OSQP solves only continuous QPs. For integer variables you need an MIQP solver.

We wrote a Python branch-and-bound wrapper around OSQP to solve MIQPs: https://github.com/oxfordcontrol/miosqp

However, there are more advanced MIQP algorithms out there such as GUROBI or CPLEX.

Did you test miosqp against standard MIQPs? For example, in QPLIB, there is a handful of convex mixed-integer QPs.

Haven't tried the MIQP extension against the QPLIB yet. It is just a prototype and it is written in Python. We first need to implement it in C to compare the timings.

You could compare number of nodes explored rather than absolute timings to see how inaccuracy solving the relaxations affects branch-and-bound.

If the variable is constrained to be an integer, the problem becomes NP-hard-as-hell (in both a complexity sense and an approximation sense), sadly. It also stops being convex, so splitting-cone methods can't be directly applied and the solution here goes out the window.

In general, you assume the variables here to be real-valued vectors.