90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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The System Dynamics (SD) methodology is a framework for modeling and simulating the dynamic behavior of socioeconomic systems. Characteristic for the description of such systems is the occurrence of feedback loops together with stocks and flows. The mathematical equations that describe the system are usually ordinary differential equations and nonlinear algebraic constraints. Therefore seemingly simple systems can show a nonintuitive, unpredictable behavior over time. Controlling a dynamical system means to specify potential interventions from outside that should keep the system on the desired track, and to define an evaluation schema to compare different controls among each other, so that a "best" control can be defined in a meaningful way. The central question is how to compute such globally optimal control for a given SD model, that allows the transition of the system into a desired state with minimum effort. We propose a mixed-integer nonlinear programming (MINLP) reformulation of the System Dynamics Optimization (SDO) problem. MINLP problems can be solved by linear programming based branch-and-bound approach. We demonstrate that standard MINLP solvers are not able to solve SDO problem. To overcome this obstacle, we introduce a special-tailored bound propagation method. We apply our new method to a predator-prey model with additional hunting activity as control, and to a mini-world model with the consumption level as control. Numerical results for these test cases are presented.
The Scenario Technique is a strategic planning method that aims to describe and analyze potential developments of a considered system in the future. Its application consists of several steps, from an initial problem analysis over an influence analysis to projections of key factors and a definition of the scenarios to a final interpretation of the results. The technique itself combines qualitative and quantitative methods and is an enhancement of the standard Scenario Technique. We use the numerical values gathered during the influence analysis, and embed them in a System Dynamics framework. This yields a mathematically rigorous way to achieve predictions of the system‘s future behavior from an initial impulse and the feedback structure of the factors. The outcome of our new method is a further way of projecting the present into the future, which enables the user of the Scenario Technique to obtain a validation of the results achieved by the standard method.
The System Dynamics (SD) methodology is a framework for modeling and simulating
the dynamic behavior of socioeconomic systems. Characteristic for the
description of such systems is the occurrence of feedback loops together with
stocks and flows. The mathematical equations that describe the system are
usually nonlinear. Therefore seemingly simple systems can show a nonintuitive,
nonpredictable behavior over time. Controlling a dynamical system means to
define a desired final state in which the system should be, and to specify
potential interventions from outside that should keep the system on the right
track. The central question is how to compute such globally optimal control for
a given SD model. We propose a branch-and-bound approach that is based on a
bound propagation method, primal heuristics, and spatial branching. We apply our
new SD-control method to a small System Dynamics model, that describes the
evolution of a social-economic system over time. We examine the problem of
steering this system on a sustainable consumption path.
Sports rankings are obtained by applying a system of rules to evaluate the
performance of the participants in a competition.
We consider rankings that result from assigning an ordinal rank to each
competitor according to their performance.
We develop an integer programming model for rankings that allows us
to calculate the number of points needed to guarantee
a team the ith position, as well as the minimum number of points
that could yield the ith place.
The model is very general and can thus be applied to many types of sports.
We discuss examples coming from football (soccer), ice hockey, and
Formula~1. We answer various questions and debunk a few myths along the way.
Are 40 points enough to avoid relegation in the Bundesliga?
Do 95 points guarantee the participation of a team in the NHL playoffs?
Moreover, in the season restructuration currently under consideration in the NHL,
will it be easier or harder to access the playoffs?
Is it possible to win the Formula~1 World Championship without winning at least one race
or without even climbing once on the podium?
Finally, we observe that the optimal solutions of the aforementioned model
are associated to extreme situations which are unlikely to happen. Thus,
to get closer to realistic scenarios, we enhance the model by adding some
constraints inferred from the results of the previous years.
Two fundamental mathematical formulations for railway timetabling are compared on a common set of sample problems, representing both multiple track high density services in Europe and single track bidirectional operations in North America. One formulation, ACP, enforces against conflicts by constraining time intervals between trains, while the other formulation, HGF, monitors physical occupation of controlled track segments. The results demonstrate that both ACP and HGF return comparable solutions in the aggregate, with some significant differences in select instances, and a pattern of significant differences in performance and constraint enforcement overall.
We consider reoptimization (i.e. the solution of a problem based on information available from solving a similar problem) for branch-and-bound algorithms and propose a generic framework to construct a reoptimizing branch-and-bound algorithm.
We apply this to an elevator scheduling algorithm solving similar subproblems to generate columns using branch-and-bound. Our results indicate that reoptimization techniques can substantially reduce the running times of the overall algorithm.
We consider a nonlinear nonconvex network design problem that arises in the extension of natural gas transmission networks. Given is such network with active and passive components, that is, valves, compressors, pressure regulators (active) and pipelines (passive), and a desired amount of flow at certain specified entry and exit nodes of the network. Besides flow conservation constraints in the nodes the flow must fulfill nonlinear nonconvex pressure loss constraints on the arcs subject to potential values (i.e., pressure levels) in both end nodes of each arc. Assume that there does not exist a feasible flow that fulfills all physical constraints and meets the desired entry and exit amounts. Then a natural question is where to extend the network by adding pipes in the most economic way such that this flow becomes feasible. Answering this question is computationally demanding because of the difficult problem structure. We use mixed-integer nonlinear programming techniques that rely on an outer approximation of the overall problem, and a branching on decision variables. We formulate a new class of valid inequalities (or cutting planes) which reduce the overall solution time when added to the formulation. We demonstrate the computational merits of our approach on test instances.
Strong branching is an important component of most variable selection rules in branch-and-bound based mixed-integer linear programming solvers.
It predicts the dual bounds of potential child nodes by solving auxiliary LPs and thereby helps to keep the branch-and-bound tree small.
In this paper, we describe how these dual bound predictions can be improved by including domain propagation into strong branching.
Computational experiments on standard MIP instances indicate that this is beneficial in three aspects: It helps to reduce the average number of LP iterations per strong branching call, the number of branch-and-bound nodes, and the overall solving time.
We present a new semidefinite representation for the trace of
a real function f applied to symmetric matrices, when a
semidefinite representation of the convex function f is known. Our construction
is intuitive, and yields a representation that is more compact than the previously known one.
We also show with the help of matrix geometric means and the Riemannian metric of the set of positive definite matrices
that for a rational number p in the interval (0,1],
the matrix X raised to the exponent p is the largest element
of a set represented by linear matrix inequalities.
We give numerical results for a problem inspired from the theory
of experimental designs, which show that the new semidefinite programming formulation
yields a speed-up factor in the order of 10.
PICOS is a user friendly interface
to several conic and integer programming solvers,
very much like YALMIP
under MATLAB.
The main motivation for PICOS is to have the possibility to
enter an optimization problem as a high level model,
and to be able to solve it with several different solvers.
Multidimensional and matrix variables are handled in a natural fashion,
which makes it painless to formulate a SDP or a SOCP.
This is very useful for educational purposes,
and to quickly implement some models and
test their validity on simple examples.
Furthermore, with PICOS you can take advantage of the
python programming language to read and write data,
construct a list of constraints by using python list comprehensions,
take slices of multidimensional variables, etc.
The industrial treatment of waste paper in order to regain valuable
fibers from which recovered paper can be produced, involves several
steps of preparation. One important step is the separation of stickies
that are normally attached to the paper. If not properly separated,
remaining stickies reduce the quality of the recovered paper or even
disrupt the production process. For the mechanical separation process
of fibers from stickies a separator screen is used. This machine has
one input feed and two output streams, called the accept and the
reject. In the accept the fibers are concentrated, whereas the reject
has a higher concentration of stickies. The machine can be controlled
by setting its reject rate. But even when the reject rate is set
properly, after just a single screening step, the accept still has too
many stickies, or the reject too many fibers. To get a proper
separation, several separators have to be assembled into a
network. From a mathematical point of view this problem can be seen as
a multi-commodity network flow design problem with a nonlinear,
controllable distribution function at each node. We present a
nonlinear mixed-integer programming model for the simultaneous
selection of a subset of separators, the network's topology, and the
optimal setting of each separator.
Numerical results are obtained via
different types of linearization of the nonlinearities and the use of
mixed-integer linear solvers, and compared with state-of-the-art
global optimization software.
The task of an elevator control is to schedule the elevators of a group such
that small waiting and travel times for the passengers are obtained. We present an exact
reoptimization algorithm for this problem. A reoptimization algorithm computes a
new schedule for the elevator group each time a new passenger arrives. Our algorithm
uses column generation techniques and is, to the best of our knowledge, the first exact
reoptimization algorithms for a group of passenger elevators. To solve the column
generation problem, we propose a Branch & Bound method.
This paper proposes a new method for probabilistic analysis of online algorithms. It is based on the notion of stochastic dominance. We develop the method for
the online bin coloring problem introduced by Krumke et al (2008). Using methods for the stochastic
comparison of Markov chains we establish the result that the performance of the online algorithm GreedyFit is stochastically better than the performance of the algorithm OneBin for any number of items processed. This result gives a more realistic
picture than competitive analysis and explains the behavior observed in simulations.
This paper provides a highly integrated solution approach for rolling stock
planning problems in the context of intercity passenger traffic. The main
contributions are a generic hypergraph based mixed integer programming
model and an integrated algorithm for the considered rolling stock rotation
planning problem. The new developed approach is able to handle a very large
set of industrial railway requirements, such as vehicle composition,
maintenance constraints, infrastructure capacity, and regularity aspects.
By the integration of this large bundle of technical railway aspects, we show
that our approach has the power to produce implementable rolling stock
rotations for our industrial cooperation partner DB Fernverkehr.
This is the first time that the rolling stock rotations at DB Fernverkehr
could be optimized by an automated system utilizing advanced mathematical
programming techniques.
Railway Track Allocation
(2012)
This article gives an overview of the results of the author's PhD thesis. The thesis deals with the
mathematical optimization for the efficient use of
railway infrastructure. We address the optimal allocation of the available
railway track capacity - the track allocation problem. This track allocation
problem is a major challenge for a railway company, independent of whether
a free market, a private monopoly, or a public monopoly is given. Planning
and operating railway transportation systems is extremely hard due to the
combinatorial complexity of the underlying discrete optimization problems,
the technical intricacies, and the immense sizes of the problem instances.
Mathematical models and optimization techniques can result in huge gains
for both railway customers and operators, e.g., in terms of cost reductions or
service quality improvements. We tackle this challenge by developing novel
mathematical models and associated innovative algorithmic solution methods
for large scale instances. We made considerable progress on solving track
allocation problems by two main features - a novel modeling approach for the
macroscopic track allocation problem and algorithmic improvements based on the
utilization of the bundle method. This allows us to produce for the first time reliable solutions for a real world instance, i.e., the Simplon corridor in Switzerland.
In this paper we give an analytical description on the structure of
solutions to the gas nomination validation problem in gas
transportation networks. These networks are assumed to contain no
active devices, only certain hypothetical pipelines, where the flow
of gas is modeled by a generalized version of the quadratic
Weymouth's equation. The purpose of considering generalized flow
formulas is to be able to adapt our results to various gas network
optimization problems involving gas flow formulas beyond Weymouth's
equation. Such formulas can appear in leaves of branch and bound
trees, or they can stem from discretization and linearization
carried out at active devices. We call a balanced supply-demand
vector a nomination, and the passive nomination validation problem
is to decide whether there exist pressures at the nodes generating a
given nomination. We prove that in our setup the pressure square
vectors generating a given nomination form a one-dimensional
connected and continuous curve in the pressure square space, and
this curve is a line for the classical Weymouth's equation. We also
present a visual approach for the easy comprehension of how this
solution curve arises; we give a short investigation of the set of
feasible nominations; and finally we give a proof that the
nomination validation problem in gas networks with active devices is
NP-complete.
In the last 20 years competitive analysis has become the main tool for
analyzing the quality of online algorithms. Despite of this,
competitive analysis has also been criticized: It sometimes cannot
discriminate between algorithms that exhibit significantly different
empirical behavior, or it even favors an algorithm that is worse from
an empirical point of view. Therefore, there have been several
approaches to circumvent these drawbacks. In this survey, we discuss
probabilistic alternatives for competitive analysis.
Die mittel- und längerfristige Planung für den Gastransport hat sich durch
Änderungen in den regulatorischen Rahmenbedingungen stark verkompliziert.
Kernpunkt ist die Trennung von Gashandel und -transport. Dieser Artikel
diskutiert die hieraus resultierenden mathematischen Planungsprobleme,
welche als Validierung von Nominierungen und Buchungen, Bestimmung der
technischen Kapazität und Topologieplanung bezeichnet werden. Diese
mathematischen Optimierungsprobleme werden vorgestellt und Lösungsansätze
skizziert.
We propose a game theoretic model for the spatial distribution of inspectors on a
transportation network.
The problem is to spread out the controls so as to enforce the payment of a transit
toll. We formulate a linear program to find
the control distribution which maximizes the expected toll revenue,
and a mixed integer program for the problem of minimizing
the number of evaders. Furthermore, we show that the problem of finding an optimal
mixed strategy for a coalition of $N$ inspectors can be solved
efficiently by a column generation procedure. Finally, we give experimental results
from an application to the truck toll on German motorways.
We study a family of combinatorial optimization problems
defined by a parameter $p\in[0,1]$, which involves spectral
functions applied to positive semidefinite matrices, and has
some application in the theory of optimal experimental design.
This family of problems tends to a generalization of the classical
maximum coverage problem as $p$ goes to $0$, and to a trivial instance
of the knapsack problem as $p$ goes to $1$.
In this article, we establish a matrix inequality which shows that the objective function is submodular for all $p\in[0,1]$, from which it follows
that the greedy approach, which has often been used for this problem, always gives a design within $1-1/e$ of the optimum.
We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors.
We prove an inequality which generalizes a classical result from the theory
of optimal designs, and allows us to give a rounding procedure with an approximation
factor which tends to $1$ as $p$ goes to $1$.