A mixed integer programming method of solving large-scale nonlinearly constrained problems

A mixed-integer programming method for solving large-scale nonlinearly constrained problems is presented in this paper. Assuming the problem convexity, the equivalence of the solution of the original problem with the solution of a finite sequence of mixed-integer linear problems, involving a finite number of constraints is first proved by using the principles of the generalized Benders decomposition method. Then the authors show that the convergence of the solutions of this problem sequence towards the solution of the original problem, is reached when two integer solutions into the sequence are identical. After a presentation of the logical steps of the algorithm, where a reduced gradient method and a mixed-integer linear programming procedure are implemented, two numerical examples are detailed. The first one is a mathematical problem involving nonlinear constraints. The second one, found in the literature, is related to the optimal design of a gas transportation network, and shows the capability of the procedure for solving large-scale problems derived from actual industrial cases.