Papers by Alexander Martin
Discrete, nonlinear and PDE constrained optimization are mostly considered as different fields of... more Discrete, nonlinear and PDE constrained optimization are mostly considered as different fields of mathematical research. Nevertheless many real-life problems are most naturally modeled as PDE constrained mixed integer nonlinear programs. For example, nonlinear network flow problems where the flow dynamics are governed by a transport equation are of this type. We present four different applications together with the derivation of the associated transport equations and we show how to model these problems in terms of mixed integer linear constraints.
Mathematics in Industry, 2021
Timetabling of railway traffic and other modes of transport is among the most prominent applicati... more Timetabling of railway traffic and other modes of transport is among the most prominent applications of discrete optimization in practice. However, it has only been recently that the connection between timetabling and energy consumption has been studied more extensively. In our joint project with VAG Verkehrs-Aktiengesellschaft, the transit authority and operator of underground transport in the German city of Nürnberg, we develop algorithms for optimal timetabling to minimize the energy consumption of the trains via more energy-efficient driving as well as increasing the usability of recuperated energy from braking. Together with VAG, we have worked extensively to establish a broad basis of operational data, for example characteristic power consumption profiles as well as travel time and dwell time distributions for the trains running in the network, to serve as input to our optimization methods. On the collected data sets, our approach has already shown significant potential to reduce energy consumption and, as a consequence, electricity costs and environmental impact. Furthermore, mathematical analysis of the polyhedral and graph structures involved in the optimization approach have enabled us to compute high-quality solutions within short time. This positive outlook motivated VAG to extend this project to include further operational constraints in the model and to adopt the resulting software planning tool in practice afterwards. It will assist timetable planners at VAG in using the available degrees of freedom in their timetable drafts to optimize the energy-efficiency of the underground system.
SIAM Journal on Control and Optimization, 2021
Mitteilungen der Deutschen Mathematiker-Vereinigung, 2015
SSRN Electronic Journal, 2010
A Discrete Optimization Approach to Large Scale Supply Networks Based on Partial Differential Equations
SIAM Journal on Scientific Computing, 2008
ABSTRACT We introduce a continuous optimal control problem governed by ordinary and partial diffe... more ABSTRACT We introduce a continuous optimal control problem governed by ordinary and partial differential equations for supply chains on networks. We derive a mixed-integer model by discretization of the dynamics of the partial differential equations and by approximations to the cost functional. Finally, we investigate numerically properties of the derived mixed-integer model and present numerical results for a real-world example.
We will present a theoretical model for an integrated planning of Eucharistic masses and Liturgie... more We will present a theoretical model for an integrated planning of Eucharistic masses and Liturgies of the Word assuming different priests, assistant priests and lay persons. This all-integer vector optimization problem can be seen as an economic, spacial and inter-temporal resource allocation problem taking the challenges in the context of parish clusters into account. Having provided some theoretical insights into the mass planning problem, numerical results show the relevance of our model for an efficient decision-making in religious practice. We demonstrate that innovative team work arrangements in the church can substantially improve the efficiency, fairness and acceptance of mass allocations.
Mathematical Methods of Operations Research, 2022
It is well known that linear prices supporting a competitive equilibrium exist in the case of con... more It is well known that linear prices supporting a competitive equilibrium exist in the case of convex markets, however, in the presence of integralities this is open and hard to decide in general. We present necessary and sufficient conditions for the existence of such prices for decentralized market problems where market participants have integral decision variables and their feasible sets are given in complete linear description. We utilize total unimodularity and the aforementioned conditions to show that such linear prices exist and present some applications. Furthermore, we compute competitive equilibria for two classes of decentralized market problems arising in energy markets and show that competitive equilibria may exist regardless of integralities.
arXiv: Optimization and Control, 2020
We consider the bipartite boolean quadric polytope (BQP) with multiple-choice constraints and ana... more We consider the bipartite boolean quadric polytope (BQP) with multiple-choice constraints and analyse its combinatorial properties. The well-studied BQP is defined as the convex hull of all quadric incidence vectors over a bipartite graph. In this work, we study the case where there is a partition on one of the two bipartite node sets such that at most one node per subset of the partition can be chosen. This polytope arises, for instance, in pooling problems with fixed proportions of the inputs at each pool. We show that it inherits many characteristics from BQP, among them a wide range of facet classes and operations which are facet preserving. Moreover, we characterize various cases in which the polytope is completely described via the relaxation-linearization inequalities. The special structure induced by the additional multiple-choice constraints also allows for new facet-preserving symmetries as well as lifting operations. Furthermore, it leads to several novel facet classes as...
The running time for solving a mixed-integer linear optimization problem (MIP) strongly depends o... more The running time for solving a mixed-integer linear optimization problem (MIP) strongly depends on the number of its integral variables. Bader et al. [Math. Progr. 169 (2018) 565–584] equivalently reformulate an integer program into an MIP that contains a reduced number of integrality constraints, when compared to the original model. Generalizing the concept of totally unimodular (TU) matrices, the reformulation is determined via a so-called affine TU decomposition of the underlying constraint matrix. In this work, we develop affine TU decompositions for two-stage resource-constrained b-matching problems that have challenging applications in runway utilization of aircraft. Mathematically, the task consists in determining an optimum two-stage bipartite b-matching in a graph where a node v can be assigned up to bv ∈ N many edges. The two stages are linked by resource restrictions modeled by specific knapsack constraints. Determining an affine TU decomposition for this problem is reduc...
In this paper, we demonstrate how to learn the objective function of a decision-maker while only ... more In this paper, we demonstrate how to learn the objective function of a decision-maker while only observing the problem input data and the decision-maker's corresponding decisions over multiple rounds. Our approach is based on online learning and works for linear objectives over arbitrary feasible sets for which we have a linear optimization oracle. As such, it generalizes previous approaches based on KKT-system decomposition and dualization. The two exact algorithms we present -- based on multiplicative weights updates and online gradient descent respectively -- converge at a rate of O(1/sqrt(T)) and thus allow taking decisions which are essentially as good as those of the observed decision-maker already after relatively few observations. We also discuss several useful generalizations, such as the approximate learning of non-linear objective functions and the case of suboptimal observations. Finally, we show the effectiveness and possible applications of our methods in a broad c...
For the problem to find an m-clique in an m-partite graph, staircase compatibility has recently b... more For the problem to find an m-clique in an m-partite graph, staircase compatibility has recently been introduced as a polynomial-time solvable special case. It is a property of a graph together with an m-partition of the vertex set and total orders on each subset of the partition. In optimization problems involving m-cliques in m-partite graphs as a subproblem, it allows for totally unimodular linear programming formulations which have shown to efficiently solve problems from different applications. In this work, we address questions concerning the recognizability of this property in the case where the m-partition of the graph is given, but suitable total orders are to be determined. While finding these total orders is NP-hard in general, we give several conditions under which it can be done in polynomial time. For bipartite graphs, we present a polynomial-time algorithm to recognize staircase compatibility, and show that staircase total orders are unique up to a small set of reorder...
Transportation Science, 2021
Over the last few years, optimization models for the energy-efficient operation of railway traffi... more Over the last few years, optimization models for the energy-efficient operation of railway traffic have received more and more attention, particularly in connection with timetable design. In this work, we study the effect of load management via timetabling. The idea is to consider trains as time-flexible consumers in the railway power supply network and to use slight shifts in the departure times from the stations to avoid too many simultaneous departures. This limits peak consumption and can help to improve the stability of the power supply. To this end, we derive efficient formulations for the problem of an optimal timetable adjustment based on a given timetable draft, two of which even allow for totally unimodular polyhedral descriptions. The proper choice of the objective function allows the incorporation of the priorities of either the train operating companies or the infrastructure manager. These include the avoidance of large peaks in average or instantaneous consumption and ...
European Journal of Operational Research, 2020
Portfolio optimization is an ongoing hot topic of mathematical optimization and management scienc... more Portfolio optimization is an ongoing hot topic of mathematical optimization and management science. Due to the current financial market environment with low interest rates and volatile stock markets, it is getting more and more important to extend portfolio optimization models by other types of investments than classical assets. In this paper, we present a mixedinteger multistage stochastic model that includes investment opportunities in irreversible and long-term infrastructure projects in the context of renewable energies, which are also subject to policy risk. On realistic time scales for investment problems of this type, the resulting instances are by far too large to be solved with today's most evolved optimization software. Thus, we present a tailored moving-horizon approach together with suitable approximations and simplifications of the model. We evaluate these approximations and simplifications in a computational sensitivity analysis and derive a final model that can be tackled on a realistic instance by our moving-horizon approach.
Industrial Mathematics and Complex Systems, 2017
We consider optimal control problems for the flow of gas or fresh water in pipe networks as well ... more We consider optimal control problems for the flow of gas or fresh water in pipe networks as well as drainage or sewer systems in open canals. The equations of motion are taken to be represented by the nonlinear isothermal Euler gas equations, the water hammer equations, or the St. Venant equations for flow. We formulate model hierarchies and derive an abstract model for such network flow problems including pipes, junctions, and controllable elements such as valves, weirs, pumps, as well as compressors. We use the abstract model to give an overview of the known results and challenges concerning equilibria, well-posedness, controllability, and optimal control. A major challenge concerning the optimization is to deal with switching on-off states that are inherent to controllable devices in such applications combined with continuous simulation and optimization of the gas flow. We formulate the corresponding mixed-integer nonlinear optimal control problems and outline a decomposition approach as a solution technique.
Computational Management Science, 2016
The purpose of this erratum is to correct a signing error in the statement of the inner approxima... more The purpose of this erratum is to correct a signing error in the statement of the inner approximation of the second-order cone L n presented in Bärmann et al. (2016). In Bärmann et al. (2016), we developed a construction for the inner approximation of L n based on the ideas of Ben-Tal and Nemirovski (2001) and Glineur (2000). We showed-using the same decomposition as in the aforementioned papers-that it suffices to find an inner approximation of L 2 , which in turn can be obtained from an inner approximation of the unit ball B 2 ⊂ R 2. However, in the statement of the latter two approximations, there was a signing error which we would like to correct here.
Mathematical Methods of Operations Research, 2016
In this article we consider combinatorial markets with valuations only for singletons and pairs o... more In this article we consider combinatorial markets with valuations only for singletons and pairs of buy/sell-orders for swapping two items in equal quantity. We provide an algorithm that permits polynomial time market-clearing and-pricing. The results are presented in the context of our main application: the futures opening auction problem. Futures contracts are an important tool to mitigate market risk and counterparty credit risk. In futures markets these contracts can be traded with varying expiration dates and underlyings. A common hedging strategy is to roll positions forward into the next expiration date, however this strategy comes with significant operational risk. To address this risk, exchanges started to offer so-called futures contract combinations, which allow the traders for swapping two futures contracts with different expiration dates or for swapping two futures contracts with different underlyings. In theory, the price is in both cases the difference of the two involved futures contracts. However, in particular in the opening auctions price inefficiencies often occur due to suboptimal clearing, leading to potential arbitrage opportunities. We present a minimum cost flow formulation of the futures opening auction problem that guarantees consistent prices. The core ideas are to model orders as arcs in a network, to enforce the equilibrium conditions with the help of two hierarchical objectives, and to combine these objectives into a single weighted objective while preserving the Several of the presented results are taken from the Ph.D. thesis of J. C. Müller (2014). Research was supported by Deutsche Börse AG.
European Journal of Operational Research, 2016
In this paper we propose a three-level computational equilibrium model that allows to analyze the... more In this paper we propose a three-level computational equilibrium model that allows to analyze the impact of the regulatory environment on transmission line expansion (by the regulator) and investment in generation capacity (by private firms) in liberalized electricity markets. The basic model analyzes investment decisions of the transmission operator (TO) and private firms in expectation of an energy only market and cost-based redispatch. In different specifications we consider the cases of one versus two price zones (market splitting) and analyze different approaches to recover network cost, in particular lump sum, capacity based, and energy based fees. In order to compare the outcomes of our multi-stage market model with the first best benchmark, we also solve the corresponding integrated planer problem. In two simple test networks we illustrate that energy only markets can lead to suboptimal locational decisions for generation capacity and thus, imply excessive network expansion. Market splitting heals those problems only partially. Those results obtain for both, capacity and energy based network tariffs, although investment slightly differs across those regimes.
Computational Management Science, 2015
Robust optimization is an important technique to immunize optimization problems against data unce... more Robust optimization is an important technique to immunize optimization problems against data uncertainty. In the case of a linear program and an ellipsoidal uncertainty set, the robust counterpart turns into a second-order cone program. In this work, we investigate the efficiency of linearizing the second-order cone constraints of the latter. This is done using the optimal linear outer-approximation approach by Ben-Tal and Nemirovski [2001] from which we derive an optimal inner approximation of the second-order cone. We examine the performance of this approach on various benchmark sets including portfolio optimization instances as well as (robustified versions of) the MIPLIB and the SNDlib.
Mathematical Programming Computation, 2012
While semidefinite relaxations are known to deliver good approximations for combinatorial optimiz... more While semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection, their practical scope is mostly associated with small dense instances. For large sparse instances, cutting plane techniques are considered the method of choice. These are also applicable for semidefinite relaxations via the spectral bundle method, which allows to exploit structural properties like sparsity. In order to evaluate the relative strengths of linear and semidefinite approaches for large sparse instances, we set up a common branch-and-cut framework for linear and semidefinite relaxations of the minimum graph bisection problem. It incorporates separation algorithms for valid inequalities of the bisection cut polytope described in a recent study by the authors. While the problem specific cuts help to strengthen the linear relaxation significantly, the semidefinite bound profits much more from separating the cycle inequalities of the cut polytope on a slightly enlarged * A conference version of this article appeared as [5]. support. Extensive numerical experiments show that this semidefinite branch-andcut approach without problem specific cuts is a superior choice to the classical simplex approach exploiting bisection specific inequalities on a clear majority of our large sparse test instances from VLSI design and numerical optimization.
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Papers by Alexander Martin