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Solving Mixed-Integer Nonlinear Programs using Adaptively Refined Mixed-Integer Linear Programs
(2017)
We propose a method for solving mixed-integer nonlinear programs (MINLPs) to global optimality by discretization of occuring nonlinearities. The main idea is based on using piecewise linear functions to construct mixed-integer linear program (MIP) relaxations of the underlying MINLP. In order to find a global optimum of the given MINLP we develope 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.
In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations (PDEs). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic PDEs and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem (MINLP). For the numerical solution of the MINLP, we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems (MIPs). Finally, we present two case studies that illustrate the applicability of our approach.