Solving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programmes

We propose a method for solving mixed-integer nonlinear programmes (MINLPs) to global optimality by discretization of occurring nonlinearities. The main idea is based on using piecewise linear functions to construct mixed-integer linear programme (MIP) relaxations of the underlying MINLP. In order to find a global optimum of the given MINLP, we develop an iterative algorithm which solves MIP relaxations that are adaptively refined. We are able to give convergence results for a wide range of MINLPs requiring only continuous nonlinearities with bounded domains and an oracle computing maxima of the nonlinearities on their domain. Moreover, the practicalness of our approach is shown numerically by an application from the field of gas network optimization.