Currently, I am using BARON through GAMS. The ususal output printed during solving is like the following. It only shows the upper and lower bound on the objective at every iteration

What is the “current feasibility of the solution”? One of the bounds give you the value of your incumbent. Some solvers also give you the number of integer infeasibilities in the log, but since BARON also needs to branch on continuous variables, this needs to be extended to the number of possible branching candidates. Not sure if this is a good measure for the user.

Okay, I see. By current feasibility, I meant simply something like the output of IPOPT or FMINCON (if you are familiar with either) where you could see the infeasibility with regards to the constraints decreases gradually with the number of iterations until it meets the stopping criteria.

Gradient methods, barrier method, simplex, etc have the concept of a “moving a point through space” and hence they can talk about statistics about this “point”. Global methods like branch-and-bound don’t have a “point” concept and hence there is no way talking about the statistics (e.g. [in]feasibility) of “the point”.