I am a computer scientist, struggling with writing a mathematical definition for a subset of a population which contains individuals that are the best scoring in the population for a set of functions. I have a population, $P$. Each individual in the population, $p \in P$, is assessed over a number of objectives, given by the functions $a_1(p),\dotsc,a_n(p)$. I wish to define a subset of $P$, which is the set of individuals in $P$ that return the maximal values for each assessment.
Currently, I have the following definition:
Let the set $P$ denote the current population, with an individual solution defined as $p \in P$. Next, for a given domain with $n$ objectives, defined by the functions $a_1(p),\dotsc,a_n(p)$, where $a: P \rightarrow \mathbb{R}$, let the set $A = \{a_1(p),a_2(p),\dotsc, a_{n}(p)\}$. Let the subset $P_{max} \subset P$ contain the maximal solutions for each assessment, where $P_{max} : \underset{a \in A, p\in P}{max} a(p)$.
I think that (if it makes any sense at all) in the way I have written it, $\underset{a \in A, p\in P}{max} a(p)$ refers to set of maximal values for each assessment, not the individuals in $P$ which return the maximal value for each assessment. I'm not sure how to express the subset correctly
The terminology Arg max is used to denote the set of points that maximize a real-valued function.
Aside: An Arg max is well defined for each of the objectives $a_1,\ldots, a_n$, but wouldn't the maximizing arguments (individuals) differ for each of the $n$ objectives? It's unclear how you would define a subset of individuals who maximize the ensemble of objectives, unless you find a way to combine the assessments over all $n$ objectives into a global score of some sort, and find the individuals who maximize that global score.