to the whole group and in particular to Enrique and also to you Tobias and probably also Dahlis

was involved. So thanks a lot. Yeah. So that is really a nice opportunity to speak about

a few of the things that I'm currently interested in. I read what I mean, I read about how you

think that seminar should be. And I understood you wanted to have quite long introduction.

So I thought I would do this. And the way we could do that is that we mention a few

applications and then also dive more into the mathematics behind and then we return

now so that and then I will also explain a little bit about the field because I think

that robust optimization is not what everybody does.

Now, so, okay, so in particular, I mean, this is in general about decision making in complex

systems. There's also data that comes into into play, of course, that you want to analyze.

And typically these complex systems, they are affected by uncertainties and you would

like to protect against this. And also, I mean, there are many different questions coming

up because there are maybe some dynamic aspects involved or some non-convex functions and

then also discrete as well as continuous decisions. So these ingredients make these questions

very difficult. Then you can have, for example, a question in logistics, say ambulance logistics

in our area here, or also some electricity networks, in particular gas networks, or also

electricity management and maybe also even in material design, say a nanoparticle design.

And the way I think about that seminar is that we mention several aspects in these applications.

And then, as I said, we dive more into the novel methodologies that then can be used.

And in particular, there are several research questions here because robust optimization

is not yet at the point where everything is known, but typically new things have to be

developed. So if you in particular in general have some optimization problem and uncertainties

are really relevant in the problem, then what can you do? Well, I mean, you can first start

ignoring them and just solve the nominal problem and you forget about uncertainties and you

hope that the nominal optimum also suffices in a perturbed system. If you recognize that

this doesn't work, then you can do some ex post, say, sensitivity analysis, where you

consider your nominal optimum and you say, OK, how long will it still be a good solution

if I start perturbing the input? And maybe this is not enough in the sense that you could

find that it is very unstable what you have obtained as an optimum solution. And then

you do some ex under protection, be it, say, with stochastic optimization or with what

we are now, then we'll speak about robust optimization or something in between. There

are very nice developments also at the interface between robustness and stochasticity. So what

do we mean with robust optimization in the modeling approach? Say, suppose this is a

linear optimization problem and this is the feasible region here, this gray shaded region,

and this is your objective, then clearly you would say, OK, if I'm maximizing, then this

would be an optimum solution. Now you say, maybe I was not clever enough in order to

determine my feasible region and maybe the true feasible region is only this red one.

Now you could say, OK, that point where I thought that was optimum is not even feasible

anymore. Now you could say, well, maybe I can round to something that is nearest and

still feasible. So you would take this red vertex then as a new solution. However, of

course, this example is made up like that. If you now look into what your optimum is,

then the now optimum solution is very far away. So then in order to actually deal with

this problem in robust optimization, what you do is already in the modeling approach

is that you say, already in the input, I take into account the typical uncertainties that

you either define via scenarios, say, from historical data or some intervals or something

like that. And among all the solutions, you only consider those that are so-called robust

feasible that have to be feasible regardless of how your uncertainty manifests itself.

And you have to do this here and now before you know how the uncertainty is realized.

And this makes it difficult, of course. And among those robust feasible solutions, you

want to determine one that has the best guaranteed solution value. Now the question is, I mean,