I have a question on Benders Decomposition (BD). Suppose I have an MILP model which can be decomposed into a master problem (MP) including integer and continuous variables and a subproblem (SP) including only integer variables. In addition, suppose that the SP generated does not hold any nice property like total unimodularity meaning that the relaxation does not do any good for me. In this case, I cannot utilize the duality theorem to generate a Benders cut.
I am familiar with Logic-Based BD (LBBD). Yet, in all the studies that I have seen using LBBD, SP becomes a feasibility problem without an objective function, which is solved by constraint programming (CP).
Now, let's further assume that the SP has a solid objective function. I was wondering if there are recent studies containing LBBD where SP is an IP with an objective function and is not solved with CP. If not, what are some viable approaches to tackle such problem settings?