Optimization methods

COURSE DESCRIPTION

As a result of this course of study, students should gain an understanding of the set of optimisation methods, the main mathematical results that form the basis for the development of optimisation methods, the domains of application of optimisation methods and their most commonly used numerical algorithms. At the same time, by developing appropriate laboratory work, students develop the ability to apply optimisation methods and their commonly used numerical algorithms to the solution of specific types of optimisation problems.

CONTENT

1. Classification of optimisation methods. Active and passive search. Probabilistic and deterministic minimisation algorithms. Use of basic results from differential calculus of one- and multi-argument functions to determine extremal values of functions; concepts of inf, sup, arg min and arg max; local and absolute extremes; determination of absolute extremes in situations with no additional conditions in the form of equations or inequalities
2. Mathematical and theoretical foundations of unimodal one-argument function minimisation, commonly used numerical algorithms - dichotomies, golden ratio, Fibonacci methods
3. The concept of a Lipschitz continuity function, numerical algorithms for minimising such one-argument functions, including the broken-line method. The tangent method
4. The most common types of linear planning tasks in applications are ration tasks, production tasks and transport tasks. Reduction possibilities of transport tasks to a transport task. Common numerical algorithms for solving linear planning problems.
5. Mathematical foundations of game theory. Reduction of a two-player zero-sum game to the solution of a linear programming problem
6. Determination of conditional absolute extrema of functions of several arguments, including the method of elimination, the method of Lagrange multipliers, the graphical method for solving such problems
7. Basic algorithms for numerical minimisation of multi-argument functions - gradient methods, Newton's method, etc.
8. Application of Bellman dynamic planning principle to optimal control problems