When developing numerical methods for solving nonlinear minimax problems, the following auxiliary problem arose: in the convex hull of a certain finite set in Euclidean space, find a point that has the smallest norm. In 1971, B. Mitchell, V. Demyanov and V. Malozemov proposed a non-standard algorithm for solving this problem, which was later called the MDM algorithm (based on the first letters of the authors' last names). This article considers a specific minimax problem: finding the smallest volume ball containing a given finite set of points. It is called the Sylvester problem and is a special case of the problem about the Chebyshev center of a set. The Sylvester problem is associated with a convex quadratic programming problem with simplex constraints. To solve this problem, it is proposed to use a variant of the MDM algorithm. With its help, a minimizing sequence of feasible solutions is constructed such that two consecutive feasible solutions differ in only two components. The indices of these components are selected based on certain optimality conditions. We prove the weak convergence of the resulting sequence of feasible solutions that implies that the corresponding sequence of vectors converges in norm to a unique solution to the Sylvester problem. Four typical examples on a plane are given.

A deep factorization of the orthogonal projection matrix onto a subspace is obtained. The LQ decomposition is used. For construction of an orthogonal matrix Q the method of successive rank reduction is applied.

The term “mathematical diagnostics” was introduced by V. F. Demyanov in the early 2000s. The simplest problem of mathematical diagnostics is to determine the relative position of some point p and the convex hull C of a finite number of given points in n-dimensional Euclidean space. Of interest is the answer to the following questions: does the point p belong to the set C or not? If p does not belong to C, then what is the distance from p to C? In the general problem of mathematical diagnostics, two convex hulls are considered. The question is whether they have common points. If there are no common points, then it is required to find the distance between these hulls. From an algorithmic point of view, the problems of mathematical diagnostics reduce to special linear- or quadratic-programming problems, which can be solved by finite methods. However, the implementation of this approach in the case of large data arrays runs into serious computational difficulties. Such situations can be dealt with by infinite but easily implemented methods, which allow one to obtain an approximate solution with the required accuracy in a finite number of iterations. These methods include the MDM method. It was developed by Mitchell, Demyanov, and Malozemov in 1971 for other purposes, but later found application in machine learning. From a modern point of view, the original version of the MDM method can be used to solve only the simplest problems of mathematical diagnostics. This article gives a natural generalization of the MDM method, oriented towards solving general problems of mathematical diagnostics. In addition, it is shown how, using the generalized MDM method, a solution to the problem of the linear separation of two finite sets, in which the separating strip has the largest width, is found.

Continuous piecewise affine functions are widely used in computational mathematics. In the one-dimensional case, such functions are called broken lines. The paper analyzes the analytical representations of broken lines both in the forms accepted in the theory of polynomial splines and in the form of the difference of the maxima of two finite families of affine functions. We establish a correlation between these representations.

Let two points a and b located to the right and left of the interval [–1, 1], respectively, be given on the real axis. The extremal problem is stated as follows: find an algebraic polynomial of the n-th degree, whose value is A at point a, it does not exceed M in absolute value in the interval [–1, 1], and takes the largest possible value at point b. This problem is connected with the second Zolotarev problem. A set of values of the parameter A, for which this problem has a unique solution, is indicated in this paper and an alternance characteristic of this solution is given. The behavior of the solution with respect to the parameter A is studied. It is found that the solution can be obtained for certain A using the Chebyshev polynomial, and can be obtained for all other admissible A with the help of the Zolotarev polynomial.

In this chapter we consider basic transforms of signals. The centerpiece are discrete Fourier transform, cyclic convolution, and cyclic correlation. We study the properties of these transforms. As an application, we provide solutions to the problems of optimal interpolation and optimal signal–filter pair. Separate sections are devoted to ensembles of signals and to the uncertainty principle in discrete harmonic analysis.

The focus of this chapter is on fast algorithms: the fast Fourier transform, the fast Haar transforms, and the fast Walsh transform. To build a fast algorithm we use an original approach stemming from introduction of a recurrent sequence of orthogonal bases in the space of discrete periodic signals. On this way we manage to form wavelet bases which altogether constitute a wavelet packet. In particular, Haar bases are wavelet ones. We pay a lot of attention to them in the book. We investigate an important question of ordering of Walsh functions. We analyze in detail Ahmed–Rao bases that fall in between Walsh basis and the exponential basis. The main version of the fast Fourier transform (it is called the Cooley–Tukey algorithm) is targeted to calculate the DFT whose order is a power of two. In the end of the chapter we show how to use the Cooley–Tukey algorithm to calculate a DFT of any order.

In this chapter we introduce discrete periodic splines and study their fundamental properties. We establish an extremal property of the interpolation splines. In terms of splines we offer an elegant solution to the problem of smoothing of discrete periodic data. We construct a system of orthogonal splines. With the aid of dual splines we solve the problem of spline processing of discrete data with the least squares method. We obtain a wavelet expansion of an arbitrary spline. We prove two limit theorems related to interpolation splines.

A fast algorithm for solving the Danskin problem is proposed. The dependence of its solution on parameters is analyzed.

The article is dedicated of memory of Professor V. F. Demyanov (1938-2014). The main scientific interests of V. F. Demyanov lay in the field of numerical optimization methods, where the notion of the direction of the steepest descent plays an important role. This notion is introduced for both smooth and nonsmooth functions, both in constrained and unconstrained cases. This paper provides a detailed analysis of methods for constructing the direction of steepest descent. In all cases, it comes down to solving the quadratic programming problem. Particular attention is paid to nonsmooth functions, in the study of which V. F. Demyanov made a significant contribution. An example of a function which is quasidifferentiable at a point is given. This function has two directions of the steepest descent and two directions of the steepest ascent.

One of the main tasks of mathematical diagnostics is the strict separation of two finite sets in a Euclidean space. Strict linear separation is widely known and reduced to the solution of a linear programming problem. We introduce the notion of strict polynomial separation and show that the strict polynomial separation of two sets can be also reduced to the solution of a linear programming problem. The objective function of the linear programming problem proposed in this paper has the following feature: its optimal value can be only zero or one, i.e., it is zero if the sets admit strict polynomial separation and one otherwise. Some illustrative examples of the strict separation of two sets on a plane with the use of fourth degree algebraic polynomials in two variables are given. The application efficiency of strict polynomial separation to binary data classification problems is analyzed.

В статье сформулирована и доказана обобщенная лемма Гиббса.Ее заключение согласовано с определением равновесия по Вардропу в транспортных сетях

A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen's inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.

The monograph published by E.Ya. Remez in 1957 addressed numerical methods with Chebyshev approximations. Particularly, the problem of the best uniform approximation of a function that is convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A.M. Vershik, V.N. Malozemov, and A.B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was found that such partition exists and generates the best piecewise polynomial approximation. In addition, a numerical method for constructing a partition with equal deviations was proposed. This paper gives three examples to describe the general approach to solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.

The construction of a rational function that is nonnegative on two intervals of which one is infinite is considered. It is assumed that the maximum deviation of the function from zero on the infinite interval takes the minimum possible value under the condition that the values of the function on the finite interval are within the given bounds. It is assumed that the rational function (fraction) has the complete alternance. In this case, the original problem is reduced to solving a system of nonlinear equations. For solving this system, a two-stage method is proposed. At the first stage, a subsystem is selected and used to find a good approximation for the complete system. At the second stage, the complete system of nonlinear equations is solved. The solution is explained in detail for the case when the order of the fraction is between one and four. Numerical results for a fraction of order ten are presented.

An extremal curve of the simplest variational problem is a continuously differentiable function. Hilbert's differentiability theorem provides a sufficient condition for the existence of the second derivative of an extremal curve. It is desirable to have a simple example in which the condition of Hilbert's theorem is violated and an extremal curve is not twice differentiable.In this paper, a cubic variational problem with the following properties is analyzed. The functional of the problem is bounded neither above nor below. There exists an extremal curve for this problem which is obtained by sewing together two different extremal curves and not twice differentiable at the sewing point. Despite this unfavorable situation, an attempt to apply the method of steepest descent (in the form proposed by V.F. Dem'yanov) to this problem is made. It turns out that the method converges to a stationary curve provided that a suitable step size rule is chosen.

Two fast orthogonal projection algorithms of a point onto the canonical simplex are analyzed. These algorithms are called the vector and scalar algorithms, respectively. The ideas underlying these algorithms are well known. Improved descriptions of both algorithms are given, their finite convergence is proved, and exact estimates of the number of arithmetic operations needed for their implementation are derived, and numerical results of the comparison of their computational complexity are presented. It is shown that on some examples the complexity of the scalar algorithm is maximal but the complexity of the vector algorithm is minimal and conversely. The orthogonal projection of a point onto the solid simplex is also considered.