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Regular version of the site
2017/2018

## Convex Optimization

Type: Optional course (faculty)
When: 3, 4 module
Language: English
ECTS credits: 5

### Course Syllabus

#### Abstract

This is an introductory course on variational methods, extremal problems, and convexity. In the first part of the course we discuss classical variational methods with examples of applications from geometry, physics, and engineering. We start with the variational calculus of Lagrange and Euler and finish with a brief discussion of the Pontryagin maximum principle. In the second (less standard) part, we try to demonstrate the power of convex analysis and explain how convexity appears in various branches of science, starting with engineering and economical applications and finishing with abstract algebraic problems.

#### Learning Objectives

• Successful participants imporve their presentation skills and prepare for participation in research projects in the subject area

#### Expected Learning Outcomes

• Successful participants imporve their presentation skills and prepare for participation in research projects in the subject area.

#### Course Contents

• Classical extremal problems from geometry and physics (isoperimetric problem, brachistochrone, geodesics, and others).
• Euler-Lagrange equations.
• Pontryagin maximum principle.
• Applications of the Pontryagin maximum principle in physics and engineering.
• Convex functions, convex bodies, convex polytopes. Support functions, Minkowski summation.
• Legendre transform. Lagrangians and Hamiltonians.
• Hamilton-Jacobi equation. Bellman principle and Bellman equation.
• Mixed volumes, geometric inequalities (Brunn—Minkowski, Alexandrov—Fenchel).
• Minimax principle. Kuhn—Tucker theorem.
• Linear programming. Kantorovich duality.
• Elements of the game theory. Von Neuman theorem.
• Some applications in graph theory: bipatrite graphs, shortest path, tropical algebra.

• Final Exam

#### Interim Assessment

• Interim assessment (4 module)
The total grade for the course is computed via the following formula: 0.2* CG+0.8*Exam

#### Recommended Core Bibliography

• Ruud, A. (2019). Convex Optimization: Theory, Methods and Applications. Hauppauge, New York: Nova. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=2043454