Min–max programming problems with addition–min constraints have been studied in the literature to model data transfer in BitTorrent-like peer-to-peer file-sharing systems. It is well known that the class of overlap functions contains various operators, including the “min” operator. The aim of this paper is to generalize the above min–max programming problem with addition–overlap function constraints. We demonstrate that this new optimization problem can be transformed into a simplified single-variable optimization problem, which makes it easier to find an optimal solution. The bisection method will be used to find this optimal solution. In addition, when the overlap function is explicitly specified, an iterative method is given to compute the optimal objective value with a polynomial time complexity. A numerical example is provided to illustrate the procedures.

Finding an optimal solution of the min-max programming problem with addition-overlap function constraints has been studied in the literature. Since the definition of overlap functions is very general and has no explicit formulations, a bisection method was proposed to yield a uniform-optimal solution for such optimization problems with a general overlap function involved in its constraint part. The uniform-optimal solution could be improved if the system manager wants extra properties such as also yielding lower cost performance. The minimal-optimal solution is proposed other than the uniform-optimal solution to answer manager’s call. In this paper, we propose an iterative method to yield at least one minimal-optimal solution. Our method starts from the uniform-optimal solution and systematically reduces the values of some non-binding variables while preserving the feasibility and optimality. To rigorously establish our method, we specify explicitly a nontrivial “min-shifted” overlap function in place of the general overlap function of the min-max programming problem. The proof of finding a minimal-optimal solution of our algorithm is given. By shifting the searching sequence of decision variables, our algorithm may find other minimal-optimal solutions which provide more flexibility for the system manager to choose. Numerical examples are provided to illustrate the procedures of the algorithm.