90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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We study online multicommodity minimum cost routing problems in networks, where commodities have to be routed sequentially. Arcs are equipped with load dependent price functions defining the routing weights. We discuss an online algorithm that routes each commodity by minimizing a convex cost function that depends on the demands that are previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is $4K/2+K$-competitive, where $K$ is the number of commodities. For the parallel arc case this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably.
In this paper, we study the efficiency of Nash equilibria for a sequence of nonatomic routing games. We assume that the games are played consecutively in time in an online fashion: by the time of playing game $i$, future games $i+1,\dots,n$ are not known, and, once players of game $i$ are in equilibrium, their corresponding strategies and costs remain fixed. Given a sequence of games, the cost for the sequence of Nash equilibria is defined as the sum of the cost of each game. We analyze the efficiency of a sequence of Nash equilibria in terms of competitive analysis arising in the online optimization field. Our main result states that the online algorithm $\sl {SeqNash}$ consisting of the sequence of Nash equilibria is $\frac{4n}{2+n}$-competitive for affine linear latency functions. For $n=1$, this result contains the bound on the price of anarchy of $\frac{4}{3}$ for affine linear latency functions of Roughgarden and Tardos [2002] as a special case. Furthermore, we analyze a problem variant with a modified cost function that reflects the total congestion cost, when all games have been played. In this case, we prove an upper bound of $\frac{4n}{2+n}$ on the competitive ratio of $\sl {SeqNash}$. We further prove a lower bound of $\frac{3n-2}{n}$ of $\sl {SeqNash}$ showing that for $n=2$ our upper bound is tight.
In this paper we study online multicommodity routing problems in networks, in which commodities have to be routed sequentially. The flow of each commodity can be split on several paths. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes each commodity by minimizing a convex cost function that only depends on the demands previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is 4K2 (1+K)2 -competitive, where K is the number of commodities. For the single-source single-destination case, this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably.
Generic Branch-Cut-and-Price
(2010)
Large neighborhood search (LNS) heuristics are an important component of modern branch-and-cut algorithms for solving mixed-integer linear programs (MIPs). Most of these LNS heuristics use the LP relaxation as the basis for their search, which is a reasonable choice in case of MIPs. However, for more general problem classes, the LP relaxation alone may not contain enough information about the original problem to find feasible solutions with these heuristics, e.g., if the problem is nonlinear or not all constraints are present in the current relaxation.
In this paper, we discuss a generic way to extend LNS heuristics that have been developed for MIP to constraint integer programming (CIP), which is a generalization of MIP in the direction of constraint programming (CP). We present computational results of LNS heuristics for three problem classes: mixed-integer quadratically constrained programs, nonlinear pseudo-Boolean optimization instances, and resource-constrained project scheduling problems. Therefore, we have implemented extended versions of the following LNS heuristics in the constraint integer programming framework SCIP: Local Branching, RINS, RENS, Crossover, and DINS. Our results indicate that a generic generalization of LNS heuristics to CIP considerably improves the success rate of these heuristics.
Convergence Analysis of Smoothing Methods for Optimal Control of Stationary Variational Inequalities
(2011)
In the article an optimal control problem subject to a stationary variational inequality
is investigated. The optimal control problem is complemented with pointwise control constraints.
The convergence of a smoothing scheme is analyzed. There, the variational inequality
is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal
control problem converge to solutions of the original one. Passing to the limit in the
optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem.
Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained.
These rates coincide with rates obtained by numerical experiments, which are included in the paper.
Many practically relevant problems can be formulated in terms of a mixed integer programming (MIP) model. MIP denotes the
optimization of a linear objective function under a certain number of linear side constraints including the need for
some of the involved variables to take integral solution values.
Applications of MIP based optimization can be found in the area of public transit,
scheduling, automatic vehicle routing,
network design, etc.
From a complexity point of view, MIP solving is known to be NP-hard and most commonly tried to be solved via
Branch-and-Bound based algorithms. Branch-and-Bound algorithms benefit from early and good feasible solutions of a MIP
in various ways.
Primal heuristics are aimed at finding new solutions during the MIP solving process. There are different types of primal heuristics:
while start heuristics are particularly
valuable to find an early solution, improvement heuristics hopefully drive a given solution further towards optimality.
This thesis focusses on primal heuristics which are part of the MIP-solving framework SCIP.
The first chapter comes with basic definitions and a brief description of SCIP and the test set which we used.
The remainder of the first chapter is an overview of the existing heuristics in SCIP which have been implemented by Achterberg
and Berthold.
In the following chapters we introduce three new heuristics which apply rounding or propagation techniques for their specific purpose,
namely the new rounding heuristic ZI Round, taken from Wallace, a 2-Opt improvement
heuristic for MIP and the propagation heuristic Shift-and-Propagate.
It is characteristic of all three heuristics that they mainly apply computationally inexpensive algorithms.
Each of them is presented in an own chapter, starting with an algorithmic description, followed by implementational details.
All chapters close with a discussion of the computational results obtained with the respective implementations in SCIP.
We propose a model for the integrated optimization of vehicle
rotations and vehicle compositions in long distance railway passenger
transport. The main contribution of the paper is a hypergraph model
that is able to handle the challenging technical requirements as
well as very general stipulations with respect to the ``regularity''
of a schedule. The hypergraph model directly generalizes network
flow models, replacing arcs with hyperarcs. Although NP-hard in
general, the model is computationally well-behaved in practice. High
quality solutions can be produced in reasonable time using high
performance Integer Programming techniques, in particular, column
generation and rapid branching. We show that, in this way,
large-scale real world instances of our cooperation partner DB
Fernverkehr can be solved.
The steel mill slab design problem from the CSPLIB is a combinatorial
optimization problem motivated by an application of the steel industry. It
has been widely studied in the constraint programming community. Several
methods were proposed to solve this problem. A steel mill slab library was
created which contains 380 instances. A closely related binpacking problem
called the multiple knapsack problem with color constraints, originated
from the same industrial problem, was discussed in the integer programming
community. In particular, a simple integer program for this problem has
been given by Forrest et al. The aim of this paper is to bring these
different studies together. Moreover, we adapt the model of Forrest et
al. for the steel mill slab design problem. Using this model and a
state-of-the-art integer program solver all instances of the steel mill
slab library can be solved efficiently to optimality. We improved,
thereby, the solution values of 76 instances compared to previous results.
Finally, we consider a recently introduced variant of the steel mill slab
design problem, where within all solutions which minimize the leftover one
is interested in a solution which requires a minimum number of slabs. For
that variant we introduce two approaches and solve all instances of the
steel mill slab library with this slightly changed objective function to
optimality.
Today the railway timetabling process and the track allocation
is one of the most challenging problems to solve by a railway infrastructure provider.
Especially due to the deregulation of the transport market in the recent years several
suppliers of railway traffic have entered the market. This leads to an increase of slot requests
and then it is natural that conflicts occur among them.
Furthermore, railway infrastructure networks consist of very expensive assets, even more
they are rigid due to the long-term upgrade process.
In order to make best use of these valuable infrastructure and to ensure economic operation,
efficient planning of the railway operation is indispensable.
Mathematical optimization models and algorithmic methodology can help
to automatize and tackle these challenges.
Our contribution in this paper is to present a renewed planning process
due to the liberalization in Europe and a general framework to
support the integration of simulation and optimization for
railway capacity allocation.
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 $1$.
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.
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.
Das vom BMBF geförderte Projekt FTTX-PLAN entwickelt mathematische Modelle und Optimierungsverfahren, um automatisiert kostenoptimierte FTTx-Netze berechnen zu können. Wir zeigen anhand einer Praxisstudie in Zusammenarbeit mit der Regensburger R-KOM, wie ein Planer von diesen Verfahren profitieren kann, um die Auswirkungen bestimmter Entscheidungen auf die Netzstruktur und -kosten zu untersuchen. Wir illustrieren dies am Beispiel eines FTTB/FTTH-Vergleichs, der Variation von Kundenanbindungsraten und der gezielten Ausnutzung existierender Leerrohre, um Tiefbau zu vermeiden.
In this paper we assess to which extent trenching costs of an FTTx network are unavoidable, even if technical side constraints are neglected. For that purpose we present an extended Steiner tree model. Using a variety of realistic problem instances we demonstrate that the total trenching cost can only be reduced by about 5 percent in realistic scenarios. This work has been funded by BMBF (German Federal Ministry of Education and Research) within the program "KMU-innovativ".
In the past few years several applications of optimal
experimental designs have emerged to optimize the measurements
in communication networks. The optimal design problems arising from
this kind of applications share three interesting properties:
(i) measurements are only available at a small number of locations of the network;
(ii) each monitor can simultaneously measure several quantities, which
can be modeled by ``multiresponse experiments";
(iii) the observation matrices depend on the topology of the network.
In this paper, we give an overview of these experimental design
problems and recall recent results for the computation of optimal
designs by Second Order Cone Programming (SOCP). New results for the
network-monitoring of a discrete time process are presented. In particular, we show
that the optimal design problem for the monitoring of an AR1 process can be reduced
to the standard form and we give experimental results.
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.