Let $0\leq L\leq 2nL<U$ and consider the convex program: $$ \begin{aligned} \min &\sum_{i=1}^n a_i^{-\frac1{2}} C_i\\ s.t. & \sum_{i=1}^n a_i \leq U\\ & a_i \leq U\\ & L\leq a_i, \end{aligned} $$ where $C_n>0$ are some fixed constants...
I know how to solve it using Lagrangian duality and Slater's condition if the last contraint is ommmited but how do I approach/solve this form of the problem?
I know that Slater's condition holds so its some sort of duality method but I can't resolve it..
So far I'm trying to setup the Lagrange dual... Clearly $\sum_{i=1}^n a_n \leq U$ implies each $a_n\leq U$ so the latter constrain is superfluous.. Therefore, the Lagrangian should be of the form: $$ L(a_n,\lambda,\mu_n)=\sum_{i=1}^n a_i^{-\frac1{2}}C_n + \lambda \left(\sum_{i=1}^n a_i - U\right) + \sum_{i=1}^n \mu_i(L-a_i)? $$ So by the KKT and Slater condition, we look for a zero of $$ 0 = \frac1{2}a_i^{-\frac{3}{2}}C_i + \lambda - \mu_i $$ which we solve (for each $i=1,\dots,n$) to obtain $$ \begin{aligned} -\frac1{2}a_i^{-\frac{3}{2}}C_i =& + \lambda - \mu_i\\ \therefore a_i^{-\frac{3}{2}} =& \frac{-\lambda +\mu_i}{2 C_i}\\ \therefore a_i^{\star} =& \left(2\frac{C_i}{\mu_i-\lambda}\right)^{\frac{2}{3}}. \end{aligned} $$
However, this depends on $\lambda$ and $\mu_n$ so I don't know how to proceed further..Do we then maximize for $\lambda $ and $\mu_i$?