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.

This course consists of lectures and exercises. All events are held in presence. If necessary, a switch to distance learning is possible on short notice.

• Lecture: All lectures are held in presence at the times given in Course dates. The first lecture (including a preview on the organization of the course) starts on 4.10.2022 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.
  
  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 in presence, 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. In case of need, the exam can be done via online video conference.

Lecture Notes (in German) can be downloaded here.