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58

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## Publications

Publications (58)

The authors consider a problem of packing circles of given types in a circular container. Circles are allowed to cross the container boundary in a predefined neighborhood that depends on the circle type (pseudo-inclusion condition). A family of circles is placed in the container under the conditions of their non-intersection, pseudo-inclusion, and...

Mathematical models of linear and Boolean linear programming for the balanced two-stage transportation problem and its two modifications are analyzed. The first modification takes into account the upper limits on the capacity of intermediate points, and the second modification allows choosing a fixed number of intermediate points, less than their t...

The quadratic optimization problem for finding the maximum k-plex in an undirected graph is constructed. Two families of superfluous quadratic constraints obtained by means of constraints of the Boolean linear programming problem for the maximum k-plex are presented. The influence of superfluous constraints on the improvement of the accuracy of Lag...

The authors consider the problem of generating spheroidal voids in a three- dimensional domain of complex geometry, with regard for the constraints on the “sparseness” of voids subject to the system balance. The problem is reduced to the optimized layout of ellipsoids of revolution in a convex container (cylinder or cuboid), taking into account the...

The authors consider the optimization problem of layout of spherical voids in three-dimensional domains bounded by cylindrical and spherical surfaces and planes. The problem is reduced to arranging spherical objects in a composite container, with regard for the constraints on their “sparseness” and balance conditions (location of the gravity center...

The paper focuses on linear programming problems with two-sided constraints whose number is much larger than the number of variables. The solution approach is based on a non-smooth convex penalty function. An appropriate penalty parameter ensures the equivalence between the original problem and the problem of minimizing the penalty function. The la...

A special form (B-form) of methods of Quasi-Newton type is discussed, which makes it easy to interpret these methods as gradient in appropriately transformed argument space. B-form of the Davidon–Fletcher–Powell method is given and compared with r-algorithms. To minimize smooth convex functions, a gradient method with space transformation is built,...

Application of a technique of dual Lagrangian quadratic bounds of N.Z. Shor to studying
the Maximum Weighted Independent Set problem is described. By the technique, two such N.Z. Shor's upper bounds are obtained. These are bounds of the graph weighted independence number �(G;w), which can be found in polynomial time. The �first bound (G;w) is assoc...

This paper describes the application of parallel computing technologies in systems with shared and distributed memory for solving optimization problems of geometric design. The first technology is based on the maximin properties of phi-functions for composite objects, and the second technology uses the multistart strategy and methods for minimizing...

A new extragradient-type method is proposed for approximate solution of variational inequalities with pseudo-monotone and Lipschitz-continuous operators acting in a finite-dimensional linear normed space. The method uses Bregman divergence (distance) instead of Euclidean distance and a new adjustment of step size, which does not require knowledge o...

We consider two subgradient methods (methods A and B) for finding the minimum point of a convex function for the known optimal value of the function. Method A is a subgradient method, which uses the Polyak’s step in the original space of variables. Method B is a subgradient method in the transformed space of variables, which uses Polyak’s step in t...

We propose and study the Levenberg–Marquardt method globalized by means of linesearch for unconstrained optimization problems with possibly nonisolated solutions. It is well-recognized that this method is an efficient tool for solving systems of nonlinear equations, especially in the presence of singular and even nonisolated solutions. Customary gl...

An algorithm with space dilation is presented, which is the circumscribed ellipsoid method under a certain choice of dilation coefficient. It is shown that its special case is the Yudin–Nemirovsky–Shor ellipsoid method. The application of the algorithm to solving a convex programming problem and the problem of finding a saddle point of a convex-con...

Functionality features of the web-based system Maneuver-New that is designed for solving integer-valued, linear, and nonlinear programming problems, and its using for finding the optimal load of power units of thermal power plants are described. The results of comparison of the obtained solution of the test task with the solution published abroad a...

Consideration is given to optimization problems of finding the best in Lp -norm parameters of regular 3D-structures and methods for their solution. It is shown that when restoring the parameters of 3D-structures with defects the least moduli method is more stable than the least squares method. The results of computational experiments for software i...

Three computational forms of r-algorithms with different amount of computation per iteration are considered. The results on the convergence of the limit variant of r-algorithms for convex smooth functions and the rμ (α)-algorithm for convex piecewise smooth functions are presented. Practical aspects of the variant of r (α)-algorithms with a constan...

Properties of three computational forms of r-algorithms differentiated by their complexities (number of calculations per iteration) are considered. The results on convergence of the limit variants of r-algorithms for smooth functions and rμ(α)-algorithm for nondifferentiable functions are presented. A variant of r(α)-algorithms with a constant coef...

The author formulates mixed Boolean linear programming problems to find the shortest route and the shortest cycle that pass through the given number of nodes in a complete graph. Their special cases provide formulations of problems for finding the shortest Hamiltonian path and the shortest Hamiltonian cycle. The problems include no more than 2n2 va...

The authors consider primal interior point algorithms to find normal solutions to systems of linear equations with bilateral constraints on variables. Analyzing this problem and the methods of its solution is important to develop the theory of mathematical modeling (in particular, to solve power engineering problems) and to create efficient computa...

The paper considers balanced packing problem of a given family of circles into a larger circle of the minimal radius as a multiextremal nonlinear programming problem. We reduce the problem to unconstrained minimization problem of a nonsmooth function by means of nonsmooth penalty functions. We propose an efficient algorithm to search for local extr...

The paper introduces a general mathematical model of the optimal layout of 3D-objects (full-spheres, right circular cylinders, right regular prisms, and right rectangular parallelepipeds) in a container (straight circular cylinder, paraboloid of revolution, truncated circular cone) with circular racks. The model takes into account the minimum and m...

In this small paper we propose a new efficient approach for numerical
calculation of equillibriums in multistage transport problems. In the very core
of our approach lies the proper combination of Universal Gradient Method
proposed by Yu. Nesterov (2013) and conception of inexact oracle
(Devolder--Glineur--Nesterov, 2011). In particular our techniq...

Two glioblastoma groups, which are distinguished from each other by expression level of 416 genes (p a parts per thousand currency sign 0.05), were determined using a mathematical model of linear Boolean programming on the basis of gene expression data, obtained by microarray analysis of the glioblastomas and available in Gene Expression Omnibus (G...

We prove that the maximum singular value of the matrix and the corresponding singular vectors are the optimal solution for a special quadratic optimization problem. We consider the economic interpretation of the optimal solution for the linear model of production and for the productive Leontief model. We relate the optimal solution to the Frobenius...

The paper is devoted to the 75th anniversary of the Kyiv mathematician Naum Shor and is focused on his three central ideas: generalized gradient descent (1962), the use of linear nonorthogonal space transformations to improve the conditionality of ravine functions (1969), and dual approach for finding bounds of the objective function in nonconvex q...

The problem of finding normalized vectors of demand and value added in a productive Leontief model is solved. These vectors
maximize the national income. It is shown that if the Leontief matrix is productive and indecomposable, then an optimal normalized
structure is determined by positive components of the eigenvectors that correspond to maximum e...

The least distance problem is considered for a convex hull of a finite family of vectors in a finite-dimensional Euclidian space. It is reduced to an equivalent nonsmooth optimization problem with a directly estimated penalty parameter for which subgradient algorithms with space dilation are proposed.
Keywords: projection problem, nonsmooth exact...

Some of the fundamental problems in coding theory can be formulated as extremal graph problems. Finding estimates of the size
of correcting codes is important from both theoretical and practical perspectives. We solve the problem of finding the largest
correcting codes using previously developed algorithms for optimization problems in graphs. We re...

Models, numerical algorithms, and database and software components aimed at decision support during the elaboration of energy-saving
measures are considered. Modern methods of nonsmooth optimization are applied to solve relevant optimization problems.
Keywordsintersectoral balance-methods of nondifferentiable optimization-Lagrangian multipliers-en...

Upper bounds for the weighted stability number of a graph are considered that are based on the approximation of its stable set polytope by linear inequalities for odd cycles and p-wheels in the graph. Algorithms are constructed for finding upper bounds on the basis of the solution of linear programming problems with a finite number of inequalities...

For a general quadratic problem, an analog is formulated as a homogeneous quadratic problem. The estimates ψ* constructed
based on Shor’s dual quadratic estimates for these problems are proved to be equal. It is shown that, for the case of a homogeneous
quadratic problem, finding ψ* is reduced to an unconstraint minimization problem for a convex fu...

New quadratic models are proposed to improve the upper-bound estimates in the maximum weighted cut problem. They are found
by two original methods for deriving redundant quadratic constraints. A well-known linear model is shown to follow from the
models proposed. Recommendations on how to develop its strengthened analogs are given.

A method of construction of functionally redundant quadratic constraints is proposed for Boolean quadratic-type optimization
problems. The method is based on an extension of a set of Boolean variables and formation of functionally redundant constraints
that relate the initial and added variables. Examples of improvement of Lagrangian dual quadratic...

Two problems of finding optimal parameters for multilayer optical coatings are considered. They are formulated as multiextremal nonlinear programming problems with a complex objective function. Finding local extrema by first-order methods is discussed. The ways of calculating the gradient of the objective function depending on the number of layers...

The problems of finding the best one-dimensional and multidimensional linear models are considered for a short selection of measurement results with a rather small number of erroneous measurements. Using an example, it is shown that the least moduli method allows one to find models more adequate to the process than the least squares method.

A version of the method of ellipsoids is proposed in the paper that has the same asymptotic rate in volume convergence as the well-known Yudin-Nemirovskii-Shor method of ellipsoids. This method can be applied to one-dimensional problems, where it guarantees a decrease in the ratio of volumes equal to 2-
$$\sqrt {{\text{Hz}}}$$
2 ˜ 0.5858 on each i...

A brief survey of nondifferentiable optimization methods developed at the Institute of Cybernetics is presented: the subgradient method, the subgradient method with space dilatation in the subgradient direction, and the r-algorithm. Applications of nondifferentiable optimization methods are considered.

Many polynomial and discrete optimization problems can be reduced to multiextremal quadratic type models of nonlinear programming. For solving these problems one may use Lagrangian bounds in combination with branch and bound techniques. The Lagrangian bounds may be improved for some important examples by adding in a model the so-called superfluous...

New results are presented concerning binary correcting codes, such as deletion-correcting codes, transposition-correction codes, and codes for the Z-channel. These codes are important due to the possibility of packet loss and corruption on internet transmissions. It is known that the problem of finding the largest correcting codes can be reduced to...

Methods of nonsmooth optimization, particularly the r-algorithm, are applied to the problem of fitting an empirical utility function to expert’s estimates of ordinal utility under certain a priori constraints. Due to these methods, the fit can be performed not only with respect to the least squares criterion but with respect to the least moduli cri...

Nonsmooth-optimization methods with space expansion are considered as applied to decomposition schemes realized in solving two-stage problems of stochastic programming in the SLP—IOR simulation systems.

In Part II one-rank linear operators for space transformations are used to study the convergence of variable metric methods for solving convex programming problems. Two Fejer-type subgradient methods for finding the minimum of a convex function with known minimum value are proposed. Numerical experiments for these methods are stated.

Finding the global minimum of a polynomial of several variables is a very difficult problem already for polynomials of the fourth degree with more than five variables. Thus, the number of local minima for fourth-degree polynomials may reach several tens already for n=6. One of the methods to solve such problems is by reduction to nonlinear programm...

A sufficiently general construction using the linear transformation of a space is considered for cutting-plane methods. It is based on the external approximation of an extremum set by simple bodies with a guaranteed decreasing of their volumes at each step of the methods. Using this construction, it is easy to prove the convergence of the algorithm...

Conclusion Transformation (6) smoothing thef(x) level lines explains the effectiveness ofr(α)-algorithms from visual geometrical considerations. It may be regarded as a satisfactory interpretation of space dilation
in the direction of the difference of two successive subgradients. On the other hand, it preserves the gradient flavor of
the method, i...

The computational scheme of a new polynomial algorithm for solving linear programming problems is given, based on the use of a non-linear projective transformation.

## Projects

Projects (2)

This project was originally aimed to develop a useful collection of computational projection algorithms for solving the generic problem of finding the least-norm element in a convex polytope. Within this context, the polytope is understood as a convex hull of a finite number of points in a finite-dimensional space.
However, with time it became clear that once you effectively solve this problem then it allows solving a least-distance problem between finitely generated convex cones and outside points. The latter problem, interesting and important per se, in turn, allows us to solve the least-norm problem for convex polyhedrons. A term polyhedron here means a not necessarily bounded set obtained by the intersection of a finite number of half-spaces, i.e., described by systems of linear inequalities.
It opens many new areas to be investigated which we plan to explore. Among those: decomposition and parallel computations in projection algorithms, projection algorithms in nondifferentiable optimization, linear optimization by projection, and others.
Taking all this into account I have decided to change the name of the project toward more general “Practical projection with applications”

The project builds up on the seminal mathematical analyses of PCMC, realised by Bertram Schefold (1976), L. L. Pasinetti (1977) and Kurz & Salvadori (1995). The background is the Theorem of Perron-Frobenius (1907, 1912). The aim is to present a modern presentaton of the material to get an operational level with a lot of calculated examples and applications up to Input-Output Tables.