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 mathematical 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:
fundamentals 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, direct vs. indirect methods, 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.

Current information on the teaching mode in the winter semester 2021/2021

This course is taught in a hybrid format. This means that there are presence and distance learning events.

• Lecture: All lectures are exclusively held as online meetings via ZOOM at the times given in Course dates. A link to the ZOOM meetings is available in the associated TUWEL course. A course registration is required to access the TUWEL course.The first lecture (including a preview on the organization of the course) starts on 5.10.2021 at 8:00.
  
  
• Exercise: All four exercises are held as presence learning events in the computer lab of the institute ACIN (room CA0426). Each exercise consists of two parts. 1) Review and discussion of prepared problems, which are the basis for the presence event and which have to be solved beforehand. 2) Solution of further problems and in parts test on lab experiments.
  
  Each exercise is scheduled for two hours and is offered with the same contents at two different dates. It is thus sufficient to attend one event per exercise. If required (e.g. because of entry restrictions at TU Wien), the exercises can be quickly switched to distance learning.
  
  All contents of the exercises are 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 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, desired format (in presence or online), your name, student ID number, and study code to steinboeck@acin.tuwien.ac.at.

Lecture Notes (in German) can be downloaded here.