I finding some difficulties in solving the below constrained problem using Lagrangian. Would be great if some one helps me with the steps.
$\min_C \sum_i \Psi(c_i)$
subject to $\sum_i c_i = 1$ and $c_i \geq 0$ for $i=1 \cdots k$ and $C=[c_i]$
Here $\Psi(x)$ is a concave function. for eg. $\Psi(x) = 2*x -x ^2$
I tried the below steps:
1. Writing Lagrangian $L = \sum_i \Psi(c_i) + \gamma (\sum_i c_i - 1) - \sum_i \alpha_i c_i$
2. Differentiating Lagrangian wrto $c_i$ and equating to zero. i.e $\psi'(c_i) + \gamma - \alpha_i = 0$
Im not sure how to proceed after this. (what value to find out and wht needs to be satisfied etc)