I'm not sure whether I'm asking very obvious/stupid question, but essentially I'm looking for references.
I am looking for the notion of independence in the context of optimization problems (I am doing LP with disjunctions over reals, but that should be irrelevant, same question applies just to LP). I know that the standard calculus course on optimization starts with "dependent" and "independent" variables, but I think that this not what I am after.
Consider a feasible maximization context $C$ (set of constraints) over a set of variables $V$ in the domain $\mathbb{R}$. Consider bounded variables $x, y$. I call $x$ and $y$ independent iff for any linear $f: \mathbb{R^2} \to \mathbb{R}$ $f(\max_C{x}, \max_C{y}) = \max_C{f(x, y)}$.
For example for a set of constraints $C: x \leq 10 \land y \leq 10$ variables $x, y$ are independent (for example $\max_C{x} + \max_C{y} = \max_C{x + y}$. However changing $C$ to $x \leq 10 \land y \leq 10 \land x + y \leq 10$ breaks this property, as $x$ and $y$ get to compete for who saturates the last constraint.
Hence the question is: what is the formal definition for the property which I am trying to capture? What would be the good references? Specifically, I am looking for algorithms to find pairs of independent variables in a given context or to at least classify a given pair of variables.
EDIT: Even more specifically, for a given pair $x, y$ and a set of constraints $C$ I would like to know whether $\max_C{x+y} = \max_C{x} + \max_C{y}$ holds.
EDIT2: Updated the notation to avoid the confusion.
