I'd like to run various optimization techniques on this variation of the stable marriage problem I formulated. Ideally, I'd be able to convert the problem I constructed into one that is more "calculus-able," so to speak, and then apply various optimizations and compare their run-times/outputted solutions. In short, my question is: does my idea/approach to this optimization problem even make sense?
Outline of Problem
The objective to be minimized is $f_0(S,R)$, which captures the information regarding overall "discontent" given a set of suite assignments.
Given a list of $n$ ranked living arrangement preferences, $X$, discontent for a given student $i$ is defined as $f_{0_i}(X,r) = $ ranking of format of $r$ in $X$. The objective function is simply the sum of all students' discontent possible:
$$f_0(S,R)=\sum_{r=0}^{s-1}\sum_{i=0}^{t-1}f_{0_i}(X,r)$$
where $R$ is a valid mapping of all suites to students, $S$ is the set of all suites, and $s$ and $t$ are the number of suites and students, respectively. The suite assignments of $r$ to student $i$ are given by $R$.
My Idea
The rankings of certain suite formats for a group of students may be represented as a matrix where entries are actual rankings ($0$ to $ n-1$) of suite formats indexed by a student and the corresponding suite number (and corresponding suite format). Such a matrix contains all of the information captured by the objective, $f_0(S,R)$, as discontent is solely determined by the format of the suite that is ultimately assigned to the student in my formulation of the problem.
One approach to applying optimization techniques to this function is to minimize the rank of the objective matrix. The objective, $f_0(S,R)$, is complicated, so the reduction of the matrix to a simpler model allows us to more easily apply the techniques learned in class to approximate the optimal suite assignment. I'm essentially trying to convert this problem into a Rank Minimization Problem (RMP). Is this a valid application of the RMP?
Definition of terms summarized here:
$f_0(S,R)=\sum_{r=0}^{s-1}\sum_{i=0}^{t-1}f_{0_i}(X,r) = $ objective function i.e. function of all possible total discontent.
$ f_{0_i}(X,r)= $ amount of discontent of student $ i $ given their preferences and assignment.
$ S = $ the set of all suites in a residential college i.e. constraint under which problem is subjected to.
$ R = $ any given mapping of the suites in $S$ to all $t$ students (properly satisfying suite formats).
$ s = |S|. $
$t = $ number of students.
$ X = $ a ranked list of a student's $ n $ most preferred suite formats.
$ r = $ a pre-specified suite assigned in $R.$ Each suite has a corresponding suite format (2-person, 4-person, 9-person, etc...).
"Discontent" = ranking of suite format of assigned suite $r$ in a student's list $ X $ (most preferred to least preferred = $ 0 $ to $n-1$).