Consider a multiobjective optimization problem $$\min\limits_{x\in \Omega} f(x),$$ where $f:\Bbb R^n \rightarrow \Bbb R^m$ and $\Omega \subseteq \Bbb R^m.$ A point $\bar{x} \in \Omega$ is said to be:
- a weakly minimal solution, if there is no $x \in \Omega$ such that $f(x) < f(\bar{x})$ (the inequality is understood componentwise),
- minimal solution, if there is no $x \in \Omega$ such that $f(x) \leq f(\bar{x})$ (again componentwise) and $f(x) \neq f(\bar{x}).$
Intuitively, when the value of $m$ is very large and the component functions are conflicting, the solution set of the problem will be a large subset of $\Omega.$ Of course this is not always the case, but perhaps it is the most natural case. I am looking for references that formalize this intuition and give sufficient conditions on $f$ and $\Omega$ to ensure that $m$ being large would make the solution set 'almost' $\Omega.$ In particular, I would be interested in the results of the type:
For a large class of functions $f$, different assumptions, and $m$ large enough, the solution set of the multiobjective problem is (or tends to be):
- $\Omega,$
- A dense subset of $\Omega,$
- A set $S$ such that $\mu(\Omega\setminus S)=0,$ where $\mu$ denotes the Lebesgue measure.
Any reference in the literature dealing with this problem??