Given $n$ variables $x_{i}$ where $i\in [0,n)$, denoted as a vector $x$  , given a linear objective function that we want to minimize

$c^{T}x$

with 2 constraints:

1. the $\sum (x_{i} )^{2} < n+1$
2. $\sum log(x_{i}) > 0$

How to solve such optimization problem  ?

The only thing I can think of is to convert the 2nd constraint to be the product of all $x$ bigger than 1. I cannot think of a way to turn the 2nd constraint into a 'common' quadratic one.

Can anyone sharesome ideas?

Given $n$ variables $x_{i}$ where $i\in [0,n)$, denoted as a vector $x$, given a linear objective function that we want to minimize $c^\top x$ with 2 constraints:

1. $\sum x_{i}^{2} < n+1$
2. $\sum\log(x_{i}) > 0$.

How can I solve this optimization problem?

The only thing I can think of is to convert the 2nd constraint to be the product of all $x$ bigger than $1$. I cannot think of a way to turn the 2nd constraint into a 'common' quadratic one.

Can anyone share some ideas?