:**This requires a chain of ${\Omega}_{0}$ consideration and requires further discussion (provided in Section 3.2.1).

**Corollary**

**2.**

**Corollary**

**3.**

#### 3.2.1. Transitive and Semi-Transitive Properties

**Theorem**

**6**

**Proof.**

#### 3.2.2. Further Illustration of Semi-Transitive Property

#### 3.3. Commonly Used Dominance Structures

#### 3.4. Identifying Generalized Non-Dominated Set

**Definition**

**9**

**Theorem**

**7**

**Proof.**

#### 3.5. Identifying Generalized Optimal Solution Set

**Theorem**

**8**

**Proof.**

- First, it can be used to test if an objective vector $\mathbf{f}$ is a potential optimal point, as discussed above, but instead of restricting the check in a finite sampled set P, every feasible point from the search space must be considered. Although it is a computationally challenging task, the concept can be used theoretically or in a geometry-based checking procedure.
- Second, ${{\Omega}^{\prime}}_{0}$ can be used to identify the entire optimal solution set for a given feasible structure $\mathbf{Z}$. This task will be useful for studies involving test problems and requires an identification of the exact optimal objective set (${\mathbf{Z}}^{*}$) from $\mathbf{Z}$ for a given dominance structure. The theoretical procedure is to identify ${{\Omega}^{\prime}}_{0}$ set for every point in $\mathbf{Z}$ systematically and by repeating the test only to ${{\Omega}^{\prime}}_{0}$ members in a nested manner. This will allow a faster computational procedure to identify the optimal solution set.
- Third, knowing one or more optimal points, ${\Omega}_{0}$ can help identify further optimal points quickly by eliminating the dominated solutions from its ${\Omega}_{0}$ set and narrowing down the search to find further optimal points. However, in such a task, often the relevant boundary points of $\mathbf{Z}$ can be tested for their optimality. Starting with extreme boundary points of $\mathbf{Z}$, ${{\Omega}^{\prime}}_{0}$ can immediately verify if the point is a member of the optimal solution set. If yes, the test can continue to the neighboring extreme boundary point, and so on. If no, the test will identify the points in the ${{\Omega}^{\prime}}_{0}$ set that dominate the extreme point and a new test can be executed on members of the ${{\Omega}^{\prime}}_{0}$ set.

**Theorem**

**9**

**Proof.**

**Corollary**

**4**

**Theorem**

**10.**

**Proof.**

**Corollary**

**5.**

**Corollary**

**6.**

#### 3.6. Theoretical and Practical Optimal Sets

**Definition**

**10**

#### 3.6.1. Search Inefficiency

#### 3.6.2. Compatibility of Dominance Structure with Discreteness in Search Space

#### 3.6.3. Implementation of Adjusted Dominance Principle

#### 3.6.4. Implementation of Grid Dominance Principle

## 4. Other Existing Dominance Structures

#### 4.1. $\alpha $-Dominance Structure

#### 4.2. Cone-Epsilon Dominance Structure

#### 4.3. CN and CN-Alpha Dominance Structures

#### 4.4. CN-$\alpha $ Domination Structure

#### 4.5. Nonlinear Dominance Structure (NLAD)

#### 4.6. D-Dominance Structure

**Definition**

**11**

#### 4.7. Strength Dominance Relationship (SDR) Structure

**Definition**

**12**

#### 4.8. $(1-k)$-Dominance Structure

**Definition**

**13.**

#### 4.9. L-Dominance Structure

**Definition**

**14**

#### 4.10. $(M-1)$-Generalized Pareto-Dominance Structure

## 5. Spatially Dependent ${\Omega}_{\mathbf{0}}$ Structure

#### 5.1. ASF-Based Dominance Structure

#### 5.2. $\theta $-Dominance or PBI-Dominance Structure

#### 5.3. Angle-Dominance Structure

**Definition**

**15**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Relationship between $\Omega \left(\mathbf{x}\right)$ and ${\Omega}_{0}$. Here, ${\Omega}_{0}$ and ${\Omega}_{0}^{\prime}$ are non-overlapping.

**Figure 7.**Repeated application of semi-transitive ${\Omega}_{0}$ creates a transitive dominance condition ${\Omega}_{0}^{\mathrm{cone}}$.

**Figure 8.**A user-specified circle dominance ${\Omega}_{0}^{\mathrm{circle}}$ results in weighted-sum dominance as an effective transitive dominance structure.

**Figure 9.**Commonly used dominance structures. In each case, ${\Omega}_{0}\cap {{\Omega}^{\prime}}_{0}=\varnothing $. For Figure 9b,d, the boundary is exclusive, whereas in the other two the boundary is inclusive.

**Figure 10.**The set ${\Omega}_{0}$ needs to be applied many times to identify a GND point, but ${{\Omega}_{0}}^{\prime}$ needs to be applied once at point O to identify if it is on the GND set.

**Figure 11.**Identification process of ${\mathbf{Z}}_{\mathrm{eff}}^{\mathrm{cone}}$ using a generalized dominance structure ${\Omega}_{0}^{\mathrm{cone}}$.

**Figure 13.**${\Omega}_{0}^{\mathrm{eps}}$ does not produce any theoretical optimal point due to the overlap between ${\Omega}_{0}$ and ${{\Omega}^{\prime}}_{0}$, although in practice it may produce an artificial optimized solution set.

**Figure 15.**Grids are fixed in objective space. ${\Omega}_{0}^{\mathrm{grid}}$ can produce a well-distributed set of optimal points.

**Figure 26.**Angle-dominance structure can be converted to angle-space, on which Pareto-dominance can be applied.

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**MDPI and ACS Style**

Deb, K.; Ehrgott, M.
On Generalized Dominance Structures for Multi-Objective Optimization. *Math. Comput. Appl.* **2023**, *28*, 100.
https://doi.org/10.3390/mca28050100

**AMA Style**

Deb K, Ehrgott M.
On Generalized Dominance Structures for Multi-Objective Optimization. *Mathematical and Computational Applications*. 2023; 28(5):100.
https://doi.org/10.3390/mca28050100

**Chicago/Turabian Style**

Deb, Kalyanmoy, and Matthias Ehrgott.
2023. "On Generalized Dominance Structures for Multi-Objective Optimization" *Mathematical and Computational Applications* 28, no. 5: 100.
https://doi.org/10.3390/mca28050100