Enrollment

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Goal

This course is designed to provide PhD students with the necessary modeling and methodological tools for analyzing and designing algorithms to solve game equilibrium problems.

Schedule

24 June 2026 10:00 – 12:30

24 June 2026 13:30 – 16:00

25 June 2026 10:00 – 12:30

25 June 2026 13:30 – 16:00

Syllabus

Introduction and motivation

Background

• Convex Optimization: Convex sets and functions. Set-valued mappings. Normal cone and tangent cone operators. Projection and proximal operators. Lagrangian duality and KKT conditions.
• Monotone Operator Theory: Fixed points, zeros, and contraction mappings; averaged and nonexpansive mappings; fixed point theorems and algorithms.

Nash equilibrium

• Nash equilibrium problem and best response mapping.
• Applications and models: Linear complementarity problems and variational inequalities.
• Existence and uniqueness of equilibria.
• Algorithms.

Generalized Nash equilibrium

• Generalized Nash equilibrium problem.
• Applications and models: Quasi-variational inequalities and mixed complementarity problems.
• Existence and uniqueness of equilibria.
• Algorithms.

Bibliography

[1] Boyd, S. P., and Vandenberghe, L. Convex Optimization. Cambridge University Press, 2004.

[2] Bauschke, H. H., and Combettes, P. L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Vol. 408. Springer, 2011.

[3] Facchinei, F., and Pang, J.-S. (eds.) Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, 2003.

[4] Osborne, M. J. An Introduction to Game Theory. Vol. 3. Oxford University Press, 2004.

[5] Basar, T., and Zaccour, G. (eds.) Handbook of Dynamic Game Theory. Springer, 2018.

— Slides and supporting material from the lecturer.