# MOEA/D

**Main idea**

approximation of the

PFcan be decomposed into a number of scalar objective optimization subproblems.

- Why This decompostion works? What’s the mechanism?

- most important, because all of the algorithm is based on this theorm

**Question**

For each Pareto optimal point $x^

$, there exists a weight vector $\lambda$ such that $x^$ is the optimal solution of $g^{te}(x|\lambda,z^)$ and each optimal solution of $g^{te}(x|\lambda,z^)$ is a Pareto optimal solution.

- is there a unique solution $x^
$ of $g^{te}(x|\lambda,z^)$ for a given $\lambda$ and $z$? making $x^$ = $Arg{g^{te}(x|\lambda,z^)}$proof

$x^$ is a pareto optimal point, $\forall x \in \Omega ,\exists$ at least one $i$, $s.t$ $f_i(x^) > fi(x)$ we choose such $\lambda$ that $\max \limits{1\leq j\leq m} {\lambda_j| f_j(x) - z_j^|} = \lambda_i|f_i(x)-z_i^|$ since $|f_i(x^)-z_i^| < |f_i(x)-z_i^|$ $min{g^{te}(x^|\lambda,z^)} = \lambda_i|f_i(x^)-z_i^|$. thus $x^$ is the optimal solution of $g^{te}(x|\lambda,z^*)$.

- but this leading to another question

for different pair of $(x^, x)$ the $i$ may be different, which means different $\lambda$. It seems that no unique $\lambda$ can hold for all condition for a given pareto optimal point $x^$

**Major motivation**

any information about these $g^{te}(x|\lambda,z^*)$ with weight vectors close to $\lambda^i$ should be helpful for optimizing $g^{te}(x|\lambda,z^*)$.

## which parts in MOEA/D need **problem-specific** method?

- when generate initial population
- when initialize referrence points
- after generating
**y’**using problem-specific repair/improvement heuristic**plan description**of MOEA/D

- every time we optimize N scalar subproblem
- the
subproblems was assgined with a $x^i$ and $\lambda^i$*ith* **update among one neighbour**- for every subporblem, consider it’s neighbour, generate a new child y from it and update all the solutions to subproblems belongs to this neighbour.
- update
**EP**

## 1. INTRODUCTION

- dominate
- decision(variable) Pspace
- attainable objective set
- Pareto optimal(objective) vector
- Pareto optimal points
- Pareto set(PS)
- Pareto front

## 2.DECOMPOSITION OF MULTIOBJECTIVE OPTIMIZATION

- A.
**Weighted Sum**Approach [1] - B.
**Tchebycheff**Approach [1] - C.
**Boundary Intersection(BI)**Approach

## 3.THE FRAMEWORK OF MOEA/D

#### A. General Framework

- neighborhood of the
*i*th subproblem

- neighborhood of the
#### B. Discussions

- Why a
**Finite Number of Subproblems**are considered? - How
**diversity**is Maintained? **Matiing restriction**and role of T

- Why a
#### C. Variants of MOEA/D

#### C. MOKP

- NP-hard

#### D. Implementations

## 5. COMPARISON WITH NSGA-II

- objective normalization

## Futher reading

- approximate the PF by using a mathematical model [5]-[8]
- cMOGA [40] Hisao