Feasibility region of LP Consider the problem
$$
\text{ Find } y \text{ s.t. } \\
\exists \text{ } x \text{ solving} \text{ }Ax\leq b, Dx=e, \text{ and } y=cx
$$
where $y$ is a $2\times 1$ vector of unknowns, $x$ is a $10\times 1$ vector of unknowns, $A,D,b,c,e$ are matrices and vectors of known values with appropriate dimensions.  

My objective is to find (plot)  the region of values of $y$ such that 
$$
\exists \text{ } x \text{ solving} \text{ }Ax\leq b, Dx=e, \text{ and } y=cx
$$
Let us call this region $\mathcal{R}$. I would like your help to investigate whether there is a way (with relative algorithm in any language) to reach my objective. 
Let me also add that the region $\mathcal{P}\equiv \{x: Ax\leq b, Dx=e\}$ is bounded.

A naive way  to go is to:
(1) rewrite the problem as a unique linear programming problem
$$
Ax\leq b\\
Dx=e\\
y=cx,
$$
(2) find the feasibility region, 
and 
(3) report only the feasible values of $y$. 
The problem is that there is no algorithm that allows me to find (plot) the feasibility region of a $12$-dimensional linear programming. The available algorithms (for example, in Matlab) allows me to plot feasibility regions of 3 dimensions at most. 
Hence, the question is: is there a simpler way to rewrite my problem in order to reach my objective?
 A: As you have noted, this (implicit) feasibility region on $y$ can be formulated as the projection (via $y=cx$) of an explicit feasibility region in $x$. I had to deal with this problem myself recently. One method, as people have commented, is to obtain the vertices from your half-space representation and then project these down; the convex hull of these projected points is the desired feasibility region. For your problem’s size, that’s probably the best way. I suggest using either SageMath’s polyhedral commands or the program ’polymake’ (both accessible online via the CoCalc cloud computation site).
In my case, this route was ultimately impractical for dimensionality reasons. A nice alternative for me, since my projection was to a low-dimensional space, was the convex hull algorithm of Lassez & Lassez (1992): https://www2.cs.duke.edu/donaldlab/Books/SymbolicNumericalComputation/103-119.pdf. For some notes on both vertex enumeration and the convex hull method, see J. Apte’s slides at https://faculty.coe.drexel.edu/jwalsh/JayantCHM.pdf.
