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.** **

**Topics****: **

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

**Textbook****s****:**

- 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.)
- D.Gale The Theory of Linear Economic Models
- S. A.
*Ashmanov*:*Linear*

**Software**** ****Mathematica, Sage**