Please note! Course description is confirmed for two academic years, which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.

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 Schedule

    Module 1, Days 1-6: Review of MCP/VI Problems+Basic Extensions; Cones+KKT Conditions for NLPs, MCPs, VIs and relations between them

    • Topic #1.1, Optimality conditions for nonlinear programs: , r

      eview of definitions, KKT optimality conditions, constraint qualifications, necessary and sufficient conditions for optimality of NLPs
    • Topic #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
Prerequisites