Explicit stationarity conditions and solution characterization for equilibrium problems with equilibrium constraints [Elektronische Ressource] / von Thomas Michael Surowiec

Thomas Michael Surowiec

Sujets

• Mathematics

Informations

• Publié par: humboldt-universitat_zu_berlin
• Publié le: 01 janvier 2010
• Nombre de lectures: 19
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Explicit Stationarity Conditions and
Solution Characterization for Equilibrium
Problems with Equilibrium Constraints
D I SS E RTATI O N
zur Erlangung des akademischen Grades
doctor rerum naturalium
im Fach Mathematik
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultät II
Humboldt-Universität zu Berlin
von
Herrn Thomas Michael Surowiec
geboren am 09.07.1982 in Passaic, New Jersey, USA
Präsident der Humboldt-Universität zu Berlin:
Prof. Dr. Dr. h.c. Christoph Markschies
Dekan der Mathematisch-Naturwissenschaftlichen Fakultät II:
Prof. Dr. Peter Frensch
Gutachter:
1. PD Dr. René Henrion
2. Prof. Dr. Werner Römisch
3. Prof. Dr. Jiří V. Outrata
eingereicht am: 12. August 2009
Tag der mündlichen Prüfung: 27. Januar 2010To my mother and father,
without whom this would
never have been possible.
iiiAcknowledgments
I would first and foremost like to thank my two academic advisors René Henrion
and Jiří V. Outrata for their unwavering support of me and my project. Their
patience and dedication, along with their deep knowledge of the theory of opti-
mization and variational analysis, allowed me to rapidly immerse myself in what
was a relatively new subject to me when I began researching three years ago.
In addition, I wish to thank Werner Römisch, whose weekly optimization sem-
inars provided me with a semesterly forum for presenting my latest results and
whose thought-provoking questions guided me along the way, as well as Boris S.
Mordukhovich and Michal Červinka, with both of whom I had many fruitful dis-
cussions. Furthermore, I would like to thank Darinka Dentcheva for first showing
me the beauty of optimization theory and Douglas I. Bauer for giving me my first
glimpses into the world of mathematics.
This dissertation is the result of three years of work at the Humboldt University
of Berlin within the international research training group RTG 1128 “Multiphase
Problems” funded by the German Research Foundation (DFG), to whom I am
indebted for the bestowment of finanical support. The research environment
made possible under the auspices of the DFG was integral in my development
during the last three years. I would personally like to thank all my colleagues for
their helpful discussions, especially Kshitij Kulshreshta, whose insight led to the
creation of Example 3.1. I also wish to thank Michael Hintermüller for providing
me with a new position at Humboldt three months before the end of my contract
with the research training group; his patience and understanding during this time
allowed for the completion of this dissertation.
Lastly, it should not go without mentioning my gratefulness to my family (es-
pecially my brother Paul for proofreading the introduction) and friends, on both
sides of the Atlantic, for all their love and support and giving me the strength to
continue striving forwards.
vZusammenfassung
Die vorliegende Arbeit beschäftigt sich mit Gleichgewichtsproblemen un-
ter Gleichgewichtsrestriktionen, sogenannten EPECs (Englisch: Equilibri-
um Problems with Equilibrium Constraints). Konkret handelt es sich um
gekoppelte Zwei-Ebenen-Optimierungsprobleme, bei denen Nash- Gleich-
gewichte für die Entscheidungen der oberen Ebene gesucht sind. Ein Ziel
der Arbeit besteht in der Formulierung dualer Stationaritätsbedingungen
zu solchen Problemen. Als Anwendung wird ein oligopolistisches Wettbe-
werbsmodell für Strommärkte betrachtet.
ZurGewinnungqualitativerHypothesenüberdieStrukturderbetrachte-
ten Modelle (z.B. Inaktivität bestimmter Marktteilnehmer) aber auch für
mögliche numerische Zugänge ist es wesentlich, EPEC-Lösungen explizit
bezüglich der Eingangsdaten des Problems zu formulieren. Der Weg dort-
hin erfordert eine Strukturanalyse der involvierten Optimierungsprobleme
(constraint qualifications, Regularität), die Herleitung von Stabilitätsre-
sultaten bestimmter mengenwertiger Abbildungen und die Nutzung von
Transformationsformeln für die sogenannte Ko-Ableitung. Weitere Schwer-
punkte befassen sich mit der Beziehung zwischen verschiedenen dualen Sta-
tionaritätstypen (S- und M-Stationarität) sowie mit stochastischen Erwei-
terungen der betrachteten Problemklasse, sogenannten SEPECs.
viAbstract
This thesis is concerned with equilibrium problems with equilibrium con-
straints or EPECs. Concretely, we consider models composed by cou-
pling together two-level optimization problems, the upper-level solutions
to which are non-cooperative (Nash-Cournot) equilibria. One of the main
goals of the thesis involves the formulation of dual stationarity conditions
to EPECs. A model of oligopolistic competition for electricity markets is
considered as an application.
In order to profit from qualitative hypotheses concerning the structure
of the considered models, e.g., inactivity of certain market participants at
equilibrium, as well as to provide conditions useful for numerical proce-
dures, the ablilty to formulate EPEC solutions in relation to the input
data of the problem is of considerable importance. The way to do this
requires a structural analysis of the involved optimization problems, e.g.,
constraints qualifications, regularity; the derivation of stability results for
certain multivalued mappings, and the usage of transformation formulae
for so-called coderivatives. Further important topics address the relation-
ship between various dual stationarity types, e.g., S- and M-stationarity,
as well as the extension of the considered problem classes to a stochastic
setting, i.e., stochastic EPECs or SEPECs.
viiContents
1 Introduction 1
2 Preliminaries and Notation 5
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Basic Variational Geometry . . . . . . . . . . . . . . . . . . . . . 5
2.3 Coderivatives of Multifunctions . . . . . . . . . . . . . . . . . . . 6
2.4 Stability Notions for . . . . . . . . . . . . . . . . . 7
2.5 Constraint Qualifications . . . . . . . . . . . . . . . . . . . . . . . 8
3 Equilibrium Problems with Equilibrium Constraints 11
3.1 Mathematical Programs withts . . . . . . . 11
3.2 Equilibrium Problems with Constraints . . . . . . . . 12
3.3 Well-posedness Issues for EPECs . . . . . . . . . . . . . . . . . . 13
3.4 EPEC modeling Oligopolistic Competition in an Electricity Spot
Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Stochastic MPECs . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Stoc EPECs . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.7 A Stochastic Spot Market EPEC . . . . . . . . . . . . . . . . . . 20
4 Dual Stationarity Concepts for MPECs and EPECs 23
4.1 Strong Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 M-Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 CM-Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Calculating the Fréchet Normal Cone via Strong Regularity . . . 31
4.5 Explicit S-Stationarity Conditions . . . . . . . . . . . . . . . . . . 37
4.6 Stationarity Conditions for EPECs . . . . . . . . . . . . . . . . . 39
5 Stability of the Perturbation Mapping 41
5.1 Calmness of P Mappings via Strong Regularity . . . . 41
5.2 of a Class of Perturbation Mappings . . . . . . . . . . . 45
6 Coderivative Transformation Formulae for Normal Cone Map-
pings 51
6.1 Polyhedral Feasible Sets . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 Nonlinearly Constrained Sets . . . . . . . . . . . . . . . . . . . . 55
6.3 Beyond Calmness . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7 Structural Properties of the Spot Market EPEC 67
7.1 The ISO problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
ixContents
7.2 A Remark on Existence of Solutions . . . . . . . . . . . . . . . . . 74
7.3 Verifying Calmness . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8 Explicit Characterizations of Solutions using Dual Stationarity
Conditions 79
8.1 Explicit M-stationarity Conditions for the Spot Market EPEC . . 79
8.2 Examples with Two Settlements . . . . . . . . . . . . . . . . . . . 82
8.3 Explicity Conditions for the Spot Market SEPEC . 90
Bibliography 99
x