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Regular version of the site
Bachelor 2022/2023

# Optimization Theory

Area of studies: Economics
When: 3 year, 3, 4 module
Mode of studies: distance learning
Online hours: 16
Open to: students of one campus
Language: English
ECTS credits: 4
Contact hours: 56

### Course Syllabus

#### Abstract

If the course is taken as a part of a BSc degree, courses which must be passed before this half course, may be attempted: Abstract mathematics. Students are also strongly encouraged to take Advanced mathematical analysis. If the course is taken as an elective discipline, the discipline which must be passed before, is Mathematica 2. The student should have knowledge and skills of calculus for functions of one and several variables, and of linear algebra, including the general theory of systems of linear algebraic equations and matrixes operations. The structure of the course includes strict abstract construction of linear spaces, metric spaces, calculus of functions of several variables, a general problem of optimization of function of several variables without restrictions and with restrictions both equalities and inequalities, finite and infinite horizon discrete dynamic optimization. The course material should teach students to understand and prove the basic formulas of Optimisation theory, and to investigate the economic problems of comparative statics and dynamic optimization within the framework of developed tools of mathematical models.

#### Learning Objectives

• enable students to obtain a rigorous mathematical background to optimisation techniques used in areas such as economics and finance;
• enable students to understand the connections between the several aspects of continuous optimisation, and about the suitability and limitations of optimisation methods for different purposes

#### Expected Learning Outcomes

• - use both the 1-st order and 2-nd order conditions in unconstrained optimization problems
• be able to use the Lagrange multipliers approach and analyze the solutions
• be able to use theoretical notions and concepts of relevant parts from real analysis, with emphasis on higher dimensions
• formulate a finite horizon dynamic program and use the backward induction
• Formulate a general problem of optimization with equalities restrictions. Understand the Lagrange theorem including its rigorous proof.
• formulate an fininite horizon dynamic program and use the Bellman equations
• formulate an optimisation problem under inequality constraints
• use both the 1-st order and 2-nd order conditions in unconstrained optimization problems
• use the Kuhn-Tucker approach for optimization problems
• use the Weierstrass’ Theorem in optimization problems
• use theoretical notions and concepts of concave analysis under solution of different optimization problems

#### Course Contents

• Mathematical preliminaries. Basic concepts of set theory. Metrics and norms. Topology. Compactness. Open and closed sets. Continuous functions.
• Weierstrass’ Theorem.
• Unconstrained optimization
• Optimisation under equality constraints. The Lagrange theorem.
• Lagrange multipliers. The constraint qualification.
• Optimisation under inequality constraints.
• Kuhn-Tucker Theorem
• Elements of convex analysis. Quasiconvex and quasiconcave functions. Pseudoconcave functions.
• Finite horizon Dynamic Programming
• Infinite horizon Dynamic Programming.

#### Assessment Elements

• home_assignments
• control work
• final exam

#### Interim Assessment

• 2022/2023 4th module
0.6 * final exam + 0.2 * home_assignments + 0.2 * control work

#### Recommended Core Bibliography

• A first course in optimization theory, Sundaram, R. K., 2011
• Mathematics for economists, Simon, C. P., 1994