I am having trouble minimizing a three variable function, that's because one of the constraining functions is an equality and the other is an inequality. Is it possible to minimize with such constrains?
Here is my function: $f(x, y, z)= 2\pi xy+ \pi x^2+\pi(x+1.5) \sqrt{(x-1.5)^2+z^2}$.
My constraining functions are:
\begin{align*} y\pi(x^2)+(\pi/3)z(x^2+1.5x + 2.25) &= 2000\\ \pi x &< 20\\ x, y, z &\geq 0. \end{align*}
I tried using the Karush Kuhn Tucker condition and the Lagrange multipliers, but I am having trouble finding a solution.
After substituting the equality constraint into the objective function, the problem can be stated as
$$ \min_{x,y}\pi \left(x^2+2 x y+(x+1.5) \sqrt{\frac{0.911891 \left(\pi x^2 y-2000.\right)^2}{(x (x+1.5)+2.25)^2}+(x-1.5)^2}\right) $$
with the constraints
$$ \cases{ \frac{3 \left(2000-\pi x^2 y\right)}{\pi \left(x^2+\frac{3 x}{2}+\frac{9}{4}\right)}\ge 0\\ 0\le x\le \frac{20}{\pi}\\ y \ge 0 } $$
or better
$$ \cases{ x\le \sqrt{\frac{2000}{\pi y}}\\ 0\le x\le \frac{20}{\pi}\\ y \ge 0 } $$
Making a level curves plot (black) with the feasible region superimposed (blue)
we can observe the tangency of a level curve with the feasible region boundary so we follow substituting $x = \frac{20}{\pi}$ and then we minimize for $y$
$$ \min_{y}\pi \left(\frac{40 y}{\pi }+7.8662 \sqrt{0.000333026 \left(\frac{400 y}{\pi }-2000.\right)^2+23.6799}+\frac{400}{\pi ^2}\right) $$
obtaining $y^* = 13.6744$ etc.