After successful completion of the course, students are able to identify, understand, analyze, formulate and graphically or mathematically solve basic static and dynamic optimization problems. They especially know about the theory, the mathematically principles and various methods for an exact or iterative solution of optimization problems. After successful completion of this course, students can moreover differentiate between unconstrained and constrained optimization problems and they can select and apply the specifically appropriate solution methods. This course strengthens and deepens engineering approaches, abstract and analytical thinking, independent solution of practical optimization problems, as well as mathematical skills.

Fundamentals of optimization:
existence of minima and maxima, gradient, Hessian, convexity, convergence

Unconstrained static optimization:
optimality conditions, computer-aided optimization, line search methods, choice of the step length, principle of nested intervals, Armijo condition, Wolfe condition, gradient method, Newton method, conjugate gradient method, Quasi-Newton method, Gauss-Newton-method, trust region method, Nelder-Mead method

Static optimization with constraints:
equality and inequality constraints, sensitivity considerations, active set method, gradient projection method, reduced gradient method, penalty and barrier functions, sequential quadratic programming (SQP), local SQP, globalization of SQP

Dynamic optimization:
basics of the calculus of variations, optimality conditions, Euler-Lagrange equations, Weierstrass-Erdmann conditions, design of optimal control solutions, minimum principle of Pontryagin,  energy-optimal, ressource-optimal, time-optimal, Bang-Bang control, singular arcs

The contents of this lecture are elaborated and discussed based on lecture notes and exercise notes (both documents freely available). The material is presented on the blackboard and with slides. To deepen, reinforce, and practically apply the material, example problems are discussed and mathematically solved. The software Matlab is used for computer-aided solution of optimization problems. In some cases, the developed solutions are practically implemented and tested on laboratory experiments.

• Exercises:
  Four exercise courses will be offered in the computer laboratory of the institute, each with a duration of 2 hours. These exercises are not mandatory but their contents is part of the final exam. The goal of the exercises is to apply the theoretical concepts and algorithms presented in the lecture to specific examples in the field of static and dynamic optimization. The focus lies on the use of numeric software (mainly Matlab). The organization of this course and the dates of the exercise courses will be discussed in the first lecture.

The performance is evaluated in an oral exam, which can take place at any time Monday to Friday from 6:00 to 20:00. To arrange a time for the examination, send an e-mail with desired dates, times or time slots, your name, student ID number, and study code to steinboeck@acin.tuwien.ac.at.

Lecture Notes (in German) can be downloaded here.