I am trying to solve the following convex optimization problem where $a_i, b_i >0$ for $i=1,2,3$. I am wondering if it is possible to get a general formula for the optimal solution without checking the $2^3$ possible values and whether there is a faster way to solve it using the special structure of the objective function.
Thanks in advance!
$$\min \quad f(x)= a_1\exp(-b_1x_1) + a_2\exp(-b_2x_2) + a_3\exp(-b_3x_3)$$ $$\text {s.t.} \quad x_1 + x_2 + x_3 = 1$$ $$x_i \geq 0$$
KKT: $$-a_ib_i\exp(-b_ix_i) + \lambda - \mu_i = 0$$ $$x_1 + x_2 + x_3 = 1$$ $$x_i \geq 0$$ $$\mu_i \geq 0$$ $$x_i\mu_i = 0$$