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} $

  • $\begingroup$ Is there supposed to also be a $-\exp\left(-{(t_3+t_1)^2\over2}\right)$ term? $\endgroup$ – saulspatz Mar 29 at 15:55
  • $\begingroup$ No, actually not. $\endgroup$ – Alex Gao Mar 29 at 16:15
  • $\begingroup$ 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? $\endgroup$ – saulspatz Mar 29 at 16:19
  • $\begingroup$ @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 $t_3$. $\endgroup$ – callculus Mar 29 at 19:31

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