The goal of this course is to investigate algebraic methods for analyzing the economic situation.  This course covers the following topics: methods for finding the optimal solution, elements of game theory,  linear programming.


  1. Convex sets, convex combination, affine subspace, cone, polyhedron, polytope etc.
  2. Farkas Lemma and its versions, proof with the simplex method and with the elimination method.  
  3. Theorem of Charathèodory.
  4. Characterizations for unboundedness.
  5. The duality theorem.
  6. The polar cone.
  7. Des­criptions of polyhedral cones.
  8. Separation of polyhedra by hyperplanes.
  9. Vertices, faces and facets of polyhedra.
  10. Dimension of a face. Characterization of a minimal face.
  11. Equivalent definitions of vertices.
  12. Every polytope is the convex combination of its vertices.
  13. The des­cription of a full-dimensional polyhedron by linear inequalities is essentially unique.
  14. The simplex method, algebraic des­cription.
  15. The min-max theorem of game theory.
  16. The Transportation Problem.
  17.  The Leontief model.


  1. Artamonov V., Latyshev V. Linear algebra and convex geometry (in Russian: Артамонов В.А., Латышев В.Н. Линейная алгебра и выпуклая геометрия. М.; Изд-во "Факториал Пресс". 2004.)
  2. D.Gale The Theory of Linear Economic Models
  3. S. A. Ashmanov: Linear Programming. (in Russian). Nauka, Moscow 1981. (Ашманов С.А. Линейное программирование. -- М.: Наука)

Software Mathematica, Sage