# Seminar Special Topics in Continuous Optimization and Optimal Control (WS 22/23)

Here you find information about the seminar on Special Topics in Continuous Optimization and Control in the winter term 2022/23.

2022-10-11

## General Information

Organizers

Kick-Off Meeting Date
Monday, 17.10.
Date
Mondays, 14:15 - 15:45
Room
Seminarroom 4 (SR 4) in INF 205

We meet in person.

Level:
Master Students (Bachelor students with relevant experience and motivation are welcome, too, though!)
Language
English
Requirements
You should have attended at least the following courses:
• Analysis 1, 2
• Linear Algebra 1
• Nonlinear Optimization

You can of course still join the seminar if you are missing one of these courses, but we strongly recommend that you attend the missing course in parallel then. The experience is that students without knowledge in these areas have problems with the topics of this seminar.

Registration
To participate in the seminar, please register for the seminar on MÜSLI. The number of participants is limited to 18.

## Kick-Off Meeting

The kick-off meeting takes place on Monday, 17.10., 14:15, in SR 4 (INF 205). Here, we

• discuss organizational matters,
• form pairs if necessary,
• distribute the topics,
• discuss the schedule.

## Topics

We have talks from three different fields on offer. We will agree on a field in the kick-off meeting. Literature suggestions can be found below.

### Methods for Nonlinear Optimization Problems

1. CG-Methods
2. Penalty and Augmented Lagrangian methods
3. SQP with Indefinite Hessian Approximations
4. BFGS-SQP Method for Nonsmooth, Nonconvex, Constrained Optimization
5. Interior-Point Filter Line-Search Algorithm for NLP
6. Global Optimization Methods
7. Robust Nonconvex Optimization

### Optimization for Data Science

2. Mirror Descent
5. Smoothing for Nonsmooth Optimization

### Model Order Reduction

1. Proper Orthogonal Decomposition (POD)
2. Empirical Interpolation Method (EIM)
3. Reduced Basis Methods (RB)
4. Dynamic Mode Decomposition (DMD)
5. Derivative Extended POD (DEPOD)
6. Balanced Truncation
7. Rational Interpolation

## Goal of the Seminar

Teach your fellow students your topic in an understandable yet professional way!

## Expectations

In order to complete the seminar successfully you have to

• attend the weekly meetings,
• prepare and do a presentation of 90min (+10min discussion) in pairs

You do not need to write an essay.

Further we expect

• a thorough understanding of your topic,
• professional display and communication of Mathematics,
• scientific literature work,
• a professional presentation (the presentation technique is up to you).

## Schedule

DateTopicSpeaker(s)
17.10.Kick-off meetigNone

The speakers will be decided in the kick-off meeting.

We primarly rate your talks based on this rubric.

## Literature

We expect you to identify and use further literature if necessary on your own! The following literature suggestions are to be understood as starting points.

You find all the following literature online when you are connected to the University network. If you are not in the University, you can connect to the network using the VPN. You find information on the VPN here.

### Methods for Nonlinear Optimization Problems

General recommendations
• Nocedal, Jorge, and Stephen J. Wright, eds. Numerical optimization. New York, NY: Springer New York, 2006.
• Ulbrich, Michael, and Stefan Ulbrich. Nichtlineare Optimierung. Springer-Verlag, 2012.
Recommendations by topics
CG-Methods
• J. Nocedal, S.J. Wright. Numerical Optimization, Springer, 2006. Ch. 5, pp. 101–133.
• Published papers about new research results on the field of nonlinear CG-methods
Penalty and augmented Lagrangian methods
• J. Nocedal, S.J. Wright. Numerical Optimization, Springer, 2006. Ch. 17, pp. 497–527.
SQP with indefinite Hessian approximations
• D. Janka. Sequential quadratic programming with indefinite Hessian approximations for nonlinear optimum experimental design for parameter estimation in differential–algebraic equations. PhD Thesis. Heidelberg. (2015).
• D. Janka, C. Kirches, S. Sager and A. Wachter. An SR1/BFGS SQP algorithm for nonconvex nonlinear programs with block-diagonal Hessian matrix (2016).
BFGS-SQP Method for nonsmooth, nonconvex, costrained optimization
• F.E. Curtis et. al. BFGS-SQP Method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles. Optimization Methods and Software (2017)
Interior-point filter line-search algorithm for NLP
• A. Wachter and L.T., Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program., Ser. A (2005).
Global optimization methods
• C.S. Adjiman et. al. A global optimization method, $$\alpha$$ BB, for general twice-differentiable constrained NLPs - I. Theoretical advances. Computers Chem. Engng Vol.2 No. 9, pp. 1137-1158. (1998).
• C. A. Meyer and C. A. Floudas. Convex underestimation of twice continuously differentiable functions by piecewise quadratic pertubation: spline $$\alpha$$ BB underestimators.
Robust nonconvex optimization
• B. Houska. Robust Optimization of Dynamic Systems. Chapter 3-4. (2011).

### Optimization for Data Science

• Sor, Naum Z., Minimization methods for non-dierentiable functions, Springer (Berlin), 1985 (in particular chapters 1,2, 4; available at the library in the Mathematikon)
• Convex Optimization Theory, Athena Scientic, 2009 by D.P. Bertsekas, MIT, Supplementary Chapter 6 on Convex Optimization Algorithms (2014) (in particular chapter 6.3)
• Convex Optimization: Algorithms and Complexity, S. Bubeck, Foundations and Trends in Machine Learning, 2015 (in particular chapter 3.1)
Mirror Descent
• Convex Optimization: Algorithms and Complexity, S. Bubeck, Foundations and Trends in Machine Learning, 2015 (in particular, chapter 4)
• Mirror descent and nonlinear projected subgradient methods for convex optimization, A. Beck and M. Teboulle, Operation Research Letters, 2002
• Amir Beck and Marc Teboulle, Gradient-Based Algorithms with Applications to Signal Recovery Problems (Convex optimization in signal processing and communications, 2009) (in particular sections 1.1 - 1.4)
• Neal Parikh and Stephen Boyd, Proximal Algorithms (Foundations and Trends in Optimization, 2014) (in particular chapters 1, 2, 4.2, 6, 7.1)
• Wright, Stephen J. “Optimization algorithms for data analysis.” The Mathematics of Data 25 (2018): 49. (in particular section 6)
Smoothing for Nonsmooth Optimization
• Amir Beck and Marc Teboulle, Smoothing and First Order methods: A Unied Framework (SIAM Journal of Optimization, 2012)
• Yuri Nesterov. Smooth minimization of non-smooth functions, (Mathematical programming 103.1, 2005)

### Model Order Reduction

General recommendations
Recommendations by topics
Proper Orthogonal Decomposition (POD)
Empirical Interpolation Method (EIM)
• Chapters 1.3.2 - 1.3.3 and chapters 5.4.2.1.1 - 5.4.2.1.3 of Model Order Reduction Volume 2
• Grepl, Martin A., et al. “Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations.” ESAIM: Mathematical Modelling and Numerical Analysis 41.3 (2007): 575-605.
• Barrault, Maxime, et al. “An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations.” Comptes Rendus Mathematique 339.9 (2004): 667-672.
• Chapter 5 of Hesthaven, Jan S., Gianluigi Rozza, and Benjamin Stamm. Certified reduced basis methods for parametrized partial differential equations. Vol. 590. Berlin: Springer, 2016.
Reduced Basis Methods (RB)
• Chapter 4 of Model Order Reduction Volume 2
• Chapter 3 of Hesthaven, Jan S., Gianluigi Rozza, and Benjamin Stamm. Certified reduced basis methods for parametrized partial differential equations. Vol. 590. Berlin: Springer, 2016.
• Quarteroni, Alfio, Andrea Manzoni, and Federico Negri. Reduced basis methods for partial differential equations: an introduction. Vol. 92. Springer, 2015.
Dynamic Mode Decomposition (DMD)
• Chapter 7 of Model Order Reduction Volume 2
• Schmid, Peter J. “Dynamic mode decomposition of numerical and experimental data.” Journal of fluid mechanics 656 (2010): 5-28.
Derivative Extended POD (DEPOD)
• Schmidt, Andreas, et al. “Derivative-extended POD reduced-order modeling for parameter estimation.” SIAM Journal on Scientific Computing 35.6 (2013)
• Schmidt, Andreas. Direct Methods for PDE-Constrained Optimization Using Derivative-Extended POD Reduced-Order Models. Diss. 2014.
Balanced Truncation
• Chapter 2 of Model Order Reduction Volume 1
Rational Interpolation
• Chapter 3 of Model Order Reduction Volume 1