MAI0124
Some Applications of Convex Analysis

Number of credits: 4 ECTS credits

Examiner: Natan Kruglyak

Course literature:
1. I. Ekeland, P. Temam, Convex Analysis and Variational Problems, SIAM, 1999.
2. J. Jost, Partial Differential Equations, Springer, 2002.
3. C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, v. 58, American Mathematical Society, 2003.
4. C. Villani, Optimal Transport, old and new, A series of Comprehensive Studies in Mathematics, v. 338, Springer, 2009.
5. Mikulas Luptacik, Mathematical optimization and Economic Analysis, Springer Optimization and Its Applications, v. 36, Springer, 2010.
6. D. Gale, The Theory of Linear Economic Models, The University of Chicago Press, 1960.
7. N. Andreasson, A. Evgrafov and M. Patriksson, An Introduction to Continuous Optimization, Studentlitteratur, Lund, 2005.

Course contents: Ekeland variational principle and its applications (Mountain pass theorem). Variational approach to Dirichlet problem, difficulties and counterexamples. Variational approach based on Sobolev spaces, Friedrichs inequality and weakly harmonic functions. Monge-Kantorovich transportation problem. Monge classical formulation and Kantorovich relaxation of Monge problem. Recent developments and new type applications (Brenier theorem). Potentials (shadow prices) in transportation problem, theorem on cycling property of optimal plan. Application of convex analysis to the economic analysis. Main type of problems. Dual problem in linear programming. Sensitivity analysis. Difficulties connected with interpretation of shadow prices. The Giffen paradox, the more-forless paradox. Optimization algorithms for convex problems.

Organisation: Lectures and seminars.

Examination: Oral presentation of assignments given during the course.

Prerequisites: Standard course on functional analysis.

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Last updated: 2022-11-15