2020, issue 1, p. 5-14
Received 07.02.2020; Revised 15.02.2020; Accepted 10.03.2020
Published 31.03.2020; First Online 26.04.2020
https://doi.org/10.34229/2707-451X.20.1.1
A STOCHASTIC SMOOTHING METHOD FOR NONSMOOTH GLOBAL OPTIMIZATION
V.I.
Norkin 1, 2 * 
1 V.M. Glushkov Institute of Cybernetics, The National Academy of Sciences of Ukraine, Kyiv, Ukraine
2 National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.
Abstract. The paper presents the results of testing the stochastic smoothing method for global optimization of a multiextremal function in a convex feasible subset of Euclidean space. Preliminarily, the objective function is extended outside the admissible region so that its global minimum does not change, and it becomes coercive. The smoothing of a function at any point is carried out by averaging the values of the function over some neighborhood of this point. The size of the neighborhood is a smoothing parameter. Smoothing eliminates small local extrema of the original function. With a sufficiently large value of the smoothing parameter, the averaged function can have only one minimum. The smoothing method consists in replacing the original function with a sequence of smoothed approximations with vanishing to zero smoothing parameter and optimization of the latter functions by contemporary stochastic optimization methods. Passing from the minimum of one smoothed function to a close minimum of the next smoothed function, we can gradually come to the region of the global minimum of the original function. The smoothing method is also applicable for the optimization of nonsmooth nonconvex functions. It is shown that the smoothing method steadily solves test global optimization problems of small dimensions from the literature.
Keywords: global optimization; Steklov smoothing; averaged functions; stochastic optimization; nonsmooth nonconvex optimization.
Cite as: Norkin V.I. A Stochastic Smoothing Method for Nonsmooth Global Optimization. Cybernetics and Computer Technologies. 2020. 1. 5–14. https://doi.org/10.34229/2707-451X.20.1.1
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2020, issue 1, p. 15-22
Received 03.03.2020; Revised 07.03.2020; Accepted 10.03.2020
Published 31.03.2020; First Online 26.04.2020
https://doi.org/10.34229/2707-451X.20.1.2
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Improving LagrangE Dual bounds for Quadratic Extremal Problems
Oleg
Berezovskyi 1 *
1 V.M. Glushkov Institute of Cybernetics, Kyiv, Ukraine
* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.
Introduction. Due to the fact that quadratic extremal problems are generally NP-hard, various convex relaxations to find bounds for their global extrema are used, namely, Lagrangian relaxation, SDP-relaxation, SOCP-relaxation, LP-relaxation, and others. This article investigates a dual bound that results from the Lagrangian relaxation of all constraints of quadratic extremal problem. The main issue when using this approach for solving quadratic extremal problems is the quality of the obtained bounds (the magnitude of the duality gap) and the possibility to improve them. While for quadratic convex optimization problems such bounds are exact, in other cases this issue is rather complicated. In non-convex cases, to improve the dual bounds (to reduce the duality gap) the techniques, based on ambiguity of the problem formulation, can be used. The most common of these techniques is an extension of the original quadratic formulation of the problem by introducing the so-called functionally superfluous constraints (additional constraints that result from available constraints). The ways to construct such constraints can be general in nature or they can use specific features of the concrete problems.
The purpose of the article is to propose methods for improving the Lagrange dual bounds for quadratic extremal problems by using technique of functionally superfluous constraints; to present examples of constructing such constraints.
Results. The general concept of using functionally superfluous constraints for improving the Lagrange dual bounds for quadratic extremal problems is considered. Methods of constructing such constraints are presented. In particular, the method proposed by N.Z. Shor for constructing functionally superfluous constraints for quadratic problems of general form is presented in generalized and schematized forms. Also it is pointed out that other special techniques, which employ the features of specific problems for constructing functionally superfluous constraints, can be used.
Conclusions. In order to improve dual bounds for quadratic extremal problems, one can use various families of functionally superfluous constraints, both of general and specific type. In some cases, their application can improve bounds or even provide an opportunity to obtain exact values of global extrema.
Keywords: quadratic extremal problem, Lagrangian relaxation, dual bound, functionally superfluous constraints.
Cite as: Berezovskyi O. Improving Lagrange Dual Bounds for Quadratic Extremal Problems. Cybernetics and Computer Technologies. 2020. 1. 15–22. (in Ukrainian) https://doi.org/10.34229/2707-451X.20.1.2
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8. Stetsyuk P.I. Functionally redundant constraints for Boolean quadratic-type optimization problems. Cybernetics and Systems Analysis. 2005. 41 (6). P. 932–935. https://doi.org/10.1007/s10559-006-0029-z
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12. Berezovskyi O.A. On the lower bound for a quadratic problem on the Stiefel manifold. Cybernetics and Systems Analysis. 2008. 44 (5). P. 709–715. https://doi.org/10.1007/s10559-008-9038-4
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2020, issue 1, p. 32-40
Received 21.01.2020; Revised 06.02.2020; Accepted 10.03.2020
Published 31.03.2020; First Online 26.04.2020
https://doi.org/10.34229/2707-451X.20.1.4
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MSC 74A15, 74F05, 74K25, 74S05
ON CERTAIN PROBLEMS OF IDENTIFICATION OF THERMAL DENSITY OF THE TEMPERATURE STATE OF THE HOLLOW CYLINDER SHELL
Albina Aralova 1 *
1 V.M. Glushkov Institute of Cybernetics, Kyiv, Ukraine
* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.
Introduction. In conditions of the active use of composite materials, as when accomplishing the tasks of extending the service life of existing structures, problems on recovering unknown parameters of their components under the known data on their surface arise. In [1–4], to solve the problems of identification of parameters of a wide range, it is proposed to construct explicit expressions of the gradients of residual functionals by means of the corresponding conjugate problems obtained from the theory of optimal control of the states of multicomponent distributed systems, which is the development of the corresponding researches of Zh. Lyons. In [5–7], this technology is extended to the problem of thermoelastic deformation of multicomponent bodies.
In this article some problems of optimal control of the temperature state of a cylindrical body with a cavity are considered.
The purpose of the paper is to show the algorithm for identifying the parameters of a cylindrical hollow shell, based on the theory of optimal control and using the gradient methods of Alifanov.
Results. Based on the theory of optimal control, the temperature control of a cylindrical shell is studied. To solve the problem of identifying the parameters of a hollow cylindrical shell, namely, finding the heat flux powers on its surfaces, based on [1, 2, 5–7], a direct and conjugate problem and gradients of non-viscous functionals are constructed. Discretization by the finite element method using piecewise quadratic functions is carried out and accuracy estimates for it are presented. The initial problem in the model examples presented is solved using gradient methods, where at each step of determining the (n + 1) the approximation of the solution, the direct and adjoint problems are solved using finite element method with the help piecewise quadratic functions by minimizing the corresponding energy functional.
A number of model examples solved.
Keywords: temperature state, gradient methods, cylindrical bodies.
Cite as: Aralova A. On Certain Problems of Identification of Thermal Density of the Temperature State of the Hollow Cylinder Shell. Cybernetics and Computer Technologies. 2020. 1. 32–40. (in Ukrainian) https://doi.org/10.34229/2707-451X.20.1.4
References
1. Sergienko I.V., Deineka V.S. System analysis of multicomponent distributed systems. Kiev: Nauk. Dumka, 2009. 640 p. (in Russian)
2. Sergienko I.V., Deineka V.S. Identification of the parameters of the stress-strain state of a multicomponent elastic body with inclusion. Prikladnaya mekhanika. 2010. 46 (4). P. 14–24. https://doi.org/10.1007/s10778-010-0319-z
3. Sergienko I.V., Deineka V.S. Optimal control of distributed systems with conjugation conditions. New York: Kluwer Academic Publishers, 2005. 400 p.
4. Deineka V.S. Models and methods for solving problems in heterogeneous environments. Kiev: Nauk. Dumka, 2001. 606 p. (in Russian)
5. Aralova A.A., Deineka V.S. Numerical solution of inverse boundary value problems of axisymmetric thermoelastic deformation of a long thick hollow cylinder. Komp'yuternaya matematika. 2011. 1. P. 3–12. (in Russian) http://dspace.nbuv.gov.ua/handle/123456789/84653
6. Aralova A.A., Deineka V.S. Optimal control of the hollow cylinder thermal stress. Dopovidi natsionalʹnoyi akademiyi nauk Ukrayiny. 2012. 5. P. 38–42. (in Russian) http://dspace.nbuv.gov.ua/handle/123456789/49803
7. Aralova A.A. Numerical solution of inverse problems of thermoelasticity for a composite cylinder. Cybernetics and System Analysis. 2014. 50 (5). P. 164–172. https://doi.org/10.1007/s10559-014-9670-0
8. Kovalenko A.D. Thermoelasticity. Kiev: Vishcha shkola, 1975. 216 p. (in Russian)
9. Motovilevets I.A., Kozlov V.I. The mechanics of related fields in structural elements. Vol. 1. Termouprugost'. Kiev: Nauk. Dumka, 1987. 264 p. (in Russian)
10. Lyons J.-L. Optimal control of systems described by partial differential equations. M .: Mir, 1972. 414 p. (in Russian)
11. Alifanov O.M., Artyukhin E.A., Rumyantsev S.V. Extreme methods for solving incorrect tasks. M .: Nauka, 1988. 288 p. (in Russian)
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2020, issue 1, p. 23-31
Received 09.02.2020; Revised 18.02.2020; Accepted 10.03.2020
Published 31.03.2020; First Online 26.04.2020
https://doi.org/10.34229/2707-451X.20.1.3
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ON CONSTRUCtion of The EXTERNAL FRANKL Nozzle COntour Using QUADRATIC CURVATure
Petro
Stetsyuk 1 * ,
Oleksandr Tkachenko 2, Olga Gritsay 2
1 V.M. Glushkov Institute of Cybernetics, Kyiv, Ukraine.
2 SE "Ivchenko-Progress", Zaporozhye, Ukraine.
* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.
The aim of the article is to develop a method, an algorithm, and appropriate software for constructing the external contour of the Frankl nozzle in the supersonic part using S-shape curves. The method is based on the problem of constructing a curve with the natural parameterization. The curve passes through two given points with the given inclination angles of the tangents and provides the given inclination angle of the tangent at the point with the given abscissa [4]. To control the inflection point of the S-shaped curve, the inclination angle of the tangent at a point with the known abscissa is used.
In the case, when the curvature is given by a quadratic function, the system of five nonlinear equations is formulated, among which three equations are integral. The system has five unknown variables – three coefficients of the quadratic function, the total length of the curve and the length of the curve to the point with a known abscissa.
The lemma on the relation between solutions of the original and the scalable systems, in which the coordinates of the points are multiplied by the same value, is proved. Due to this lemma, it becomes possible, using the obtained solution of the well-scalable system, to find easily the corresponding solution of a bad-scalable (singular) system.
To find a solution to the system, we suggest to use the modification of the r-algorithm [5] solving special problem on minimization of the nonsmooth function (the sum of the modules of the residuals of the system), under controlling of the constraints on unknown lengths, in order to guarantee their feasible values.
The algorithm is implemented using the multistart method and the ralgb5a octave function [6]. It finds the best local minimum of nonsmooth function by starting the modification of the r-algorithm from a given number of starting points. The algorithm uses an analytical computation of generalized gradients of the objective function and the trapezoid rule to calculate the integrals.
The computational experiment was carried out to design the fragment of supersonic part in the external contour of a Frankl-type nozzle. The efficiency of the algorithm, developed for constructing S-shape curves, is shown.
Keywords: nozzle contour, natural parameterization, curvature, nonsmooth optimization, r-algorithm
Cite as: Stetsyuk P., Tkachenko O., Gritsay O. On Construction of the External Frankl Nozzle Contour Using Quadratic Curvature. Cybernetics and Computer Technologies. 2020. 1. 23–31. (in Ukrainian) https://doi.org/10.34229/2707-451X.20.1.3
References
1. Alemasov V.E. Theory of rocket engines: Textbook for students of higher technical educational institutions/ Alemasov V.E., A.F. Dregalin, A.P. Tishin; Ed. V.P. Glushko. M. Mashynostroenie, 1989. 464 p. (in Russian)
2. Rashevskii P.K. Differential Geometry Course. 4 edition. M. Gostekhizdat, 1956. 420 p. (in Russian)
3. Borysenko V., Agarkov O.,Pal’ko K, Pal’ko M. Modeling of plane curves in natural parameterization. Geometrychne modeliuvania ta informaciini tekhnologii. 2016. 1. P. 21 – 27. (in Ukrainian) http://nbuv.gov.ua/UJRN/gmtit_2016_1_6
4. Borysenko V. D., Ustenko S.A., Ustenko I.V. Geometric modeling of s-shaped skeletal lines of axial compressor blades profiles. Vestnik dvigatelestroeniia. 2018. 1. P. 45 – 52. (in Russian) https://doi.org/10.15588/1727-0219-2018-1-7
5. Shor N.Z., Stetsyuk P.I. Modified r-algorithm to find the global minimum of polynomial functions. Cybernetics and Systems Analysis. 1997. 33(4) P. 482 – 497. https://doi.org/10.1007/BF02733104
6. Stetsyuk P.I. Computer program "Octave program ralgb5a: r(α)-algorithm with adaptive step". Svidotstvo pro rejestratsiju avtorskogo prava na tvir 8501. Ukraine. Ministerstvo ekonomichnogo rozvytku I torgivli. Derzhavnyi department intelektualnoji vlasnosti. Data reiestratsii 29.01.2019. (in Ukrainian)
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2020, issue 1, p. 41-52
Received 27.02.2020; Revised 02.03.2020; Accepted 10.03.2020
Published 31.03.2020; First Online 26.04.2020
https://doi.org/10.34229/2707-451X.20.1.5
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CONSTRUCTING THE APPROXIMATE SOLUTION OF AXISYMMETRIC PROBLEM ON THE DYNAMICS OF ANISOTHERMAL MOISTURE TRANSFER
Olga Marchenko 1, Tetiana Samoilenko 1 *
1 V.M. Glushkov Institute of Cybernetics, Kyiv, Ukraine
* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.
Introduction. Calculation of dynamics of the anisothermal moisture transfer processes in axisymmetric formulation is essential in the study of wet soils condition around, for example, vertical drains, wells, piles, etc. In this paper, we formulate the initial boundary value problem for the system of moisture and heat transfer nonstationary equations. The problem is considered for isotropic medium in cylindrical coordinate system under the inhomogeneous mixed boundary conditions.
The obtained results are important for future research in cylindrical coordinates of problems that model the migration of moisture during the seasonal freezing of the soil, taking into account phase transitions from unfrozen water to ice in the entire volume of the soil mass without highlighting the crystallization front. In this case moisture exchange and heat transfer characteristics appear as functions of the total humidity. Consequently, the equation of moisture transfer is written relative to the "fictitious" moisture content. Because of the main direction of moisture migration relative to the freezing/melting front, the convective heat transfer along the vertical coordinate axis is considered to be essential that leads to sufficient coincidence with the experimental data.
The purpose of the paper is to formulate the appropriate generalized problem in the Galorkin form for the axisymmetric initial-boundary value problem. The important goal is to investigate the accuracy of the continuous in time and completely discrete approximate generalized solutions based on the finite elements method.
Results. The algorithm for constructing of approximate generalized solution of the axisymmetric initial-boundary value problem for the system of filtration and heat transfer equations is proposed. The estimates of the convergence rate for the continuous in time and discrete approximate solutions based on the finite elements method are obtained.
Keywords: moisture transfer, heat transfer, axisymmetric initial boundary value problem, generalized solution, finite elements method, Crank-Nicolson scheme.
Cite as: Marchenko O., Samoilenko T. Constructing the Approximate Solution of Axisymmetric Problem on the Dynamics of Anisothermal Moisture Transfer. Cybernetics and Computer Technologies. 2020. 1. 41–52. (in Ukrainian) https://doi.org/10.34229/2707-451X.20.1.5
References
1. Pavlov A.R., Permyakov P.P. Mathematical model and computer calculation algorithms of heat and mass transfer during soil freezing. Journal of Engineering Physics. 1983. 44 (2). P. 311–316. https://doi.org/10.1007/BF00826153
2. Marchenko O.A., Lezhnina N.А. Appoximate solution of the problem of moisture and heat transfer in freezing soils by the finite elements method. Computer mathematics. 2002. 1. P. 24–33. (in Russian)
3. Marchenko O.A., Samoilenko T.A. Analyzing an approximate solution of a quasilinear parabolic-hyperbolic problem. Cybern Syst Anal. 2012. 48 (5). P. 142–154. (in Russian) https://doi.org/10.1007/s10559-012-9455-2
4. Shastunova U. Yu. Experimental study and a mathematical model of the processes in frozen soil under a reservoir with a hot heat-transfer agent / A. A. Kislitsyn, U. Yu. Shastunova, Yu. F. Yanbikova. Journal of Engineering Physics and Thermophysics. 2018. 91 (2). Р. 507–514. https://doi.org/10.1007/s10891-018-1771-6
5. Bogaenko V.A., Marchenko O.A., Samoilenko T.A. An analysis of numerical non-isothermal processes in a soil massif modeling. Computer mathematics. 2016. 2. P. 3–11. (in Russian) http://nbuv.gov.ua/UJRN/Koma_2016_2_3
6. Deineka V.S., Sergienko I.V., Skopetsky V.V. Mathematical models and computation methods for problems with discontinuous solutios. Kyiv: Naukova dumka, 1995. 262 p. (in Russian)
7. Rektorys K. Variational Methods in Mathematics, Science and Engineering. Springer Netherlands, 1977. 571 p. https://doi.org/10.1007/978-94-011-6450-4
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