2021-09-13

The department of Decision-Making Theory organizes a Mini-Symposium dedicated to three exceptional speakers: Silvia Carpitella, Tobias Boege, and Michael Mandlmayr. The Mini-Symposium is organized in a hybrid form. It is possible to participate either in person or connect virtually using the Zoom application.

The symposium will be held at UTIA, in Lecture hall No. 3 accessible from the lobby, on September 13, 2021, starting at 14:00.

**14:00 - 15:00**Silvia Carpitella:*How to make consistent decisions under evaluation criteria often conflicting with each other? (Zoom Link)*

ABSTRACT - The complexity of the decision-making problems included under the paradigm called Multi-Criteria Decision-Making (MCDM) has favored the proliferation of many schools of thought and, as a consequence, of many varied methodologies. At present, it has not been possible to prove the supremacy of any of these schools or philosophies over the others. Moreover, in some cases, it is difficult to combine the theoretical validity of the approximations with their practical appropriateness. It seems that rigor and applicability are two confronted concepts, something that should not be so. It is our responsibility to bridge the gap between the two. To reduce the gap between theory and practice, that is, to use effective methodological approaches, it is necessary to combine the rigor and objectivity of traditional science with the realism and subjectivity of human behavior. The Analytic Hierarchy Process (AHP) perfectly combines a classical axiomatic foundation, which provides the objectivity of the traditional scientific method, with an excellent adaptation to the real behavior of individuals and systems in decision making, which connects with the behavioral subjectivity. To help achieve this harmony between theoretical foundation and applicability, it is possible to make use of a framework built within the AHP that provides a mechanism of consistency improvement based on a process of linearization. This provides the rigor counterpart of our scheme being, at the same time, a really flexible approach suitable for dealing with a wide number of engineering contexts as well as other diverse domains.**15:10 - 15:30 coffee break****15:30 - 16:30**Tobias Boege:*The Gaussian CI inference problem (Zoom link)*

ABSTRACT - Conditional independence is a ternary relation on subsets of a finite vector of random variables*x*. A CI statement*(ij|K)*asserts that whenever the outcome of all the variables*x*, is known, learning the outcome of_{k}, k ∈ K*x*provides no further information on_{i}*x*. These relations are highly structured, in particular under assumptions about the joint distribution. The goal is to describe this by_{j}*CI inference rules*: given that certain CI statements hold, which other (disjunctions of) CI statements are implied under the distribution assumption?

In this talk, regular Gaussian distributions are assumed. Conditional independence has then an algebraic characterization in terms of subdeterminants

of the covariance matrix and inference becomes a geometric incidence problem in the space of positive-definite matrices. I first show that the inference problem for Gaussians is just as difficult as deciding whether a polynomial system with integer coefficients has a solution over the real numbers. This leads to the resolution of a question posed by Petr Šimeček about rational points on CI models. Then, I present some approximations to the inference problem which use the special finite structure of the covariance subdeterminants. In these formulations, SAT solvers and linear programming are able to prove some (but not all) valid inference rules and they terminate much faster than a general method. I also address the property of*selfadhesivity*previously observed for entropic polymatroids and present some computational results on the way of finding all realizable Gaussian CI structures on five random variables.**16:40 - 17:00 coffee break****17:00 - 18:00**Michael Mandlmayr:*Semismooth* Newton Methods Applications (Zoom link)*ABSTRACT - We present the application of the general semismooth* Newton method, introduced by Gfrerer and Outrata, to three challenging problems:

- Quasi-Variaitonal Inequalities
- Tresca Friction
- Coloumb Friction

The novelty of this method is, that these problems are interpreted as generalized equations. This means that for a set valued function we are interested in finding a point, such that the image contains zero.

As the method is called a “Newton”-method, to fulfill the expectations some linearization has to happen at some point of the graph. The linearization of a set valued mapping is done via a so-called normalcones to the graph, loosely speaking this normalcone can be seen as a generalization of normals. Furthermore, we need to have suitable point for the linearization. In contrast to the well known newton method for functions, where for an iteration point

*x*we just take*(x,F(x))*, for set valued mappings the choice is not trivial and is done in a so-called Approximation step.So this method consists of two parts:

- An approximation step that chooses a point of the graph
- A newton step, where we solve a linearized problem

We will show that under suitable assumptions this method converges locally superlinear to the solution. Moreover, we will illustrate the construction of such a method for the Quasi-Variational Inequalities and Coloumb Friction. Also, we will present numerical evidence for this convergence speed.

**19:00 symposium dinner**