List of all possible x for a convex optimisation problem

Can we have all possible x (list of all possible candidates) in the valid search space instead of just one x that satisfies y=Ax while solving minimization of L1 norm of x subject to Ax==y?

No for Cvx. There are other tools that can do it.

Thanks. Which tools exactly?

I think you can find names in answers at

https://or.stackexchange.com/