What is the minimum for this function of $x_1,x_2, \ldots, x_n$:
$$\sum_{i=1}^n c_i \log x_i + \lambda \; \sum_{i=1}^n d_i x_i, $$ where $\lambda$, $c$ and $d$ series are positive constants, $x_i \in (0.02,1]$ and $\sum x_i = 1$.
What is the minimum for this function of $x_1,x_2, \ldots, x_n$:
$$\sum_{i=1}^n c_i \log x_i + \lambda \; \sum_{i=1}^n d_i x_i, $$ where $\lambda$, $c$ and $d$ series are positive constants, $x_i \in (0.02,1]$ and $\sum x_i = 1$.
Let $ x_i=1/m $ certainly $ x_i \in (0,1] $ for all $i$ and all natural numbers $m$. Then your function becomes $-\log m \sum_{i=1}^{n}{c_i} + \frac{\lambda}{m} (\sum_{i=1}^{n}{d_i})$ which can be written as $-m A + B/m $ where $A,B$ are positive constants. This can be made arbitrarily large and negative, so there is no minimum. If this is part of some work on Lagrangian multipliers you need to think again about your restrictions on $\lambda$.