Hi, I want to formulate a maximization problem in CVX. The objective function is concave but there exists a constraint in the following form:

\sum_{i=1}^K \dfrac{a_i}{1+e^{-b_i x_i}} \geq c_i

where x_i, i=1,..., K are variables to be optimized and we also have the upper and lower limit of each x_i as

l_i \leq x_i \leq u_i.

Can I formulate this problem in CVX?

Presuming a_i, b_i > 0, the LHS of the constraint is convex for x <= 0 and concave for x >= 0. Therefore, the constraint is only convex for x >= 0. Therefore, this is only a convex optimization problem if l_i >= 0. If that is the case, I am not optimistic (other than for K = 1) this can be formulated in CVX, because the LHS is not concave over the entirety of its natural domain. Your only hope is if a (Mosek) wizard comes along and proves my pessimism to be misplaced.

Thanks for your answer. Yes, a_i, b_i > 0 and l_i \geq 0.

So, if CVX cannot be used to solve this problem, is there any other tool which can be used to do so? Because the problem is not straightforward to be solved analytically.

You can use a tool such as YALMIP, which with a nonlinear solver, should handle it, even if not in conic form.