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Equilibrium Modeling

PhD School at the Faculty of SCIENCE at University of Copenhagen


The modeling of equilibrium is crucial for the understanding of many real-world phenomena, e.g. functioning of energy markets and congestion in traffic networks. As a result, equilibrium problems arise in numerous areas of engineering and economics. This course will introduce the state-of-the-art concepts of equilibrium problems, including applications in nonlinear programming and game theory and a number of applications in energy and transportation.

The curriculum covers both one-level equilibrium problems expressed as mixed complementarity problems (MCP’s) and variational inequalities (VI’s), and two-level equilibrium problems, expressed as mathematical programs with equilibrium constraints (MPECs).

Day 1: Review of nonlinear programming. The Karush-Kuhn-Tucker optimality conditions and mixed complementarity problems.
Days 2-3: Mixed complementarity problems: Source problems, connections to optimization, game theory.
Day 4: Two-level optimization problems and mathematical programs with equilibrium constraints (MPECs).
Day 5: Discussion of course project and assistance with model formulations, data, etc.

Formel requirements

Prerequisites: A graduate-level course in optimization, e.g. OR2. Some exposure to the GAMS software is preferred.

Learning outcome

The students will become well acquainted with the theory of equilibrium problems and a wide range of applications. Furthermore, they will reinforce the theoretical concepts through hands-on modeling and implementation exercises. By the end of the course, they will be able to independently characterize, formulate, solve and analyze real-world equilibrium problems.


Excerpts of literature 

  1. S.A. Gabriel, A.J. Conejo, J.D. Fuller, B.F. Hobbs, C. Ruiz. 2013. Complementarity Modeling in Energy Markets, Springer.
  2. F. Facchinei and J.-S. Pang. 2007. Finite-dimensional Variational Inequalities and Complementarity Problems, Springer
  3. P. T. Harker and J. S. Pang. 1990. “Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms, and applications,” Mathematical Programming, 48, 161-220.
  4. M. C. Ferris and J. S. Pang. 1997. “Engineering and economic applications of complementarity problems,” SIAM Review, vol. 39, No.4, 669-713.
  5. B. H. Ahn, and W. W. Hogan. 1982. “On convergence of the PIES algorithm for computing equilibria,” Operations Research, 30, 281-300.
  6. S. A. Gabriel, A. S. Kydes, P. Whitman, 2001. "The National Energy Modeling System: A Large-Scale Energy-Economic Equilibrium Model," Operations Research, 49 (1), 14-25.
  7. S. A. Gabriel, S. Kiet, J. Zhuang, 2005. "A Mixed Complementarity-Based Equilibrium Model of Natural Gas Markets", Operations Research, 53(5), 799-818.
  8. H. Z. Aashtiani and T. Magnanti, 1981. “Equilibria on a Congested Transportation Network,” SIAM Journal on Algebraic and Discrete Methods, 2, 213-226.
  9. T. L. Friesz, 1985. “Transportation Network Equilibrium, Design, and Aggregation: Key Developments and Research Opportunities,'” Transportation Research A, 19A, 413-427. 10. S. A. Gabriel and D. Bernstein, 1997. “The traffic equilibrium problem with nonadditive costs,” Transportation Science, 31, 337-348.

Teaching and learning methods

Preparation and self-study, lectures, exercises (including modeling and implementation), in-class discussion of student projects.


Steven A. Gabriel (, University of Maryland, Full Professor, Dept. of Mechanical Engineering, and Full Professor, Applied Math & Statistics, & Scientific Computation Program,

Other Appointments include Adjunct Professor, Dept. of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Trondheim, Norway,, Energy Transition Programme, and Research Professor, DIW (German Institute for Economic Research), Berlin.

The lecturer is an internationally leading expert within optimization and equilibrium/game theory, algorithmic development and applications in energy, environmental issues, transportation and other networks, as also evidenced by his long list of publications in the most prominent academic journals of the area.


Course dates
22 July 2019 - 26 July 2019
Department of Mathematical Sciences
Universitetsparken 5
København Ø
2,5 points

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