# Optimisation

## Subject description

Convex sets and functions, convex programming. Lagrange duality, dual problem, weak and strong duality. Slater's condition, the Karush-Kuhn-Tucker theorem.

Linearly constrained optimization problems, quadratic and semidefinite programming with generalizations. Numerical procedures, penalty functionsA short overview of software tools for solving optimization problems.Convex sets and functions, convex programming. Lagrange duality, dual problem, weak and strong duality. Slater's condition, the Karush-Kuhn-Tucker theorem.

Linearly constrained optimization problems, quadratic and semidefinite programming with generalizations. Numerical procedures, penalty functions. Integer programming.

A short overview of software tools for solving optimization problems.

## Objectives and competences

Students encounter the fundamental types of problems in mathematical programming, with emphasis on the convex ones. They get to know the basic mathematical tools for tackling these problems, using appropriate software packages for solving them in practice.

## Teaching and learning methods

Lectures, seminar, exercises, homework, consultations, and independent work by the students

Part of the pedagogical process will be carried out with the help of ICT technologies and the opportunities they offer.

## Expected study results

Knowledge and understanding:  Students are able to model various important applied problems accurately. They are familiar with the basic techniques and software tools that can be used to solve the resulting optimization problems efficiently.

Application: Solving optimization problems which appear in practice.

Reflection: The importance of representing practical problems in a formal way which helps to solve them efficiently and adequately.

Transferable skills:  Ability to model practical problems as mathematically formulated optimization problems, to distinguish between computationally feasible and infeasible problems, to construct models on one's own and to analyze them by means of appropriate software tools.

## Basic sources and literature

• S. Boyd, L. Vandenberghe: Convex Optimization, Cambridge Univ. Press, Cambridge, 2004.
• B. H. Korte, J. Vygen: Combinatorial Optimization: Theory and Algorithms, 3. izdaja, Springer, Berlin, 2006.

## Stay up to date

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