# MOEA/D

## Main idea

approximation of the PF can 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?

1. when generate initial population
2. when initialize referrence points
3. after generating y’ using problem-specific repair/improvement heuristic

## plan description of MOEA/D

• every time we optimize N scalar subproblem
• the ith subproblems was assgined with a $x^i$ and $\lambda^i$
• 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
• #### B. Discussions

• Why a Finite Number of Subproblems are considered?
• How diversity is Maintained?
• Matiing restriction and role of T
• #### C. Variants of MOEA/D

• step 2.2 is not a must in MOEA/D
• EP is an option

• NP-hard

## 5. COMPARISON WITH NSGA-II

• objective normalization