2
$\begingroup$

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)

  • 0
    I don't think I got the meaning of $C=[c_i]$... Can you explain?2011-12-18
  • 0
    Its a vector i.e $C = [c_1...c_k]$2011-12-18
  • 0
    So you want to solve: $$\min \left\{ \sum_{i=1}^k \Psi (c_i),\ \text{under constraints } c_1,\ldots c_k\geq 0 \text{ and } \sum_{i=1}^k c_i=1\right\}\; .$$ Am I right?2011-12-18
  • 0
    yes. here $\psi(c_i)$ is a concave function2011-12-19

1 Answers 1