MOEA/DMOEA/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 $max limits{1leq jleq m} {lambda_j| f_j(x) - z_j^|} = lambda_i|f_i(x)-z_i^|$ 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 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