We will examine a few scenarios where aggregation methods can aid in the construction, analysis, and evaluation of tools related to data clustering, including linkage criteria, partition similarity measures, and cluster validity indices. We indicate some noteworthy challenges for both theoretical and practical future research endeavours.

The usual definitions of fuzzy set operations assume that the involved membership functions have the same domain of definition. The common way of dealing with fuzzy sets defined on different domains is to fill in zeros (or other suitable values) outside the domains of definition. It can be shown, however, that such modification may introduce spurious values into the results unless the domains of membership functions are explicitly taken into account. As a remedy, I will compare several methods of explicitly marking the domains of membership functions and discuss several ways of extending fuzzy set operations to accommodate the explicit domains. I will present selected results on such variable-domain operations on fuzzy sets and fuzzy relations and hint at a few applications in function approximation and fuzzy rules. (Most of the presented results come from joint work with Martina Daňková.)

Nowadays, real world optimization problems more and more frequently optimize black-box objective functions, which are not evaluated analytically, neither explicitly nor implicitly, but rather empirically by simulations, measurements or experiments. Most successful in their optimization are stochastic optimization methods, especially evolution strategies, as they make nearly no assumptions about the black-box objective, tend to find its global optimum, and need only its empirically obtained values, in as many points as possible. On the other hand, the empirical evaluation of a black-box objective is often costly and/or time consuming, which makes desirable to evaluate it in as few points as possible. To alleviate that dilemma, a machine-learning-based approach has been used for nearly two decades: a model is learned using data from previous iterations of the optimization method and serves as a surrogate of the true black-box objective functions in most of its new evaluations. The talk will give a survey of successful surrogate models, as well as a survey of strategies how to control when to evaluate the true objective and when its surrogate.

Galois connections are pervasive structures which have found application in several areas. In our case, we found them in an important construction in (Fuzzy) Formal Concept Analysis.

We focus on the problem of characterizing the existence of a fuzzy (isotone/antitone) Galois connection under certain circumstances. Specifically, given a set with a certain structure (fuzzy preposet/poset/T-digraph) and a function from it to an unstructured set, we look for necessary and sufficient conditions to give structure to the codomain and find the adjoint of the function.

The path followed aims at considering a properly fuzzy Galois connection in its most general sense. This leads to a twofold objective: on the one hand, the components of the connection should be fuzzy functions (in some approaches, although the structure of the domain is fuzzy, the pair constituting the connection consists of crisp functions); on the other hand, the underlying fuzzy structure requires certain minimal conditions so that the construction do not collapse.

We will present a survey of results obtained in different instances of the problem above.

The theory of aggregations is nowadays an important part of mathematics. In the contribution, we will introduce the concept of conditional aggregation operators. We define novel survival functions and the generalized Choquet integral based on them. We will discuss their properties and indicate their possible advantages in applications.

Institute for Research and Applications of Fuzzy Modeling of the University of Ostrava 2019.

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