On one extremal problem of adaptive machine learning for detection of anomalies

An adaptive algorithm to solve a wide range of problems of unsupervised learning by constructing a sequence of interrelated extremal principles was proposed. The least squares method with a priori defined weights used as a starting point enabled determination of the “center” of learning sample. Next, a natural passage from the least squares method to more flexible extremal principle enabling adaptive determination of both the “center” and weights of the learning sample events was performed. Finally, a universal extremal principle enabling determination of the scaling coefficient of the membership function in addition to the “center” and weights was constructed.     

Singular extremal control problem with time delay

In this article, the singular extremal control problem with time delay is studied from the viewpoint of the calculus of variation and matrix theory. Some sufficient conditions for the non-negativity of the second variation are obtained for an optimal control problem with a delayed state vector. The hypothesis of Hass is first assumed, and the total singular situation of Jacobson and Bell is also considered. A second-order necessary condition for optimality is also presented.     

Partitioning the parameter domain in a Chebyshev extremal problem

We consider the problem of global extremum of the Stieltjes integral of a given function dependent on parameters, with variable integrating function from the class of distribution functions with two given power moments.Translated from Kibernetika, No. 2, pp. 65–69, March–April, 1990.     

Prelimit distribution functions and limit polynomials in a chebyshev extremal problem

Cybernetics and Systems Analysis -     

基于极值动力学的自组织优化算法求解TSP问题

旅行商问题(traveling salesman problem, TSP)具有很强的理论研究和工程应用价值. 在定义离散状态变量和局部适应度的基础上, 分析了TSP优化解的微观特征; 将自组织临界(self-organized criticality, SOC)的概念引入到组合优化领域, 提出了一种基于极值动力学的自组织优化算法. 该算法利用快速下降和间断涨落的动态搜索过程, 高效地遍历解空间中的局部最优解. 针对TSPLIB中典型实例, 计算结果表明其求解效率和优化性能均优于模拟退火和遗传算法等优化方法. 文中算法提供了一种全新的思路, 有助于从系统角度理解组合优化问题的复杂性, 并分析合理的优化动力学过程.     

Algorithm for one classification problem

Conclusions We have examined a class of optimal classification problems with an objective function based on a loss functional defined directly on the set of all subsets of the given set of objects without resorting to distance between objects.A fast algorithm MATCHING was proposed for approximate solution of the optimal class fication problem. The algorithm is iterative, and the number of partitioning classes analyzed is roughly halved from iteration to iteration. The assignment problem is solved in each iteration.Computer experiments have shown that in many cases (over 75% for problems of dimension from 4 to 15) a symmetric initial matrix in the assignment problem produces a symmetric optimal assignment.The relative error of the proposed algorithm investigated for some model problems does not exceed 12% in most of the cases (Figs. 4 and 5) and, in general, depends on the dimension of the problem (n) and the location of k^* on the interval [1, n], where k^* is the number of classes in the optimal classification.Translated from Kibernetika, No. 6, pp. 90–95, November–December, 1988.     

On one variational Butkovskii problem

One of the unsolved problems of the theory of optimal control, the Butkovskii variational problem which is of importance for applications, was solved.     

Abelian combinatorics on words: A survey

We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on abelian equivalence, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these results.     

On one problem of integer optimization

A recognition problem of the following form is studied: to find put for the prescribed polyhedron whether the maximum of the linear objective function is achieved at its integral point. It is established that this problem is NP-hard in the general case and polynomially solvable in the class of rooted semimetric polyhedra.     

Numerical control optimization methods in one pursuit-evasion problem

This paper is devoted to the construction and study of different numerical suboptimal strategies of the pursuing player in a nonlinear three-dimensional pursuit-evasion game problem. The dynamics of the objects correspond to the known game of two cars [1]. The distinction of this problem from those considered before is the cost function structure, which contains the angular position of the velocity vectors with respect to the line of sight in addition to the distance between the players. In order to construct the numerical strategies of the pursuing player, we use the results of [2–6], where the elements of optimal control for simplified problems are obtained by using the Pontryagin maximum principle and the basic Isaacs-Bellman equation. On the basis of the strategies presented, we propose the algorithm for numerical construction of a suboptimal synthesis in the entire phase space of the initial problem. This algorithm follows an idea formulated in [7]. The numerical results of the suboptimal synthesis in a certain subdomain of the phase space are performed for some specified values of the parameters.     

Remark on the star-height-problem

A method is presented which allows to construct general regular expressions of star-height 1 for a class of rather ‘complicated’ regular events. (General regular expressions include an operation symbol for complement.) If events of greater general star-height exist (which is still open), they must be more complex than those accessible by this method. An event which seems to be of this kind is suggested at the end of the paper.     

关于凝聚式Quantale的注记

主要给出了凝聚式Quantale的具体构造,讨论了它们之间的关系;证明了凝聚式Quantale范畴是m-半格范畴的反射子范畴。