Optimization 1

This course provides an introduction to continuous optimization in finite dimensional vector spaces.

Topics to be discussed are: First and second order necessary and sufficient (Karush-Kuhn-Tucker) optimality conditions for unconstrained and constrained optimization problems in finite-dimensional vector spaces. Basics of convex analysis and convex duality theory and their application to optimization problems and algorithms. An overview of modern optimization techniques and algorithms for smooth problems (including Newton and quasi-Newton methods for unconstrained optimization; algorithms for linear programming; SQP). Basic algorithms for non-smooth convex optimization problems. Introduction to vector optimization.

The student successfully meeting the learning objectives of the course will be able to:

1. assess the existence and uniqueness of solutions to a given optimization problem;
2. validate convexity of functions, sets, and optimization problems;
3. derive necessary and sufficient optimality conditions for a given optimization problem;
4. understand and use the duality concept in optimization;
5. understand solution concepts in vector optimization;
6. solve small optimization problems analytically;
7. explain the underlying principles and limitations of modern techniques and algorithms for optimization;
8. estimate the rate of convergence and complexity requirements of various optimization algorithms;
9. implement optimization algorithms on a computer;
10. apply optimization algorithms to model problems in engineering and natural sciences.

Lectures and project. The final grade is composed of a written exam (70%) and a portfolio of project work (30%).