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# Optimization Theory

2020/2021
ENG
Instruction in English
4
ECTS credits
Course type:
Elective course
When:
3 year, 3, 4 module

### 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

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

#### 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
• UoL exam
The UoL exam’s mark in “Theory of optimisation” will be used for UoL Diploma for students of specialization "Mathematics and Economics" only.

#### Interim Assessment

• Interim assessment (4 module)
0.2 * control work + 0.6 * final exam + 0.2 * home_assignments

#### Recommended Core Bibliography

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