I'm very new to linear programming, so please bear with me:
I have a problem where I want to maximize the amount of money I can return for $d$ products to a group of members that are split into 3 groups arbitrarily per product. The amount I can give per product has an upper bound, $b \in \mathbb{R}^d$, and each grouping has an amount of money I'm wanting to return a percentage on. The formulation I have so far looks like this:
Maximize: $$c^Tx$$ Subject to: $$ c_1x_1 + c_2x_2 + c_3x_3 \le b_1 \\ \vdots \\ c_{3d-2}x_{3d-2} + c_{3d-1}x_{3d-1} + c_{3d}x_{3d} \le b_d \\ $$
There are additional bounds on the $x$'s to define a range of values that they can take. Is there a way to constrain only some of the $x$'s such that: $$ x_1 \le x_2 \le x_3 \\ \vdots \\ x_{3d-2} \le x_{3d-1} \le x_{3d} \\ $$ so that there's some kind of partial ordering due to the products? How would I go about formalizing this?
Your constraint already look formal. You can rewrite them to $x_1−x_2\leq 0$, $x_2-x_3\leq 0$, etc. Or, more formally: $$x_i - x_j \leq 0 \quad \forall (i,j)\in S$$ where $S$ is the set of pairs $(i,j)$ such that $x_i \leq x_j$.