LEARNING OUTCOMES
Credits: 5
Schedule: 07.01.2025 - 24.01.2025
Teacher in charge (valid for whole curriculum period):
Teacher in charge (applies in this implementation): Steven Gabriel
Contact information for the course (applies in this implementation):
This is a Ph.D/masters level course where we cover both the theory as well as the modeling of optimization and equilibrium problems. We meet in the mornings during three weeks in January. Applications include engineering and economics and multi-agent systems (e.g., markets, networks) that typically appear in energy, transportation, and other infrastructure networks.
You can contact me at sgabriel@umd.edu if you have questions about taking this course in January 2025.
Kiitos paljon.
Steve Gabriel
http://stevenagabriel.umd.edu/index.html
CEFR level (valid for whole curriculum period):
Language of instruction and studies (applies in this implementation):
Teaching language: English. Languages of study attainment: English
CONTENT, ASSESSMENT AND WORKLOAD
Content
applies in this implementation
Legend
- NLP= nonlinear programs (i.e, nonlinear optimization problems)
- MCP=mixed complementarity problem: generalization of the Karush-Kuhn-Tucker (KKT) optimality conditions for optimization problems and includes additionally non-cooperative game theory (e.g., Nash models), problems in mechanical engineering, etc. all considered as "equilibrium problems"
- VI=variational inequality problem, similar to MCPs
- MPEC=mathemtical programs with equilibrium constrraints, this includes to some extent bilevel optimization problems under certain conditions
- EPEC=equilibrium problem with equilibrium constraints (equilibrium at the top and at the bottom level)
Approximate ScheduleModule 1, Days 1-6: Review of MCP/VI Problems+Basic Extensions; Cones+KKT Conditions for NLPs, MCPs, VIs and relations between themTopic #1.1, Optimality conditions for nonlinear programs: , r
eview of definitions, KKT optimality conditions, constraint qualifications, necessary and sufficient conditions for optimality of NLPsTopic #1.2, Mixed Complementarity Problems (MCPs): problem definitions, KKT conditions as special case, other examples in engineering and economics
- Topic
#1.3, Variational inequalities (VI), Principle of Symmetry, Theory of KKT
Conditions, LCP Theory:
variational principle for optimization=>MCP/VI, Cones + proof of necessary direction for KKT conditions for optimization plus KKT relationship for MCP/VI, Principle of Symmetry connecting optimization and equilibrium problems, Specialization of existence and uniqueness theory for linear complementarity problems (LCPs), GAMS/Julia exercise
Module 2, Days 7-11: Generalized Nash Equilibria, QVI, MPECs, EPECs
- Topic #2.1, Generalized Nash equilibria: problem definitions, examples, theorems related to the quasi-variational inequality problem and Lagrange multipliers
- Topic #2.2, Bilevel problems: MPECs- problem definitions and examples, several approaches for solving MPECs, EPECs definition and examples, diagonalization method, GAMS/Julia exercises
Module 3, Days 12-13: DC-MCPs
- Introduction to Discretely Constrained MCPs (DC-MCPs): problem definitions, connection with logic/equity-enforcement constraints and equilibrium problems
Assessment Methods and Criteria
applies in this implementation
There will be an evaluation based on:
Course participation
Class project (can be 1-3 students)
Workload
applies in this implementation
Readings, in-class homeworks, some voluntary homeworks occasionally, class project with about 1-2 months to finish it.
DETAILS
Study Material
applies in this implementation
These will be provided in pdfs for the readings and the lectures. Students should download either GAMS software (https://gams.com/) for optimization/MCP or Julia ahead of time.
Substitutes for Courses
valid for whole curriculum period:
Prerequisites
valid for whole curriculum period: