Sinkhorn algorithms and linear programming solvers for optimal partial transport problems

In this note, we generalize the classical optimal partial transport (OPT) problem by modifying the mass destruction/creation term to function-based terms, introducing what we term ``generalized optimal partial transport'' problems. We then discuss the dual formulation of these problems and the associated Sinkhorn solver. Finally, we explore how these new OPT problems relate to classical optimal transport (OT) problems and introduce a linear programming solver tailored for these generalized scenarios.