90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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Bus rapid transit systems in developing and newly
industrialized countries often consist of a trunk with a path
topology. On this trunk, several overlapping lines are operated
which provide direct connections. The demand varies heavily over the
day, with morning and afternoon peaks typically in reverse
directions. We propose an integer programming
model for this problem, derive a structural property of line plans
in the static (or single period) ``unimodal demand'' case, and
consider approaches to the solution of the multi-period version that
rely on clustering the demand into peak and off-peak service
periods. An application to the Metrobüs system of Istanbul is
discussed.
Optimization-based bound tightening (OBBT) is one of the most effective procedures to reduce variable domains of nonconvex mixed-integer nonlinear programs (MINLPs). At the same time it is one of the most expensive bound tightening procedures, since it solves auxiliary linear programs (LPs)—up to twice the number of variables many. The main goal of this paper is to discuss algorithmic techniques for an efficient implementation of OBBT.
Most state-of-the-art MINLP solvers apply some restricted version of OBBT and it seems to be common belief that OBBT is beneficial if only one is able to keep its computational cost under control. To this end, we introduce three techniques to increase the efficiency of OBBT: filtering strategies to reduce the number of solved LPs, ordering heuristics to exploit simplex warm starts, and the generation of Lagrangian variable bounds (LVBs). The propagation of LVBs during tree search is a fast approximation to OBBT without the need to solve auxiliary LPs.
We conduct extensive computational experiments on MINLPLib2. Our results indicate that OBBT is most beneficial on hard instances, for which we observe a speedup of 17% to 19% on average. Most importantly, more instances can be solved when using OBBT.
Generic Branch-Cut-and-Price
(2010)
Modern MIP solving software
incorporates dozens of auxiliary algorithmic components for supporting
the branch-and-bound search in finding and improving solutions and in strengthening the relaxation.
Intuitively, a dynamic solving strategy with an appropriate emphasis on different solving components and strategies is desirable during the search process.
We propose an adaptive solver behavior that dynamically reacts
on transitions between the three typical phases of a MIP solving process:
The first phase objective is to find a feasible solution. During the second phase,
a sequence of incumbent solutions gets constructed
until the incumbent is eventually optimal. Proving
optimality is the central objective of the remaining third phase.
Based on the MIP-solver SCIP, we demonstrate
the usefulness of the phase concept both with an exact recognition of the optimality of a solution, and provide
heuristic alternatives to make use of the concept in practice.
The operation of a railway network as large as Deutsche Bahn's Intercity Express (ICE) hinges on a number of factors, such as the availability of personnel and the assignment of physical vehicles to a timetable schedule, a problem known as the rolling stock rotation problem (RSRP). In this paper, we consider the problem of creating an alternative timetable in the case that there is a long-term disruption, such as a strike, and the effects that this alternative timetable has on the resulting vehicle rotation plan. We define a priority measure via the Analytic Hierarchy Process (AHP) to determine the importance of each trip in the timetable and therefore which trips to cancel or retain. We then compare our results with those of a limited timetable manually designed by Deutsche Bahn (DB). We find that while our timetable results in a more expensive rotation plan, its flexibility lends itself to a number of simple improvements. Furthermore, our priority measure has the potential to be integrated into the rolling stock rotation optimization process, in particular, the Rotation Optimizer for Railways (ROTOR) software, via the cost function. Ultimately, our method provides the foundation for an automated way of creating a new timetable quickly, and potentially in conjunction with a new rotation plan, in the case of a limited scenario.
This paper describes how we solved 12 previously unsolved mixed-integer program-
ming (MIP) instances from the MIPLIB benchmark sets. To achieve these results we
used an enhanced version of ParaSCIP, setting a new record for the largest scale MIP
computation: up to 80,000 cores in parallel on the Titan supercomputer. In this paper
we describe the basic parallelization mechanism of ParaSCIP, improvements of the
dynamic load balancing and novel techniques to exploit the power of parallelization
for MIP solving. We give a detailed overview of computing times and statistics for
solving open MIPLIB instances.
We present two algorithms to solve a 3-objective optimization problem arising in telecommunications access network planning, the k-Architecture Connected Facility Location Problem. The methods can also be used to solve any 3-objective integer linear programming model and can be extended to the multiobjective case. We give some exemplary computations using small and medium-sized instances for our problem.
Duty rostering problems occur in different application contexts and come in different flavors. They give rise to very large scale integer programs which ypically have lots of solutions and extremely fractional LP relaxations. In such a situation, heuristics
can be a viable algorithmic choice. We propose an mprovement method of the Lin-Kernighan type for the solution of duty rostering problems. We illustrate its versatility and solution quality on three different applications in public transit, vehicle routing, and
airline rostering with a focus on the management of preferences, fairness, and fatigue, respectively.
Railway transportation and in particular train timetabling is one of the basic and source application areas of combinatorial optimization and integer programming. We will discuss two well established modeling techniques for the train timetabling problem. In this paper we focus on one major ingredient - the bounding by dual relaxations. We compare two classical dual relaxations of large scale time expanded train timetabling problems - the Lagrangean Dual and Lagrangean Decomposition. We discuss the convergence behavior and show limitations of the Lagrangean Decomposition approach for a configuration based model. We introduce a third dualization approach to overcome those limitations. Finally, we present promising preliminary computational experiments that show that our new approach indeed has superior convergence properties.
Rising traffic in telecommunication networks lead to rising energy costs for the network operators. Meanwhile, increased flexibility of the networking hardware may help to realize load-adaptive operation of the networks to cut operation costs. To meet network operators’ concerns over stability, we propose to switch network configurations only a limited number of times per day. We present a method for the integrated computation of optimal switching times and network configurations that alternatingly solves mixed-integer programs and constrained shortest cycle problems in a certain graph. Similarly to the Branch & Bound Algorithm, it uses lower and upper bounds on the optimum value and allows for pivoting strategies to guide the computation and avoid the solution of irrelevant subproblems. The algorithm can act as a framework to be adapted and applied to suitable problems of different origin.
We study an extension of the shortest path network interdiction problem and present a novel real-world application in this area. We consider the problem of determining optimal locations for toll control stations on the arcs of a transportation network. We handle the fact that drivers can avoid control stations on parallel secondary roads. The problem is formulated as a mixed integer program and solved using Benders decomposition. We present experimental results for the application of our models to German motorways.
The selection of a good branching variable is crucial for small search trees in Mixed Integer Programming. Most modern solvers employ a strategy guided by history information, mainly the variable pseudo-costs, which are used to estimate the objective gain. At the beginning
of the search, such information is usually collected via an expensive look-ahead strategy called strong-branching until variables are considered reliable.
The reliability notion is thereby mostly based on fixed-number thresholds, which may lead to ineffective branching decisions on problems with highly varying objective gains.
We suggest two new notions of reliability motivated by mathematical statistics that take into account the sample variance of the past observations on each variable individually. The first method prioritizes additional strong-branching look-aheads on variables whose pseudo-costs
show a large variance by measuring the relative error of a pseudo-cost confidence interval. The second method performs a two-sample Student-t test for filtering branching candidates with a high probability to be better than the best history candidate.
Both methods were implemented in the MIP-solver SCIP and computational results on standard MIP test sets are presented.
Recently, there have been many successful applications of optimization algorithms that solve a sequence of quite similar mixed-integer programs (MIPs) as subproblems. Traditionally, each problem in the sequence is solved from scratch. In this paper we consider reoptimization techniques that try to benefit from information obtained by solving previous problems of the sequence. We focus on the case that subsequent MIPs differ only in the objective function or that the feasible region is reduced. We propose extensions of the very complex branch-and-bound algorithms employed by general MIP solvers based on the idea to ``warmstart'' using the final search frontier of the preceding solver run. We extend the academic MIP solver SCIP by these techniques to obtain a reoptimizing branch-and-bound solver and report computational results which show the effectiveness of the approach.
We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery.
We apply customized versions of the ε-constraint Method and the Two-Phase Method to a problem originating in access network planning. We introduce various notions of quality measures for approximated/partial sets of nondominated points, utilizing the concept of hypervolume for biobjective problems. We report on computations to assess the performance of the two methods in terms of these measures.
Planning and operating railway transportation systems is an extremely
hard task due to the combinatorial complexity of the underlying discrete
optimization problems, the technical intricacies, and the immense size of
the problem instances. Because of that, however, mathematical models
and optimization techniques can result in large gains for both railway cus-
tomers and operators, e.g., in terms of cost reductions or service quality
improvements. In the last years a large and growing group of researchers
in the OR community have devoted their attention to this domain devel-
oping mathematical models and optimization approaches to tackle many
of the relevant problems in the railway planning process. However, there
is still a gap to bridge between theory and practice, with
a few notable exceptions. In this paper we address three success stories,
namely, long-term freight train routing (part I), mid-term rolling stock
rotation planning (part II), and real-time train dispatching (part III). In
each case, we describe real-life, successful implementations. We will dis-
cuss the individual problem setting, survey the optimization literature,
and focus on particular aspects addressed by the mathematical models.
We demonstrate on concrete applications how mathematical optimization
can support railway planning and operations. This gives proof that math-
ematical optimization can support the planning of rolling stock resources.
Thus, mathematical models and optimization can lead to a greater effi-
ciency of railway operations and will serve as a powerful and innovative
tool to meet recent challenges of the railway industry.
We prove a mathematical programming characterisation of approximate partial D-optimality under general linear constraints. We use this characterisation with a branch-and-bound method to compute a list of all exact D-optimal designs for estimating a pair of treatment contrasts in the presence of a nuisance time trend up to the size of 24 consecutive trials.
Energy storages can be of great value when added to power grids. They introduce the possibility to store and release energy whenever this is favorable. This is particularly relevant, for example, if power supply is volatile (as is the case with renewable energy) and the network is small (so that there are few other nodes that might balance fluctuations in consumption or production). We present models and methods from mathematical optimization for computing an optimized storage schedule for this purpose. We look at alternative optimization objectives, such as smallest possible peak load, low energy costs, or the close approximation of a prescribed load curve. The optimization needs to respect general operational and economic constraints as well as limitations in the use of storage, which are imposed by the chosen storage technology. We therefore introduce alternative approaches for modeling the non-linear properties of energy storages and study their impact on the efficiency of the optimization process. Finally, we present a computational study with batteries as storage devices. We use this to highlight the trade-off between solution quality and computational tractability. A version of the model for the purpose of leveling peaks and instabilities has been implemented into a control system for an office-building smart grid scenario.
One quarter of Europe’s energy demand is provided by natural gas distributed through a vast pipeline network covering the whole of Europe. At a cost of 1 million Euros per kilometer the extension of the European pipeline network is already a multi billion Euro business. Therefore, automatic planning tools that support the decision process are desired. We model the topology extension problem in gas networks by a mixed-integer nonlinear program (MINLP). This gives rise to a so-called active transmission problem, a continuous nonlinear non-convex feasibility problem which emerges from the MINLP model by fixing all integral variables. In this article we offer novel sufficient conditions for proving the infeasibility of this active transmission problem. These conditions can be expressed in the form of a mixed-integer program (MILP), i.e., the infeasibility of a non-convex NLP can be certified by solving an MILP. These results provide an efficient bounding procedure in a branch-and-bound algorithm. Our computational results demonstrate a substantial speed-up for the necessary computations.
The Graduate-Level Research in Industrial Projects (G-RIPS) Program provides an
opportunity for high-achieving graduate-level students to work in teams on a
real-world research project proposed by a sponsor from industry or the public
sector. Each G-RIPS team consists of four international students (two from
the US and two from European universities), an academic mentor, and an industrial sponsor.
This is the report of the Rail-Lab project on the definition and integration of
robustness aspects into optimizing rolling stock schedules. In general, there is
a trade-off for complex systems between robustness and efficiency. The ambitious
goal was to explore this trade-off by implementing numerical simulations and
developing analytic models.
In rolling stock planning a very large set of industrial railway requirements,
such as vehicle composition, maintenance constraints, infrastructure capacity,
and regularity aspects, have to be considered in an integrated model. General
hypergraphs provide the modeling power to tackle those requirements.
Furthermore, integer programming approaches are able to produce high quality
solutions for the deterministic problem.
When stochastic time delays are considered, the mathematical programming problem
is much more complex and presents additional challenges. Thus, we started with a
basic variant of the deterministic case, i.e., we are only considering
hypergraphs representing vehicle composition and regularity.
We transfered solution approaches for robust optimization
from the airline industry to the setting of railways and attained a
reasonable measure of robustness. Finally, we present and discuss different
methods to optimize this robustness measure.
The Cycle Embedding Problem
(2014)
Given two hypergraphs, representing a fine and a coarse "layer", and a cycle cover of the nodes of the coarse layer, the cycle embedding problem (CEP) asks for an embedding of the coarse cycles into the fine layer. The CEP is NP-hard for general hypergraphs, but it can be solved in polynomial time for graphs. We propose an integer rogramming formulation for the CEP that provides a complete escription of the CEP polytope for the graphical case. The CEP comes up in railway vehicle rotation scheduling. We present computational results for problem instances of DB Fernverkehr AG that justify a sequential coarse-first-fine-second planning approach.
We propose a new coarse-to-fine approach to solve certain linear programs by column generation. The problems that we address contain layers corresponding to different levels of detail, i.e., coarse layers as well as fine layers. These layers are utilized to design
efficient pricing rules. In a nutshell, the method shifts the pricing of a fine linear program to a coarse counterpart. In this way, major decisions are taken in the coarse layer, while minor
details are tackled within the fine layer. We elucidate our methodology by an application to a complex railway rolling stock rotation problem. We provide comprehensive computational results that demonstrate the benefit of this new technique for the solution of large scale problems.
We study System Dynamics models with several free parameters that can be altered by the user. We assume that the user's goal is to achieve a certain dynamic behavior of the model by varying these parameters. In order to the find best possible combination of parameter settings, several automatic parameter tuning methods are described in the literature and readily available within existing System Dynamic software packages. We give a survey on the available techniques in the market and describe their theoretical background. Some of these methods are already six decades old, and meanwhile newer and more powerful optimization methods have emerged in the mathematical literature. One major obstacle for their direct use are tabled data in System Dynamics models, which are usually interpreted as piecewise linear functions. However, modern optimization methods usually require smooth functions which are twice continuously differentiable. We overcome this problem by a smooth spline interpolation of the tabled data. We use a test set of three complex System Dynamic models from the literature, describe their individual transition into optimization problems, and demonstrate the applicability of modern optimization algorithms to these System Dynamics Optimization problems.
We provide an overview of new theoretical results that we obtained while further investigating multiband robust optimization, a new model for robust optimization that we recently proposed to tackle uncertainty in mixed-integer linear programming. This new model extends and refines the classical Gamma-robustness model of Bertsimas and Sim and is particularly useful in the common case of arbitrary asymmetric distributions of the uncertainty. Here, we focus on uncertain 0-1 programs and we analyze their robust counterparts when the uncertainty is represented through a multiband set. Our investigations were inspired by the needs of our industrial partners in the research project ROBUKOM.
We investigate the Robust Multiperiod Network Design Problem, a generalization of the classical Capacitated Network Design Problem that additionally considers multiple design periods and provides solutions protected against traffic uncertainty.
Given the intrinsic difficulty of the problem, which proves challenging even for state-of-the art commercial solvers, we propose a hybrid primal heuristic based on the combination of ant colony optimization and an exact large neighborhood search. Computational experiments on a set of realistic instances from the SNDlib show that our heuristic can find solutions of extremely good quality with low optimality gap.
A central assumption in classical optimization is that all the input data of a
problem are exact. However, in many real-world problems, the input data are subject
to uncertainty. In such situations, neglecting uncertainty may lead to nominally optimal
solutions that are actually suboptimal or even infeasible. Robust optimization
offers a remedy for optimization under uncertainty by considering only the subset of
solutions protected against the data deviations. In this paper, we provide an overview
of the main theoretical results of multiband robustness, a new robust optimization
model that extends and refines the classical theory introduced by Bertsimas and Sim.
After introducing some new results for the special case of pure binary programs, we
focus on the harvest scheduling problem and show how multiband robustness can
be adopted to tackle the uncertainty affecting the volume of produced timber and
grant a reduction in the price of robustness.
We present a novel heuristic algorithm to identify feasible solutions of a mixed-integer nonlinear programming problem arising in natural gas transportation: the selection of new pipelines to enhance the network's capacity to a desired level in a cost-efficient way. We solve this problem in a linear programming based branch-and-cut approach, where we deal with the nonlinearities by linear outer approximation and spatial branching. At certain nodes of the branching tree, we compute a KKT point for a nonlinear relaxation. Based on the information from the KKT point we alter some of the integer variables in a locally promising way. We describe this heuristic for general MINLPs and then show how to tailor the heuristic to exploit our problem-specific structure. On a test set of real-world instances, we are able to increase the chance of identifying feasible solutions by some order of magnitude compared to standard MINLP heuristics that are already built in the general-purpose MINLP solver SCIP.
We consider a system dynamics model that describes the effect of human activity on natural resources. The central stocks are the accumulated profit, the industry structures, and the water resources. The model can be controlled through two time-dependent parameters. The goal in this paper is to find a parameter setting that leads to a maximization of a performance index, which reflects both environmental and economic aspects. Thus, the goal is to identify the most sustainable stock of industry structures within the model's constraints and assumptions. In order to find a proven global optimal parameter set, we formulate the System Dynamics Optimization model as a mixed-integer nonlinear problem that is accessible for numerical solvers. Due to the dynamic structure of the model, certain steps of the solution process must be handled with greater care, compared to standard non-dynamic problems. We describe our approach of solving the industry structure model and present computational results. In addition, we discuss the limitations of the approach and next steps.
A common technique in the solution of large or complex optimization problems is the use of micro-macro transformations. In this paper, we carry out a theoretical analysis of such transformations for the track allocation problem in railway networks. We prove that the cumulative rounding technique of Schlechte et al. satisfies two of three natural optimality criteria and that this performance cannot be improved. We also show that under extreme circumstances, this technique can perform inconvieniently by underestimating the global optimal value.
We provide an overview of our main results about studying Linear Programming Problems whose coefficient matrix is subject to uncertainty and the uncertainty is modeled through a multi-band set. Such an uncertainty set generalizes the classical one proposed by Bertsimas and Sim and is particularly suitable in the common case of arbitrary non-symmetric distributions of the parameters. Our investigations were inspired by practical needs of our industrial partner in ongoing projects with focus on the design of robust telecommunications networks.
The rapid technological evolution of telecommunication networks demands service providers to regularly update their technology, with the aim of remaining competitive in the marketplace. However, upgrading the technology in a network is not a trivial task. New hardware components need to be installed in the network and during the installation network connectivity may be temporarily compromised. The Wavelength Division Multiplexing (WDM) technology, whose upgrade is considered in here, shares fiber links among several optical connections and tearing down a single link may disrupt several optical connections at once. When the upgrades involve large parts of a network, typically not all links can be upgraded in parallel, which may lead to an unavoidable longer disruption of some connections. A bad scheduling of the overall endeavor, however, can dramatically increase the disconnection time of parts of the networks, causing extended service disruption. In this contribution, we study the problem of finding a schedule of the fiber link upgrades that minimizes the total service disruption time. To the best of our knowledge, this problem has not yet been formalized and investigated. The aim of our work is to close this gap by presenting a mathematical optimization model for the problem and an innovative solution algorithm that tackles the intrinsic difficulties of the problem. Computational experience on realistic instances completes our study. Our original investigations have been driven by real needs of DFN, operator of the German National Research and Education Network and our partner in the BMBF research project ROBUKOM (http://www.robukom.de/).
We propose a novel optimization model for resource assignment in heterogeneous wireless network. The model adopts two objective functions maximizing the number of served users and the minimum granted utility at once. A distinctive feature of our new model is to consider two consecutive time slots, in order to include handover as an additional decision dimension.
Furthermore, the solution algorithm that we propose refines a heuristic solution approach recently proposed in literature, by considering a real joint optimization of the considered resources. The simulation study shows that the new model leads to a significant reduction in handover frequency, when compared to a traditional scheme based on maximum SNR.
Recently, parallel computing environments have become significantly popular. In order to obtain the benefit of using parallel computing environments, we have to deploy our programs for these effectively. This paper focuses on a parallelization of SCIP (Solving Constraint Integer Programs), which is a MIP solver and constraint integer programming framework available in source code. There is a parallel extension of SCIP named ParaSCIP, which parallelizes SCIP on massively parallel distributed memory computing environments. This paper describes FiberSCIP, which is yet another parallel extension of SCIP to utilize multi-threaded parallel computation on shared memory computing environments, and has the following contributions: First, the basic concept of having two parallel extensions and the relationship between them and the parallelization framework provided by UG (Ubiquity Generator) is presented, including an implementation of deterministic parallelization. Second, the difficulties to achieve a good performance that utilizes all resources on an actual computing environment and the difficulties of performance evaluation of the parallel solvers are discussed. Third, a way to evaluate the performance of new algorithms and parameter settings of the parallel extensions is presented. Finally, current performance of FiberSCIP for solving mixed-integer linear programs (MIPs) and mixed-integer non-linear programs (MINLPs) in parallel is demonstrated.
The Rolling Stock Rotation Problem is to schedule rail vehicles in order to cover timetabled trips by a cost optimal set of vehicle rotations. The problem integrates several facets of railway optimization, i.e., vehicle composition, maintenance constraints, and regularity aspects.
In industrial applications existing schedules often have to be re-optimized to integrate timetable changes or construction sites.
We present an integrated modeling and algorithmic approach for this task as well as computational results for industrial problem instances of DB Fernverkehr AG.
The planning of a communication network is inevitably depending on the quality of both the planning tool and the demand forecast used. In this article, we show exemplarily how the emerging area of Robust Optimization can advance the network planning by a more accurate mathematical description of the demand uncertainty. After a general introduction of the concept and its application to a basic network design problem, we present two applications: multi-layer and mixed-line-rate network design. We conclude with a discussion of extensions of the robustness concept to increase the accuracy of handling uncertainties.
This thesis represents a game-theoretic investigation of the allocation of inspectors in a transportation network, comparing Nash and Stackelberg equilibrium strategies to a strategy in which inspections are conducted proportionally to the traffic volume. It contains specifications for the integration of space and time dependencies and extensive experimental tests for the application on the transportation network of German motorways using real data. Main results are that - although the formulated spot-checking game is not zero-sum - we are able to compute a Nash equilibrium using linear programming and secondly, that experimental results yield that a Nash equilibrium strategy represents a good trade-off for the Stackelberg equilibrium strategy between efficiency of controls and computation time.
We consider the following freight train routing problem (FTRP). Given is a
transportation network with fixed routes for passenger trains and a
set of freight trains (requests), each defined by an origin and
destination station pair. The objective is to calculate a feasible
route for each freight train such that a sum of all expected delays and
all running times is minimal. Previous research concentrated on
microscopic train routings for junctions or inside major stations. Only
recently approaches were developed to tackle larger corridors or even
networks. We investigate the routing problem from a strategic
perspective, calculating the routes in a macroscopic transportation
network of Deutsche Bahn AG. Here macroscopic refers to an aggregation of
complex real-world structures are into fewer network elements. Moreover, the
departure and arrival times of freight trains are approximated.
The problem has a strategic
character since it asks only for a coarse routing through the network
without the precise timings. We give a mixed-integer nonlinear programming~(MINLP)
formulation for FTRP, which is a multi-commodity flow model on a time-expanded
graph with additional routing constraints. The model's nonlinearities are due to
an algebraic approximation of the delays of the trains on the arcs of
the network
by capacity restraint functions. The MINLP is reduced to a mixed-integer linear model~(MILP)
by piecewise linear approximation. The latter is solved by a state of the art MILP solver for various real-world test instances.
We consider a nonlinear nonconvex network flow problem that arises, for example, in natural gas or water 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. The problem is how to numerically compute this flow and pressures. We review an existing approach of Maugis (1977) and extend it to the case of networks with active elements (for example, compressors). We further examine different ways of relaxations for the nonlinear network flow model. We compare different approaches based on nonlinear optimization numerically on a set of test instances.
In this paper we study the cost-optimal deployment of optical access networks considering variants of the problem such as fiber to the home (FTTH), fiber to the building (FTTB), fiber to the curb (FTTC), or fiber to the neighborhood (FTTN). We identify the combinatorial structures of the most important sub-problems arising in this area and model these, e.g., as capacitated facility location, concentrator location, or Steiner tree problems. We discuss modeling alternatives as well. We finally construct a “unified” integer programming model that combines all sub-models and provides a global view of all these FTTx problems. We also summarize computational studies of various special cases.
We introduce a new variant of the connected facility location problem that allows for modeling mixed deployment strategies (FTTC/FTTB/FTTH) in the design of local access telecommunication networks. Several mixed integer programming models and valid inequalities are presented. Computational studies on realistic instances from three towns in Germany are provided.
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.