I'm trying to find a reasonable way to tackle this awful optimization problem, any suggestions?
I'm optimizing an equation of $3$ variables using Lagrange multipliers, and have taken all partials. Here is the resulting system:
$\frac{\partial\mathcal{L}}{\partial K_1}=-\lambda + \frac{K_2\big(\frac{(1-K_1-K_2)^{K_1+K_2-1}}{K_1^{K_1}K_2^{K_2}}\big)^h}{4\pi |h|(K_1K_2-K_1^2K_2-K_1K_2^2)^{3/2}} \cdot\frac{((2hK_1^2+(2hK_2-2h)K_1)\text{ln}(x)+(2h-2hK_2)\text{ln}(1-K_1-K_2)-2h\text{ln}(1-K_1-K_2)K_1^2+((2h-2hK_2)\text{ln}(1-K_1-K_2)+2)}{4\pi |h|(K_1K_2-K_1^2K_2-K_1K_2^2)^{3/2}}$
$\frac{\partial\mathcal{L}}{\partial K_2}=-2\lambda + \frac{K_1\big(\frac{(1-K_2-K_1)^{K_2+K_1-1}}{K_2^{K_2}K_1^{K_1}}\big)^h}{4\pi |h|(K_2K_1-K_2^2K_1-K_2K_1^2)^{3/2}} \cdot\frac{((2hK_2^2+(2hK_1-2h)K_2)\text{ln}(x)+(2h-2hK_1)\text{ln}(1-K_2-K_1)-2h\text{ln}(1-K_2-K_1)K_1^2+((2h-2hK_1)\text{ln}(1-K_2-K_1)+2)}{4\pi |h|(K_2K_1-K_2^2K_1-K_2K_1^2)^{3/2}}$
$\frac{\partial\mathcal{L}}{\partial h}=-\frac{\text{ln}(K_1^{K_1}(1-K_1-K_2)^{1-K_1-K_2}K_2^{K_2})h+1}{2\pi \sqrt{-K_1K_2^2-K_1^2K_2+K_1K_2}(K_1^{K_1}(1-K_1-K_2)^{1-K_1-K_2}K_2^{K_2})^hh|h|}$
$\frac{\partial\mathcal{L}}{\partial \lambda}= K_1 + 2K_2 - 1$
And of course, I need to solve for when $\nabla \mathcal{L}=0$.