1. komponenta

Lecture type	Total
Lectures	45

COURSE AIMS AND OBJECTIVES: Getting acquainted the analytical and numerical approach to optimization problems and the application of basic results on convex sets and functions in optimization. In particular, for linear problems, learn the basic results of the geometry of polyhedral sets, the meaning of duality theory and numerical methods: the simplex method and the interior point method.

COURSE DESCRIPTION AND SYLLABUS:
1. Convex sets. Affine and convex sets, convex cones, projection of a point to a set, separation of convex sets, separation of a point and a finitely generated cone
2. Linear programming. Simplex algorithm, initialization, degeneration, finiteness of simplex algorithm with Bland pivoting, complexity of simplex algorithm, duality in linear programming: complementarity conditions, sensitivity, dual simplex method
3. Polyhedral geometry. Farkas-Minkowski-Weyl theorem, representation of a polyhedral set, extreme points and extreme recessive directions, solvability of a linear programming problem
4. Non-linear programming. Convex functions, subdifferential, nonlinear and convex programming, Karush-Kuhn-Tucker conditions, John conditions, Farkas lemma
5. Numerical methods. Gradient methods of unconditional minimization, Nesterov accelerated gradient method, Newton method, interior point method

1. Geometrija linearnog programiranja, L. Čaklović, Element, 2010.
2. Linear and nonlinear programming, D. Luenberger, Y. Ye, Springer, 2008.
3. Introduction to linear optimization, D. Bertsimas, J. Tsitsiklis, Athena, 1997.
4. Introductory Lectures on Convex Optimization: A Basic Course, Y. Nesterov, Kluwer, 2004.
5. Convexity and Optimization, A.-L. Lindahl, Lecture Notes, 2015.
6. Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with MATLAB, A. Beck, SIAM, 2014.
7. Nonlinear Programming: Theory and Algorithms, M. S. Bazaraa, H. D. Sherali, C. M. Shetty, John Wiley, 2006.

Mandatory course - Regular study - Applied Mathematics