Because the constraint in my problem is maximizing a convex function within a bounded region, I am considering it as a feasibility problem without constraints.
What I am trying is to get a feasible set to that problem and understand that first. However, there is only one solution returned. I checked previous discussions; it seems to me that different solvers will give different solutions to under-determined linear systems by default. However, I am wondering whether there are any solver that will return me the set instead.
In a way, this question doesn’t make sense. After all, what would the set look like? How would the computer describe it? Arguably, the very same way that you are describing it to CVX: as a set of equations and inequalities. So you already have a description of the feasible set.
There are some ways that you can explore the feasible set, however. For instance, you can try solving the problem with two different objectives for each variable:
minimize(x(i))
maximize(x(i))
This will give you a box that is guaranteed to contain the set. You can construct a polyhedral approximation by solving multiple objectives in this form: