Alexander Martin

We propose an approach based on quadratic approximations for solving general Mixed-Integer Nonlinear Programming (MINLP) problems. Specifically, our approach entails the global approximation of the epigraphs of constraint functions by means of paraboloids, which are polynomials of degree two with univariate quadratic terms, and relies on a Lipschitz property only. These approximations are then integrated into the original problem. To this end, we introduce a novel approach to compute globally valid epigraph approximations by paraboloids via a Mixed-Integer Linear Programming (MIP) model. We emphasize the possibility of performing such approximations a-priori and providing them in form of a lookup table, and then present several ways of leveraging the approximations to tackle the original problem. We provide the necessary theoretical background and conduct computational experiments on instances of the MINLPLib. As a result, this approach significantly accelerates the solution process of MINLP problems, particularly those involving many trigonometric or few exponential functions. In general, we highlight that the proposed technique is able to exploit advances in Mixed-Integer Quadratically-Constrained Programming (MIQCP) to solve MINLP problems.

We propose an approach based on quadratic approximations for solving general Mixed-Integer Nonlinear Programming (MINLP) problems. Specifically, our approach entails the global approximation of the epigraphs of constraint functions by means of paraboloids, which are polynomials of degree two with univariate quadratic terms, and relies on a Lipschitz property only. These approximations are then integrated into the original problem. To this end, we introduce a novel approach to compute globally valid epigraph approximations by paraboloids via a Mixed-Integer Linear Programming (MIP) model. We emphasize the possibility of performing such approximations a-priori and providing them in form of a lookup table, and then present several ways of leveraging the approximations to tackle the original problem. We provide the necessary theoretical background and conduct computational experiments on instances of the MINLPLib. As a result, this approach significantly accelerates the solution process of MINLP problems, particularly those involving many trigonometric or few exponential functions. In general, we highlight that the proposed technique is able to exploit advances in Mixed-Integer Quadratically-Constrained Programming (MIQCP) to solve MINLP problems.

In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz constant. In recent literature, methods based on non-convex outer approximation are proposed for tackling one-dimensional equality constraints that are Lipschitz with respect to the maximum norm. To the best of our knowledge, however, there does not exist a non-convex outer approximation method for a general problem class. We introduce a meta-level solution framework to solve such problems and tackle the underlying theoretical foundations. Considering the feasible domain without the constraint function as manageable, our method relaxes the multidimensional constraint and iteratively refines the feasible region by means of norm-induced cuts, relying on an oracle for the resulting subproblems. We show the method’s correctness and investigate the problem complexity. In order to account for discussions about functionality, limits, and extensions, we present computational examples including illustrations.

Over the last few years, the efficient design of processes in hospitals and medical facilities has received more and more attention, particularly when the improvement of the processes is aimed at relieving the workload of medical staff. To this end, we have developed a method to determine optimal allocations of intra-hospital transports to hospital transport employees. When optimizing transport plans in hospitals, there are various optimization goals to strive for. Therefore, we used a lexicographic approach to solve this multi-criteria optimization problem. To calculate optimal transport plans in a sufficiently short computation time, we have decomposed the problem at hand with the Dantzig–Wolfe reformulation and solved the resulting pricing subproblem with an enumerative column generation approach. To improve the efficiency of the column generation process, we have investigated and implemented different pruning methods, dominance rules and a column reuse mechanism for the online setting of the application at hand. In an extensive computational study, we first evaluated the efficiency of the different pruning methods before we compared our solution approach with the standard branch-and-bound column search approach, classical column generation methods and the solution of an integrated MIP model solved by a commercial solver. Finally, we present performance indicators of the transport optimization tool, which was developed from our method and is now productively used in a German hospital.

We consider a problem from the context of energy-efficient underground railway timetabling, in which an existing timetable draft is improved by slightly changing departure and running times. In practice, synchronization between accelerating and braking trains to utilize regenerative braking plays a major role for the energy efficiency of a timetable. Because deviations from a planned timetable may lead to unnecessarily high energy consumption during actual operation, we include operational uncertainties in our model to create a timetable that remains energy efficient, even if deviations from the nominal timetable occur. To solve the problem, we use a scenario expansion model in conjunction with a Benders decomposition approach. As an alternative to solving the Benders subproblems, we present a heuristic sparse cut that can be computed efficiently. The resulting sparse-cut heuristic produces high-quality solutions on a set of real-world instances stemming from the Nürnberg underground system, outperforming the integrated mixed-integer programming approach as well as the basic Benders approach. Additionally, we evaluate two static recovery strategies—shortening dwell times as well as shortening dwell and running times—to determine the cost and benefit of handling delays using a simple static rule. In our experiments, we are able to reduce the energy consumption by up to 9.4% and confirm that delay recovery via shortening dwell times is an energy-efficient and effective way to increase punctuality at low cost in terms of energy. Funding: This research was supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy through the Center for Analytics – Data – Applications (ADA-Center) within the framework of “BAYERN DIGITAL II” (20-3410-2-9-8).