# Thoughts on how to solve this optimization problem using KKT?

I came across this optimization problem but I did not learn optimization theory previously. Could anyone provide some thoughts on how to solve them?

\begin{align} \text{minimize} & \quad e^{-\frac{{t_1}^2}{2}} + e^{-\frac{{t_2}^2}{2}} + e^{-\frac{{t_3}^2}{2}} - e^{-\frac{(t_1+t_2)^2}{2}} - e^{-\frac{(t_2+t_3)^2}{2}} + e^{-\frac{(t_1+t_2 + t_3)^2}{2}}\\ \text{subject to} & \quad t_1 + t_2 + t_3 \leq K, \\ & \quad t_1, t_2, t_3 \geq 0 \end{align}

• Is there supposed to also be a $-\exp\left(-{(t_3+t_1)^2\over2}\right)$ term? – saulspatz Mar 29 at 15:55
• No, actually not. – Alex Gao Mar 29 at 16:15
• It's a smooth function on a compact set. The minimum occurs at a critical point in the interior, or at some point on the boundary. Have you trie to find the critical points? – saulspatz Mar 29 at 16:19
• @AlexGao Here is an image of the KKT-conditions. It can be seen that you have to be able to calculate the partial derivatives w.r.t. $t_1,t_2$ and $t_3$. – callculus Mar 29 at 19:31