I encountered an optimization problem \begin{align} f(x)=(-1)^{N-1}\sum_{i=1}^M\frac{\ln x_{i}}{x_{i}^N}\prod_{j\neq i}\frac{x_i}{x_i-x_j} \end{align} where $N$ is a positive integer, $x_i>0$ for all $i=1,2,\cdots,M$, and with the constraint $\sum_{i=1}^Mx_i=M$. Seems to me that this problem has no optimal values, since by the AM-GM inequality the optimal value seems to be at the singularity of the function (correct me if I am wrong). While my question is if this function has no optimal value, in which directions the function is increasing or decreasing?
My second question related with this form is that now if I have $c_1,\cdots, c_M$ with the constraint $c_i > 0$ for all $i=1,\cdots, M$, and $\sum_{i=1}^M c_i = M$. And $y$ is a transformation of $x$ such that $y_i=c_ix_i$, then can we compare the value of $f(y)$ with that of $f(x)$?
Thanks!