Same values constraint and grouping of variables

Question

In a linear program, I would like some variables to: 1. Take the same values 2. Group some variables i.e. some variables should take same values or lie within certain percentage. 3. All different values

The objective function is given by:

Answer 1

(1) This is correct, and there's nothing wrong with having a whole lot of constraints that each require $x_6 = x_j$ for some $j$. But if you know in advance that these variables will all equal each other, why not just define a new variable that equals all of them? That is, create a variable $x_{6-10}$ that equals $x_6$ through $x_{10}$ and use this variable everywhere any of the $x_6$ through $x_{10}$ variables appear?

(2) Using the logic described here, you can create a binary variable $y_6$ that equals 1 if $x_6 \ge 0$, and another binary variable $y_8$ that equals 1 if $x_8 \ge 0$. Then you can enforce the "if-then" implications using the logic described here.

(3) Some solvers have a built-in feature that allows you to specify that certain variables must be different from each other; see Matrix in ampl: constraint that the values ​are all different. But I think this only works for integer variables. For continuous variables, I think you need to use big-Ms for this, although maybe others will chime in with better ideas.