@article{SchmidtSirventWollner2018, author = {Schmidt, Martin and Sirvent, Mathias and Wollner, Winnifried}, title = {The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities}, series = {Optimization Letters}, journal = {Optimization Letters}, pages = {16}, year = {2018}, abstract = {It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function---in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute approximate global optimal solutions in finite time and we also provide a worst-case iteration bound. In some first numerical experiments we show that the ``cost of not knowing enough'' is rather high by comparing our approach with the open-source global solver SCIP. This reveals that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.}, language = {en} } @unpublished{EggerKuglerWollner, author = {Egger, Herbert and Kugler, Thomas and Wollner, Winnifried}, title = {Numerical optimal control of instationary gas transport with control and state constraints}, abstract = {We consider the optimal control of a nonlinear hyperbolic system of balance laws on a one-dimensional network which arises in the context of gas transport in pipeline systems. State constraints, which are required for the safe operation of the system, are incorporated by a barrier method. We discuss the well-posedness of the governing system of partial differential-algebraic equations and investigate the existence of minimizers. For the numerical solution, we then consider the approximation of the state equation by mixed finite elements in space and a particular linear implicit time integration scheme that can be interpreted as a discontinuous Galerkin approximation. We establish well- posedness of this discretization scheme and prove the existence of minimizers for the corresponding discretized optimal control problem and discuss its numerical solution by a projected Gauß-Newton method. The efficient realization of the Jacobian and Hessian of the quadratic approximations that have to be minimized in every iteration of the Gauß-Newton method can be obtained via the solution of discretized sensitivity and adjoint equations. These are obtained by formal differentiation and transposition of the Galerkin methods employed for the discretization of the state equations. All approximations obtained after discretization can thus be interpreted as functions on the continuous level and, since the functional analytic setting is not changed by the Galerkin discretization, we observe mesh independence of the resulting fully discrete methods. For illustration of our theoretical results and to demonstrate the efficiency of the proposed method, we present numerical results for two test problems that model typical situations that may arise in the daily operation of gas networks.}, language = {en} } @unpublished{BeckersBehrensWollner2017, author = {Beckers, Susanne and Behrens, J{\"o}rn and Wollner, Winnifried}, title = {Duality Based Error Estimation in the Presence of Discontinuities}, pages = {21}, year = {2017}, abstract = {Goal-oriented mesh adaptation, in particular using the dual-weighted residual (DWR) method, is known in many cases to produce very efficient meshes. For obtaining such meshes the (numerical) solution of an adjoint problem is needed to weight the residuals appropriately with respect to their relevance for the overall error. For hyperbolic problems already the weak primal problem requires in general an additional entropy condition to assert uniqueness of solutions; this difficulty is also reflected when considering adjoints to hyperbolic problems involving discontinuities where again an additional requirement (reversibility) is needed to select appropriate solutions. Within this article, an approach to the DWR method for hyperbolic problems based on an artificial viscosity approximation is proposed. It is discussed why the proposed method provides a well-posed dual problem, while a direct, formal, application of the dual problem does not. Moreover, we will discuss a further, novel, approach in which the forward problem need not be modified, thus allowing for an unchanged forward solution. The latter procedure introduces an additional residual term in the error estimation, accounting for the inconsistency between primal and dual problem. Finally, the effectivity of the extended error estimator, assessing the global error by a suitable functional of interest, is tested numerically; and the advantage over a formal estimator approach is demonstrated.}, language = {en} } @article{SchmidtSirventWollner2017, author = {Schmidt, Martin and Sirvent, Mathias and Wollner, Winnifried}, title = {A Decomposition Method for MINLPs with Lipschitz Continuous Nonlinearities}, series = {Mathematical Programming}, journal = {Mathematical Programming}, number = {178(1)}, pages = {449 -- 483}, year = {2017}, abstract = {Many mixed-integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly. This is, e.g., the case for problems constrained by differential equations or for models that rely on black-box simulation runs. For these problem classes, we present, analyze, and test algorithms that solve mixed-integer problems with only Lipschitz continuous nonlinearities. Our theoretical results depend on the assumptions made on the (in)exactness of function evaluations and on the knowledge of Lipschitz constants. If Lipschitz constants are known, we prove finite termination at approximate globally optimal points both for the case of exact and inexact function evaluations. If only approximate Lipschitz constants are known, we prove finite termination and derive additional conditions under which infeasibility can be detected. A computational study for gas transport problems and an academic case study show the applicability of our algorithms to real-world problems and how different assumptions on the constraint functions up- or downgrade the practical performance of the methods.}, language = {en} }