Winter school of MTR, 2025

The Winter School of the Department of Decision-Making Theory is a well-established gathering of students and researchers from the department, collaborating researchers from several Czech institutions, and international guests. This year, it is organized by Tomáš Roubal, M. Studený, and J. Outrata. The scientific program reflects the research interests of the MTR UTIA department and our invited guests. The event takes place at UTIA's chalet.

Useful information

• Date: February 22 - March 2, 2025.
• Registration: To register, please send an email to Tomáš Roubal. Please specify how long you wish to stay, the title of your talk, and the preferred duration.
• Location: The workshop takes place in Máriánská (near Jáchymov) in the Czech Republic. The venue is about 20 km from Karlovy Vary (Karlsbad) and approximately 70 km from Chemnitz, Germany (see the map below).
• Presentation: Each participant is invited to give a presentation of approximately 60 minutes.
• Accommodation: Participants will be accommodated in the chalet. Please note that this is not a hotel—facilities include shared kitchens and toilets. Internet is available via Eduroam. Remember to bring your slippers.

• Catering: The organizers do not provide catering. You can prepare your own breakfast in the chalet’s kitchens—please remember to bring your own food. We plan to have joint dinners at a nearby restaurant.

   

Talks:

• Milan Studený: Four different views on finite posets

  • Abstract: In connection with the intention to describe faces of the cone of supermodular games over a finite variables set N in combinatorial way, a subtask arose what are possible alternative views on posets (= partially ordered sets) on N. I will present three alternative views on these posets, which appear to play an important role in this context. Graphical approach uses geodetically convex sets of enumerations (= permutations) of N, geometric approach uses particular polyhedral cones in RN known as braid cones. A combinatorial alternative uses (finite) topologies on N which distinguish points.

• Václav Kratochvíl: Difficulties in Using the Jeffrey Interval for Learning Belief Functions

  • Learning belief functions from data remains a relatively unexplored area. The Jeffrey interval provides lower and upper bounds on the probability of an event, computed from data—these bounds also reflect the amount of data available (there is a difference between observing 1 out of 1, 10 out of 10, or 1000 out of 1000 cases). One interpretation of a belief function is that it represents a lower bound on probability, while plausibility serves as the upper bound. But what happens if we try to construct a basic probability assignment from belief and plausibility functions derived in this way?

• Martin Šmíd: Non-authentic behavior in social networks: case of Telegram

• Marián Fabian: Fréchet versus Gateaux differentiability – counter-example

  • Abstract: Based on a forthcoming joint paper by Jan Kolář and me, we investigate several statements from differential calculus, what happens if Fréchet differentiability is replaced by Gateaux differentiability. The culminating result is an example showing non-validity of the "Gateaux" form of chain rule: There exists an involution f : R2 → R2, C∞-smooth off the origin, Gateaux differentiable at (0, 0), whose derivative f ′(0, 0) is singular, and hence f ′(0, 0) ◦ f ′(0, 0) is not the identity mapping. On the other hand, we show that the chain rule formula holds true for the Gateaux derivatives in an important special case of mappings between Banach spaces

• Jiří Outrata: On the implicit programming approach in a class of MPECs with conic constraints

  • Abstract: We consider a mathematical program with equilibrium constraints, where the lower-level equilibrium is governed by a generalized equation with conic constraints. Based on the generalized semismoothness property and the theory of SCD (subspace containing derivative) mappings, we suggest an efficient procedure for the numerical solution of such problems via a bundle method for nonsmooth optimization. The approach is implemented to problems, where the conic constraints involve the Lorentz (ice cream) cones and one may employ the respective Jordan algebra

• Mirek Pištěk: Solution to the Byzantine Generals’ Problem with Real-World Applications

  • Abstract: We will briefly introduce the Two Generals’ Problem and the Byzantine Generals’ Problem (BGP), two classical models of distributed systems from the 1980s, both posing fundamental challenges in achieving reliable consensus. We then examine an innovative anti-spam technique from the 1990s that introduced computational work as a physical constraint, effectively bridging the laws of physics and computing for the first time. This concept ultimately contributed to a pioneering solution to BGP in the late 2000s, leading to important real-world applications.

• Tomáš Roubal: On Josephy-Halley method2

  • Abstract: We discuss a generalization of the Halley method for solving generalized equations involving single-valued and set-valued mappings in Banach spaces. The focus is on local and semilocal convergence analysis, extending classical results. Numerical experiments confirm the cubic convergence rate and illustrate the method’s effectiveness in theoretical and applied contexts. 

• Stefan Krömer: Magnetostatics and convex conjugates

  • Abstract: We will discuss the cross-connection between two equivalent models of magnetostatics in solids. While convex conjugates appear in the transition, the link does not seem to be standard Fenchel duality. 

• Marouan Handa: Arrow decomposition of polynomial matrix inequalities combined with the moment SOS hierarchy

• Robert Baier: A Priori Estimates and Numerical Approximation of Reachable Sets by a Subdivision Scheme

  • Keywords: reachable sets, control problems, differential inclusions, Runge-Kutta methods, optimization, Lipschitz bounds.

• Helmut Gfrerer: On a globally convergent SCD semismooth∗ Newton method in composite convex optimization

  • Abstract: Very recently, a semismooth∗ Newton method based on SC (subspace containing) derivatives for solving variational inequalities of the second kind has been introduced. In this talk we describe how this approach can be used to efficiently solve nonsmooth composite convex optimization problems. 

• Martin Šmíd: Solution od multi-stage stochastic optimization problems by smoothed quantization and Markov SDDP

• Anna Doležalová: Limits of Sobolev homeomorphisms in models of Nonlinear Elasticity

  • Abstract: In this talk, we will focus on Sobolev spaces in the context of models of Nonlinear Elastcity. Several nice properties can be studied, e.g. the sign of the Jacobian (representing that the deformation is sense-preserving), different notions of injectivity (representing that nterpenetration of matter does not occur) or whether the mapping can be approximated by more regular ones.

• Jiří Vomlel: Dividing Lines in Czech Society: Beyond the Standard Statistical Analysis