Search

Abstract The local equivalence of singular and ill-posed problems in a class of sufficiently smooth mappings is shown, which justifies the use of the p-factor regularization method to solve them. The main constructions in p-regularity theory that are necessary for stable solution of approximate problems are described, and estimation theorems for regularizing algorithms are proved.

New properties of convex infinitely differentiable functions related to extremal problems are established. It is shown that, in a neighborhood of the solution, even if the Hessian matrix is singular at the solution point of the function to be minimized, the gradient of the objective function belongs to the image of its second derivative. Due to this new property of convex functions, Newtonian methods for solving unconstrained optimization problems can be applied without assuming the nonsingularity of the Hessian matrix at the solution of the problem and their rate of convergence in argument can be estimated under fairly general assumptions.

Various choices of a finite-difference scheme for approximating the heat diffusion equation in solving a three-dimensional coefficient inverse problem were studied. A comparative analysis was conducted for several alternating direction schemes, such as locally one-dimensional, Douglas–Rachford, and Peaceman–Rachford schemes, as applied to nonlinear problems for the three-dimensional heat equation with temperature-dependent coefficients. Each numerical method was used to compute the temperature distribution inside a parallelepiped. The methods were compared in terms of the accuracy of the resulting solution and the computation time required for achieving the prescribed accuracy on a computer.

Some fundamental optimization results are proved in new ways, which are not traditional and provide a new view of well-known results. Constructions of $$p$$ -regularity theory are used to justify the facts under consideration, and the 2-factor method is applied to solve singular problems.

A numerical method combining a gradient technique with the projection onto a linear manifold is proposed for solving systems of linear inequalities. It is shown that the method converges in a finite number of iterations and its running time is estimated as a polynomial in the space dimension and the number of inequalities in the system.

The inverse problem under consideration is to determine a temperature-dependent thermal conductivity coefficient from experimental observations of the temperature field in the studied substance and (or) the heat flux on the surface of the object. The study is based on the Dirichlet boundary value problem for the nonstationary heat equation stated in the general n -dimensional formulation. For the numerical solution of the problem an algorithm based on the modern fast automatic differentiation technique is proposed.

Results obtained by the authors in solving inverse coefficient problems are overviewed. The inverse problem under consideration is to determine a temperature-dependent thermal conductivity coefficient from experimental observations of the temperature field in the studied substance and (or) the heat flux on the surface of the object. The study is based on the Dirichlet boundary value problem for the nonstationary heat equation stated in the general $$n$$-dimensional formulation. For this general case, an analytical expression for the cost functional gradient is obtained. The features of solving the inverse problem and the difficulties encountered in the solution process are discussed.

A Newton-type method is proposed for numerical minimization of convex piecewise quadratic functions, and its convergence is analyzed. Previously, a similar method was successfully applied to optimization problems arising in mesh generation. It is shown that the method is applicable to computing the projection of a given point onto the set of nonnegative solutions of a system of linear equations and to determining the distance between two convex polyhedra. The performance of the method is tested on a set of problems from the NETLIB repository.

A new method is proposed for computing the first variation of a functional in the case when the considered domain or its boundary contains singular points. In contrast to earlier proposed techniques, this method is simpler and applies to a larger class of equations governing optimization processes and to more general singularities occurring in the domain or on its boundary.

A fireproof composition based on ammonium polyphosphate and pentaerythritol with a number of functional additives was developed and studied. The additives are able to form a protective char during fire exposure below the thermal decomposition temperature of the polymer composites. The decrease in the char formation temperature of the fire-protective coating provides a molar excess of ammonium polyphosphate with respect to the mole fractions of pentaerythritol and aluminum hydroxide. Introducing the latter in the composition of the flame-retardant coating also contributes to the decrease in the char formation temperature. The fire-protection coating can be used to protect various combustible materials, e.g., wood, laminates, plastics, etc.

The classical Farkas theorem of the alternative is considered, which is widely used in various areas of mathematics and has numerous proofs and formulations. An entirely new elementary proof of this theorem is proposed. It is based on the consideration of a functional that, under Farkas’ condition, is bounded below on the whole space and attains a minimum. The assertion of Farkas’ theorem that a vector belongs to a cone is equivalent to the fact that the gradient of this functional is zero at the minimizer.

For the minimization problem for a differentiable function on a set defined by inequality constraints, a simple proof of the Kuhn–Tucker theorem in the Fritz John form is presented. Only an elementary property of the projection of a point onto a convex closed set is used. The approach proposed by the authors is applied to prove Farkas’ theorem. All results are extended to the case of Banach spaces.

In order to solve an underdetermined system of linear equations with nonnegative variables, the projection of a given point onto its solutions set is sought. The dual of this problem—the problem of unconstrained maximization of a piecewise-quadratic function—is solved by Newton’s method. The problem of unconstrained optimization dual of the regularized problem of finding the projection onto the solution set of the system is considered. A connection of duality theory and Newton’s method with some known algorithms of projecting onto a standard simplex is shown. On the example of taking into account the specifics of the constraints of the transport linear programming problem, the possibility to increase the efficiency of calculating the generalized Hessian matrix is demonstrated. Some examples of numerical calculations using MATLAB are presented.

In this paper, we consider new sufficient conditions of optimality of the second-order for equality constrained optimization problems, which essentially enhance and complement the classical ones and are constructive. For example, they establish equivalence between sufficient conditions in the equality constrained optimization problems and sufficient conditions for optimality in equality constrained problems by reducing the latter to equalities with the help of introducing slack variables. Previously, when using the classical sufficient optimality conditions, this fact was not considered to be true, that is, the existing classical sufficient conditions were not complete, so the proposed optimality conditions complement the classical ones and close the question of the equivalence of the problems with inequalities and the problems with equalities when reducing the first to the second by introducing slack variables.

A new method for solving the inequality constrained optimization problem is proposed for the case when the system of necessary optimality conditions of Kuhn—Tucker is degenerate. This situation occurs for example in the case when strict complementarity conditions fails in solution point. The reduction of the inequalities con- strained optimization problem to the equalities constrained problem is substantiated and the use of a new 2-fac- tor Newton method for the effective solution of the obtained degenerate system of optimality conditions is shown.

The classical Farkas theorem of the alternative is considered, which is widely used in various areas of mathematics and has numerous proofs and formulations. An entirely new elementary proof of this theorem is proposed. It is based on the consideration of a functional that, under Farkas’ condition, is bounded below on the whole space and attains a minimum. The assertion of Farkas’ theorem that a vector belongs to a cone is equivalent to the fact that the gradient of this functional is zero at the minimizer.

A new method is proposed for solving constrained optimization problems with inequality constraints in the case when the system of Kuhn–Tucker necessary optimality conditions is degenerate. This situation arises, for example, when the strict complementarity conditions do not hold for the solution. A reduction of the inequality-constraint problem to an equality-constraint one and the use of a new 2 factor Newton method for efficiently solving the resulting degenerate system of optimality conditions are justified.

The connection is established between theorems of the alternative for linear systems of equations and/or inequalities and duality theorems in linear programming. We give new versions of theorems of the alternative in which the alternative systems have different matrices of various sizes.

The problem of approximating the set of all solutions to a system of nonlinear inequalities is studied. A method based on the concept of nonuniform coverings is proposed. It allows one to obtain an interior and exterior approximation of this set with a prescribed accuracy. The efficiency of the method is demonstrated by determining the workspace of a parallel robot.

A method for solving the following inverse linear programming (LP) problem is proposed. For a given LP problem and one of its feasible vectors, it is required to adjust the objective function vector as little as possible so that the given vector becomes optimal. The closeness of vectors is estimated by means of the Euclidean vector norm. The inverse LP problem is reduced to a problem of unconstrained minimization for a convex piecewise quadratic function. This minimization problem is solved by means of the generalized Newton method.

A new efficient technique intended for the numerical solution of a broad class of optimal control problems for complicated dynamical systems described by ordinary and/or partial differential equations is investigated. In this approach, canonical formulas are derived to precisely calculate the objective function gradient for a chosen finite-dimensional approximation of the objective functional.

The space of one-dimensional stationary Schrodinger equations with a real-valued periodic potential and periodic boundary conditions is considered. An analytic and topological description of its foliation by hypersurfaces defined by the condition that the nth spectral lacuna has fixed length is given. The case when a lacuna degenerates into a point gives the Schwarzian derivative and the Arnold manifold. In the nondegenerate case, the linking number of the loop formed by potentials with shifted argument and an Arnold manifold is calculated. Bibliography: 12 titles.

A new proof of the Kuhn–Tucker theorem on necessary conditions for a minimum of a differentiable function of several variables in the case of inequality constraints is given. The proof relies on a simple inequality (common in textbooks) for the projection of a vector onto a convex set.

The paper provides some examples of mutually dual unconstrained optimization problems originating from regularization problems for systems of linear equations and/or inequalities. The solution of each of these mutually dual problems can be found from the solution of the other problem by means of simple formulas. Since mutually dual problems have different dimensions, it is natural to solve the unconstrained optimization problem of the smaller dimension.

Application of the non-uniform coverage method to the multicriteria optimization problems was considered, and the concept of the ɛ-Pareto set was formulated and studied. An algorithm to construct a ɛ-Pareto set with a guaranteed accuracy ɛ was described. Efficient implementation of this approach was described, and the results of experiments were presented.

A dual problem of linear programming is reduced to the unconstrained maximization of a concave piecewise quadratic function for sufficiently large values of a certain parameter. An estimate is given for the threshold value of the parameter starting from which the projection of a given point to the set of solutions of the dual linear programming problem in dual and auxiliary variables is easily found by means of a single solution of the unconstrained maximization problem. The unconstrained maximization is carried out by the generalized Newton method, which is globally convergent in an a finite number of steps. The results of numerical experiments are presented for randomly generated large-scale linear programming problems.

Given a system of nonlinear equations, a formula is derived for the family of its approximate solutions of up to the pth order of smallness. A formula approximating an implicit function up to the third order of smallness is presented. Iterative methods converging with the pth order are constructed for solving systems of nonlinear equations. These results are extended to the degenerate case. Examples of applying the results are given.

The paper describes a method for solving multi-objective optimization problems with box constraints. Unlike existing approaches, the proposed method not only constructs a finite approximation of Pareto frontier, but also proves its ϵ-optimality. The paper gives a detailed explanation of basic theoretical concepts behind the method and describes the algorithmic implementation. A practically important application of the proposed method to finding the working space of a robotic manipulator is presented.

The nonuniform covering method is applied to multicriteria optimization problems. The ?-Pareto set is defined, and its properties are examined. An algorithm for constructing an ?-Pareto set with guaranteed accuracy ? is described. The efficiency of implementing this approach is discussed, and numerical results are presented.

This article reviews the latest advances in the production and investigation of the properties of TiO2-based nanomaterials as photocatalysts for water decomposition into molecular oxygen and hydrogen, a component of disinfecting, photooxidative, and self-cleaning surfaces, and materials for sensors and catalysts.

The nonuniform covering method for global optimization of functions of several variables is extended to nonlinear programs. It is shown that this method can be used for solving problems that, in addition to conventional constraints, involve partial integrality conditions. Estimates for the accuracy of the solution and for the number of steps required for finding a minimum with a prescribed tolerance are derived. New minorants based on an estimate for the spectrum of the Hessian matrix of the objective function and the constraints are given. New formulas for covering sets improving the efficiency of the method are obtained. Examples of solving nonlinear programs with the use of the proposed approach are presented.

A brief review is given of insulating systems for electric machines and impregnating compounds as one of the key insulating components is given. Possibilities of producing and processing various types of domestic compounds are considered. Properties of glass mica tapes that contain an impregnating compound for cohesion are described.

The reactions of the formation of monosubstituted titanium ethylene, diethylene, and triethylene glycolates based on the interaction of glycols with tetrabutoxytitanium in solutions were investigated. It was shown that ethylene glycol predominantly forms cyclic structures: diethylene glycol, a mixture of cyclic and linear structures, and triethylene glycol, mainly linear structures. The comparative analysis of the reaction products showed that the most promising precursor for the synthesis of TiO2 is a titanium compound based on triethylene glycol.

Reaction of epoxy-diane resin with complex based on monoamide of phosphonic acid and ammonium chloride (MODAF) is studied. It is established that the MODAF equivalent in the reaction with epoxy resin is equal to 1/5 of its molecular weight. Epoxy resin is cured via the reactions of both monoamide and ammonium chloride. The dependence of the epoxide and hydroxyl groups has a stepped character over time, which is probably due to the existence of certain threshold concentrations of the curing coefficients. The investigated reaction is a new example of macroscopic coherence manifestation.