Constraint
2(l + w) = L = 1
Residual r(l,w) = 2(l+w) − 1
A = l · w
Residual r = 2(l+w) − 1
0.0000
Perimeter P = 2(l+w)
1.0000
Feasible optimum (on r=0)
A*=0.062500 at l=w=0.25
Right-hand plot is in (r,A) coordinates.
The dot is your current operating point (r(l,w), A(l,w)).
The dashed line is feasibility: r=0.
Left: rectangle. Right: (residual r, area A) with current point.
(c) Fayyaz Minhas
Rectangle view (aspect ratio l:w)
Text includes A and r.
Feasible means r=0.
Clean plot in (r,A): constraint violation vs area
x-axis: r = 2(l+w) − 1 (constraint violation). y-axis: A = l·w.
On r=0, the best achievable area is A=0.0625 at l=w=0.25.
If you allow r>0, you can get larger areas, but you are exceeding L.