Maximize the objective function $f(x)=\sum_{i=1}^n (1-e^{-k_ix_i})$, where $(k_i >0) \in \mathbb{R}, (0\leq x_i \leq 1) \in \mathbb{R} $ .
$\max\sum_{i=1}^n (1-e^{-k_ix_i})$
Subject to
i) $\sum_{i=1}^n x_i(1-x_i) \leq A$
ii) $x_i \geq 0$, $\forall i=1\cdots n$
I have followed some examples given in Internet and have tried to solve.
My attempt :
$L({x}, {\mu}) = (1-e^{-k_ix_i}) - \mu_0\left(\underset{i=1}{\overset{n}{\sum}}x_i(1-x_i) - A\right) - \underset{i=1}{\overset{n}{\sum}} \mu_i(-x_i)$
$\nabla_{x_i}L({x}, {\mu}) = k_ie^{-k_ix_i} +\mu_0 (2\sum_{i=1}^n x_i - n)+\mu_i= 0$
$\mu_0 = - \frac{k_ie^{-k_ix_i}}{2\sum_{i=1}^n x_i - n}$
Is my approach correct? For some values $\mu_0 > 0$ and for some values $\mu_0 <0$ .
How to find i) $\mu_0,\mu_i$ ii) $x_i$ iii) and final set of equations.