Published Papers

It is well known that the theory of monotone systems transforms clustering from a global optimization problem (which is often NP-hard) into a successive elimination problem solvable in polynomial time. The proposed approach requires minimal a priori information: specifying only the relationship measure of one object with a subset of objects. The algorithm guarantees an exact solution to the stated extremal problem. The approach is based on the concept of a monotone system $<A,F>$, where $A$ is a finite set of objects, and $F(X)$ is an importance (or weight) function defined on subsets $(X \subseteq A)$. The monotonicity condition is: $F(X\setminus\{a\}) > F(X)$. We consider a generator of monotone systems. The proposed procedure for generating a family of monotone systems consists of two stages: i) constructing a set of transformation operators for a monotone system of a sufficiently general form, defined on the same initial set $W$, $|W| =N$; ii) constructing a set of basic functions on the set W. The desired generator is considered as a structure that generates compositional chains of operators over monotone systems selected as basis ones. The proposed extension of the class of basic functions for monotone systems is implemented in a class of Estimate Calculation Algorithms (ECA). A problem statement is formulated for defining a set of three types of basic functions in a monotone system in a class of estimator algorithms. Changing the sets of operators in the basic systems generates a family of monotone systems, which has a wide range of applications, for example, in genetic network analysis, natural language processing (NLP), image processing, and, in general, as a new tool for solving complex problems of structuring large data sets.