## 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING

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#### Keywords

We consider the modeling of operation modes for complex compressor stations (i.e., ones with several in- or outlets) in gas networks. In particular, we propose a refined model that allows to precompute tighter relaxations for each operation mode. These relaxations may be used to strengthen the compressor station submodels in gas network optimization problems. We provide a procedure to obtain the refined model from the input data for the original model. This procedure is based on a nontrivial reduction of the graph representing the gas flow through the compressor station in an operation mode.

Frankl’s (union-closed sets) conjecture states that for any nonempty finite union-closed (UC) family of distinct sets there exists an element in at least half of the sets. Poonen’s Theorem characterizes the existence of weights which determine
whether a given UC family ensures Frankl’s conjecture holds for all UC families which contain it. The weight systems are nontrivial to identify for a given UC family, and methods to determine such weight systems have led to several other open questions and conjectures regarding structures in UC families.
We design a cutting-plane method that computes the explicit weights which imply the existence conditions of Poonen’s Theorem using computational integer programming coupled with redundant verification routines that ensure correctness. We find over one hundred previously unknown families of sets which ensure Frankl’s conjecture holds for all families that contain any of them. This improves significantly on all previous results of the kind.
Our framework allows us to answer several open questions and conjectures regarding structural properties of UC families, including proving the 3-sets conjecture of Morris from 2006 which characterizes the minimum number of 3-sets that ensure Frankl’s conjecture holds for all families that contain them. Furthermore, our method provides a general algorithmic road-map for improving other known results and uncovering structures in UC families.

Improving branching for disjunctive polyhedral models using approximate convex decompositions
(2018)

Disjunctive sets arise in a variety of optimization models and much esearch has been devoted to obtain strong relaxations for them. This paper focuses on the evaluation of the relaxation during the branch-and-bound search process. We argue that the branching possibilities (\ie binary variables) of the usual formulations are unsuitable to obtain strong bounds early in the search process as they do not capture the overall shape of the the entire disjunctive set. To analyze and exploit the shape of the disjunctive set we propose to compute a hierarchy of approximate convex decompositions and show how to extend the known formulations to obtain improved branching behavior.

Gas networks are an important application area for optimization. When considering long-range transmission, compressor stations play a crucial role in these applications. The purpose of this report is to collect and systematize the models used for compressor stations in the literature. The emphasis is on recent work on simple yet accurate polyhedral models that may replace more simplified traditional models without increasing model complexity. The report also describes an extension of the compressor station data available in GasLib (http://gaslib.zib.de/) with the parameters of these models.

The topic of this thesis is the examination of an optimization model
which stems from the clustering process of non-reversible markov processes.
We introduce the cycle clustering problem und formulate it as a mixed
integer program (MIP).
We prove that this problem is N P-hard and discuss polytopal aspects
such as facets and dimension. The focus of this thesis is the development of
solving methods for this clustering problem. We develop problem specific
primal heuristics, as well as separation methods and an approximation
algorithm. These techniques are implemented in practice as an application
for the MIP solver SCIP.
Our computational experiments show that these solving methods result
in an average speedup of ×4 compared to generic solvers and that our
application is able to solve more instances to optimality within the given
time limit of one hour.

The Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. Although the different Steiner tree problem variants are usually strongly related, solution approaches employed so far have been prevalently problem-specific. Against this backdrop, the solver SCIP-Jack was created as a general-purpose framework that can be used to solve the classical Steiner tree problem and 11 of its variants. This versatility is achieved by transforming various problem variants into a general form and solving them by using a state-of-the-art MIP-framework. Furthermore, SCIP-Jack includes various newly developed algorithmic components such as preprocessing routines and heuristics. The result is a high-performance solver that can be employed in massively parallel environments and is capable of solving previously unsolved instances. After the introduction of SCIP-Jack at the 2014 DIMACS Challenge on Steiner problems, the overall performance of the solver has considerably improved. This article provides an overview on the current state.

This article describes new features and enhanced algorithms made available in version 5.0 of the SCIP Optimization Suite. In its central component, the constraint integer programming solver SCIP, remarkable performance improvements have been achieved for solving mixed-integer linear and nonlinear programs. On MIPs, SCIP 5.0 is about 41 % faster than SCIP 4.0 and over twice as fast on instances that take at least 100 seconds to solve. For MINLP, SCIP 5.0 is about 17 % faster overall and 23 % faster on instances that take at least 100 seconds to solve. This boost is due to algorithmic advances in several parts of the solver such as cutting plane generation and management, a new adaptive coordination of large neighborhood search heuristics, symmetry handling, and strengthened McCormick relaxations for bilinear terms in MINLPs. Besides discussing the theoretical background and the implementational aspects of these developments, the report describes recent additions for the other software packages connected to SCIP, in particular for the LP solver SoPlex, the Steiner tree solver SCIP-Jack, the MISDP solver SCIP-SDP, and the parallelization framework UG.

Current linear energy system models (ESM) acquiring to provide sufficient detail and reliability frequently bring along problems of both high intricacy and increasing scale. Unfortunately, the size and complexity of these problems often prove to be intractable even for commercial state-of-the-art linear programming solvers. This article describes an interdisciplinary approach to exploit the intrinsic structure of these large-scale linear problems to be able to solve them on massively parallel high-performance computers. A key aspect are extensions to the parallel interior-point solver PIPS-IPM originally developed for stochastic optimization problems. Furthermore, a newly developed GAMS interface to the solver as well as some GAMS language extensions to model block-structured problems will be described.

The analysis of infeasibility plays an important role in solving satisfiability problems (SAT) and mixed integer programs (MIPs). In mixed integer programming, this procedure is called conflict analysis. So far, modern MIP solvers use conflict analysis only for propagation and improving the dual bound, i.e., fathoming nodes that cannot contain feasible solutions. In this short paper, we present a new approach which uses conflict information to improve the primal bound during a MIP solve. To derive new improving primal solutions we use a conflict driven diving heuristic called conflict diving that uses the information obtained by conflict analysis. Conflict diving pursues a twofold strategy. By using conflict information the new diving approach is guided into parts of the search space that are usually not explored by other diving heuristics. At the same time, conflict diving has a fail-fast-strategy to reduce the time spent if it cannot find a new primal solution. As a byproduct, additional valid conflict constraints can be derived, from which a MIP solver can gain benefit to improve the dual bound as well. To show the added-value of conflict diving within a MIP solver, conflict diving has been implemented within the non-commercial MIP solver SCIP. Experiments are carried out on general MIP instances from standard public test sets, like MIPLIB2010 or Cor@l.

Compressor machines are crucial elements in a gas transmission network, required to compensate for the pressure loss caused by friction in the pipes. Modelling all physical and technical details of a compressor machine involves a large amount of nonlinearity, which makes it hard to use such models in the optimization of large-scale gas networks. In this paper, we are going to describe a modelling approach for the operating range of a compressor machine, starting from a physical reference model and resulting in a polyhedral representation in the 3D space of mass flow throughput as well as in- and outlet pressure.