Bilevel Programming Problems Theory, Algorithms and Applications to Energy Networks

332 Pages·2015·5.883 MB·English

by  Dempe, Stephan;Kalashnikov, Vyacheslav;Kalashnykova, Nataliya;Pérez-Valdés, Gerardo A

Energy Systems
Stephan Dempe
Vyacheslav Kalashnikov
Gerardo A. Pérez-Valdés
Nataliya Kalashnykova
Bilevel
Programming
Problems
Theory, Algorithms and Applications to
Energy Networks
Energy Systems
Series editor
Panos M. Pardalos, Gainesville, USA
More information about this series at http://www.springer.com/series/8368
Stephan Dempe Vyacheslav Kalashnikov
(cid:129)
é é
Gerardo A. P rez-Vald s
(cid:129)
Nataliya Kalashnykova
Bilevel Programming
Problems
Theory, Algorithms and Applications
to Energy Networks
123
StephanDempe Gerardo A.Pérez-Valdés
Fak.für Mathematik undInformatik Department of IndustrialEconomics
TU Bergakademie Freiberg Norwegian University ofScience and
Freiberg,Sachsen Technology(NTNU)
Germany Trondheim
Norway
Vyacheslav Kalashnikov
Department of Systemsand Industrial Nataliya Kalashnykova
Engineering Facultad CienciasFísico-Mathemáticas
Tecnologico deMonterrey (ITESM) (FCFM)
Monterrey, NL Universidad Autónomade NuevoLeón
Mexico (UANL)
San Nicolásde losGarza
and Mexico
Department of SocialModeling
Central Economicsand Mathematics
Institute(CEMI)
RussianAcademy ofSciences
Moscow
RussianFederation
and
Deparment ofComputing
Sumy StateUniversity
Sumy
Ukraine
ISSN 1867-8998 ISSN 1867-9005 (electronic)
Energy Systems
ISBN 978-3-662-45826-6 ISBN 978-3-662-45827-3 (eBook)
DOI 10.1007/978-3-662-45827-3
LibraryofCongressControlNumber:2014958578
SpringerHeidelbergNewYorkDordrechtLondon
©Springer-VerlagBerlinHeidelberg2015
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Preface
Bileveloptimizationisavitalfieldofactiveresearch.Dependingonitsformulation
it is part of nonsmooth or nondifferentiable optimization, conic programming,
optimization with constraints formulated as generalized equations, or set-valued
optimization. The investigation of many practical problems as decision making in
hierarchicalstructures,orsituationswherethereactionofnatureonselectedactions
needstoberespected,initiatedmodelingthemasbileveloptimizationproblems.In
this way, new theories have been developed with new results obtained.
A first attempt was the use of the Karush-Kuhn-Tucker conditions in situations
when they are necessary and sufficient optimality conditions for the lower level
problem, or dual problems in case strong duality holds to model the bilevel opti-
mization problem. The result is a special case of the mathematical program with
equilibrium constraints (MPEC), or complementarity constraints (MPCC). The
latterhasmotivatedtheinvestigationofoptimalityconditionsandthedevelopment
of algorithms solving such problems. Unfortunately, it has been shown very
recently that stationary points of an MPEC need not be related to stationary solu-
tions of the bilevel optimization problem. Because of that, the solution algorithms
must select the Lagrange multipliers associated with the lower level problem very
carefully. Another option is to avoid the explicit use of Lagrange multipliers
resulting in the so-called primal KKT transformation, which is an optimization
problem with a generalized equation as the constraint. Violation of the constraint
qualifications,oftenusedtoverifytheoptimalityconditionsandconvergenceofthe
solution algorithms, at every feasible point are other challenges for research.
Theideaofusingtheoptimalvaluefunctionofthelowerlevelproblemtomodel
the bilevel optimization problem is perhaps self-explanatory. The result yet is a
nondifferentiable equality constraint. One promising approach here is based on
variational analysis, which is also exploited to verify the optimality conditions for
theMPCC.So,bileveloptimizationinitiatedsomeadvancesinvariationalanalysis,
too.
Applications often force the use of integer variables in the respective models.
Besides suitable formulations, mixed-integer bilevel optimization problems renew
the question of existence of an optimal solution, leading to the notion of a weak
v
vi Preface
solution. Surprisingly, adding some constraints that are inactive at a global opti-
mum of the continuous bilevel problem, as well as replacing a discrete bilevel
problem with its continuous relaxation can destroy the global optimality of a fea-
sible point.
Theseandotherquestionsarethetopicofthefirstpartofthemonograph.Inthe
second part, certain applications are carefully investigated, especially a natural gas
cash-out problem, an equilibrium problem in a mixed oligopoly, and a toll
assignment problem. For these problems, besides the formulation of solution
algorithms, results of the first numerical experiments with them are also reported.
Bilevel optimization is a quickly developing field of research with challenging
andpromisingcontributionsfromdifferenttopicsofmathematicslikeoptimization,
aswellasfromotherscienceslikeeconomics,engineering,orchemistry.Itwasnot
a possible aim of the authors to provide an overview of all the results available in
thisarea.Ratherthanthat,weintendedtoshowsomeinteractionswithothertopics
ofresearch,andtoformulateouropinionaboutsomedirectionsforexplorationsin
the future.
Stephan Dempe
Vyacheslav Kalashnikov
Gerardo A. Pérez-Valdés
Nataliya Kalashnykova
Acknowledgment
KalashnikovV.andKalashnykovaN.,wassupportedbytheSEP-CONACYTgrant
CB-2013-01-221676 (Mexico).
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Bilevel Optimization Problem. . . . . . . . . . . . . . . . . . . . . . 1
1.2 Possible Transformations into a One-Level Problem. . . . . . . . . . 2
1.3 An Easy Bilevel Optimization Problem: Continuous
Knapsack Problem in the Lower Level. . . . . . . . . . . . . . . . . . . 8
1.4 Short History of Bilevel Optimization. . . . . . . . . . . . . . . . . . . . 10
1.5 Applications of Bilevel Optimization . . . . . . . . . . . . . . . . . . . . 12
1.5.1 Optimal Chemical Equilibria. . . . . . . . . . . . . . . . . . . . . 12
1.5.2 Optimal Traffic Tolls. . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.3 Optimal Operation Control of a Virtual Power Plant . . . . 14
1.5.4 Spot Electricity Market with Transmission Losses . . . . . . 15
1.5.5 Discrimination Between Sets. . . . . . . . . . . . . . . . . . . . . 16
1.5.6 Support Vector Machines . . . . . . . . . . . . . . . . . . . . . . . 18
2 Linear Bilevel Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 21
2.1 The Model and First Properties . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Optimality Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Solution Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.1 Computation of a Local Optimal Solution. . . . . . . . . . . . 33
2.3.2 A Global Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Reduction of Bilevel Programming to a Single Level Problem . . . . 41
3.1 Different Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Parametric Optimization Problems . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Convex Quadratic Lower Level Problem. . . . . . . . . . . . . . . . . . 52
3.4 Unique Lower Level Optimal Solution . . . . . . . . . . . . . . . . . . . 54
3.4.1 Piecewise Continuously Differentiable Functions. . . . . . . 55
3.4.2 Necessary and Sufficient Optimality Conditions. . . . . . . . 59
3.4.3 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
ix
x Contents
3.5 The Classical KKT Transformation . . . . . . . . . . . . . . . . . . . . . 62
3.5.1 Stationary Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.2 Solution Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6 The Optimal Value Transformation . . . . . . . . . . . . . . . . . . . . . 84
3.6.1 Necessary Optimality Conditions. . . . . . . . . . . . . . . . . . 85
3.6.2 Solution Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.7 Primal KKT Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.8 The Optimistic Bilevel Programming Problem. . . . . . . . . . . . . . 100
3.8.1 One Direct Approach. . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.8.2 An Approach Using Set-Valued Optimization . . . . . . . . . 104
3.8.3 Optimality Conditions Using Convexificators . . . . . . . . . 113
4 Convex Bilevel Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1 Optimality Conditions for a Simple Convex
Bilevel Program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1.1 A Necessary but Not Sufficient Condition. . . . . . . . . . . . 117
4.1.2 Necessary Tools from Cone-Convex Optimization. . . . . . 119
4.1.3 A Solution Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2 A Penalty Function Approach to Solution of a Bilevel
Variational Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.2 An Existence Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 126
4.2.3 The Penalty Function Method. . . . . . . . . . . . . . . . . . . . 128
4.2.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Mixed-Integer Bilevel Programming Problems. . . . . . . . . . . . . . . . 133
5.1 Location of Integrality Conditions in the Upper or Lower
Level Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 Knapsack Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3 Weak Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.3.1 Regions of Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.3.2 Properties of the Solution Sets. . . . . . . . . . . . . . . . . . . . 143
5.3.3 Extended Solution Sets. . . . . . . . . . . . . . . . . . . . . . . . . 144
5.3.4 Solution Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.3.5 Weak Solution Functions . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.6 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3.7 Computation of Optimal Solutions. . . . . . . . . . . . . . . . . 156
5.4 Optimality Conditions Using a Radial-Directional Derivative. . . . 158
5.4.1 A Special Mixed-Discrete Bilevel Problem. . . . . . . . . . . 158
5.4.2 Some Remarks on the Sets ΨDðxÞ and RðyÞ. . . . . . . . . . 161
5.4.3 Basic Properties of ϕbðxÞ. . . . . . . . . . . . . . . . . . . . . . . . 163
5.4.4 The Radial-Directional Derivative . . . . . . . . . . . . . . . . . 166