Minimize the number of unique elements in a vector

Question

I was wondering if there is a simple or known way to minimize the number of unique elements in a decision variable (vector). Note that I'm not asking for minimization of nonzero elements (rank constraint). In particular I'm searching for a penalization (soft constraint) or hard constraint in the form $$f(x) <= n_{max} $$ where f() is what I'm searching for, x is the decision vector and n_max is the maximum number of unique elements that x is allowed to have.

EDIT: I am not trying to minimize the number of unique elements in a vector " per se", which of course is trivial. The question is related to minimizing the unique element in a vector of decision variables that is the solution to another optimization problem. Eg: $$argmin_{x} g(x) $$ s.t. $$x \in \mathbb{X} $$ $$\sum(unique(x))<n_{max }$$ Thank you for your time, Lorenzo

Answer 1

Inspired by the well-known approach of employing a regularization (penalty) term $\gamma ||\mathbf{x}||_1$ as a means of relaxing the requirement of minimum non-zero elements of vector $\mathbf{x}$ (e.g., see here), you could try using a regularization of the from $$ \gamma ||\mathbf{x} - c\mathbf{1}||_1, $$ where $\mathbf{1}$ is the all-ones vector and $c\in \mathbb{R}$ (assuming the domain is the set of real-valued vectors). You would then solve the resulting problem with respect to both $\mathbf{x}$ and $c$.