# how to interpret local minima of combinatorial optimization

I am having a difficult time trying to interpret and visualize the local minima of a combinatorial optimization objective function.

Here's a rough sketch of my problem:

I have $m$ points $x_1,\dots,x_m$ in $\mathbb{R}^d$. My objective function is only a function of any $n$ of these points where $n < m$ and $n$ is fixed. The goal is to find the set of poins $x_{i_1},\dots,x_{i_n}$ which minimizes the objective function. We may denote the solution point to the optimization problem by $(x_{i_1},\dots,x_{i_n}).$ The order of these $n$ points is not important and so there are many local minima, assuming it exists.

Clearly, this is a combinatorial optimization problem with $m$ choose $n$ possible combinations for the solution. Thus the set of possible solution points is finite.

What I'm finding hard to grasp is the meaning of a local minima in this context. Suppose that $x^* =(x_{i_1},\dots,x_{i_n})$ is a "local minima". If we are to use the classical definition of local minima, how do we define the neighborhood of $x^*$? Can we ensure that such a neighborhood belongs to the $m$ points where we started with?

Assume that $n = 1$ and $d=2.$How do you plot the objective function if you have discrete inputs?

any insight appreciated!