¿How to visualize inequalities with plot?

Scilab beginners
1

I have the following inequalities:

\begin{align*} x+y&\leq 3\\ 5x+4y&\geq 20\\ x&\geq 0\\ y&\geq 0 \end{align*}

How can I visualize them and its feasible region?

2

Hello, welcome to Scilab’s Discourse !

First, are you sure of the inequalities ? Because they are describing a non feasible region.

S.

3

You’re absolutely right. I’ll get other ones:

\begin{align*} 2x-y&\leq 6\\ 2x+y&\leq 10\\ x&\geq 0\\ y&\geq 0 \end{align*}
4

A simple algorithm for a domain in \mathbb{R}^2 can be the following: enumerate each pair of saturated constraints, and if the corresponding solution (x,y) verifies other inequalities, keep it as a vertex. Then your admissible domain is the convex hull of these vertices. In Scilab, I would code the algorithm like this:

// domain is described in standard form
// by A*x <= b
A = [2 -1; 2 1; -1 0; 0 -1];
b = [6; 10; 0; 0];

x = [];
m = size(A,1);
for i = 1:m-1
    for j = i+1:m
        if rank(A([i j],:)) == 2
            u =  A([i j],:)\b([i j]);
            // check that vertex verifies other inequalities
            k = [1:i-1 i+1:j-1 j+1:m];
            if and(A(k,:)*u - b(k) < %eps)
                x = [x u];
            end
        end
    end
end

if ~isempty(x)
    // compute convex hull of vertices
    [_, hull] = mesh2d(x(1,:),x(2,:));
    clf
    h = plot(x(1,hull),x(2,hull),'-o');
    h.fill_mode = 1;
    h.background=color("gray");
end

q

5

Ingenious, but I am not sure, if I understood…
Heinz

6

Let us hope that @mtlb will !

By the way, playing with the above code, it is quite easy to illustrate the fact that in \mathbb{R}^2 a non-linear constraint such as

\Vert x \Vert^2 \leq 1

can be approximated (arbitrarily close) with a sequence of linear constraints under the form Ax\leq b. To illustrate this, replacing the first two lines of code above with

t = linspace(0,2*%pi,32)(1:$-1)';
A = [cos(t) sin(t)];
b = ones(t);

and re-running the script yields the following graph:
q