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Volume 61 >>> № 4 July — August 2025

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UDC 519.85

P.I. Stetsyuk¹

¹ V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

The scientific heritage of Academician Shor

Abstract. An overview of the scientific heritage of Academician N.Z. Shor is provided. His fundamental scientific ideas are considered, including the method of sequential analysis of variants (1961), subgradient method (1962), the use of linear non-orthogonal space transformations for improving the properties of ravine functions (1969), and a dual approach for obtaining and refining objective function estimates in non-convex quadratic models (1985). Examples of implementing these ideas in the methods and algorithms developed at the V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine are presented.

Keywords: non-smooth (non-differentiable) function, subgradient method, space dilation operator, ellipsoid method, r-algorithm, extremal quadratic problem.

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REFERENCES

1. Shor N.Z. Methods of Minimization of Non-Differentiable Functions and Their Applications. Kyiv: Nauk. Dumka, 1979. 200 p.

2. Shor N.Z. Minimization Methods for Non-Differentiable Functions. Berlin: Springer-Verlag, 1985. 178 p.

3. Mikhalevich V.S., Trubin V.A., Shor N.Z. Optimization Problems of Production and Transport Planning. Models, Methods, Algorithms. Moscow: Nauka, 1986. 260 p.

4. Shor N.Z. Nondifferentiable Optimization and Polynomial Problems. Boston; Dordrecht; London: Kluwer Academic Publ., 1998. 412 p.

5. Sergienko I.V., Stetsyuk P.I. On Three Scientific Ideas of N.Z. Shor. Kibernetika i sistemnyj analiz. 2012. No. 1. Pp. 4–22.

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7. Shor N.Z. Methods of Minimization of Non-Smooth Functions and Matrix Optimization Problems: Collection of Selected Papers. Chisinau: Eureka, 2009. 240 p.

8. Shor N.Z. Algorithms for Sequential and Nonsmooth Optimization: Collection of Selected Papers. Chisinau: Evrika, 2012. 272 p.

9. Mikhalevich V.S., Shor N.Z. Method of Sequential Analysis of Variants in Solving Variational Problems of Management, Planning, and Design. Abstract of the Report of the 4th All-Union Mathematical Congress (July 3–12, 1961, Leningrad). 1961. P. 91.

10. Mikhalevich V.S., Shor N.Z. Numerical Solutions of Multivariant Problems by the Method of Sequential Analysis of Variants. Scientific and Methodological Materials of the Economic and Mathematical Seminar. Moscow, 1962. Issue 1. P. 15–42. (Rotapr. / USSR Academy of Sciences. LEMI).

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12. Mikhalevich V.S., Shor N.Z., Galustova L.A. et al. Computational methods for selecting optimal design solutions. Kyiv: Nauk. dumka, 1977. 178 p.

13. Shor N.Z. Application of the gradient descent method for solving a network transport problem. Proc. of the scientific. seminar on theoretical and applied problems of cybernetics and operations research. Scientific council on cybernetics of the Ukrainian Academy of Sciences. Kyiv, 1962. Iss. 1. P. 9–17.

14. Polyak B.T. Introduction to Optimization. Moscow: Nauka, 1983. 384 p.

15. Shor N.Z., Biletsky V.I. Space Stretching Method for Accelerating Convergence in Ravine-Type Problems. Proc. of the Seminar of the Scientific Council of the Ukrainian Academy of Sciences on Cybernetics "Theory of Optimal Solutions". Kyiv, 1969. No. 2. P. 3–18.

16. Shor N.Z. Using the Space Stretching Operation in Minimizing Convex Functions. Kibernetika. 1970. No. 1. P. 6–12.

17. Shor N.Z. Cutting Method with Space Stretching for Solving Convex Programming Problems. Kibernetika. 1977. No. 1. P. 94–95.

18. Shor N.Z. Methods of minimization of non-differentiable functions and their applications: author's abstract. diss. ... doctor of physical and mathematical sciences. Kyiv, 1970. 44 p.

19. Shor N.Z., Zhurbenko N.G. Minimization method using the operation of space stretching in the direction of the difference of two successive gradients. Kibernetika. 1971. No. 3. Pp. 51–59.

20. Shor N.Z., Zhurbenko N.G., Likhovid A.P., Stetsyuk P.I. Development of non-differentiable optimization algorithms and their applications. Kibernetika i sistemnyj analiz. 2003. No. 4. Pp. 80–94.

21. Shor N.Z., Stetsyuk P.I. Using a modification of the r-algorithm to find the global minimum of polynomial functions. Kibernetika i sistemnyj analiz. 1997. No. 4. Pp. 28–49.

22. Stetsyuk P.I. Theory and software implementations of Shor's -algorithms. Kibernetika i sistemnyj analiz. 2017. Vol. 53, No. 5. Pp. 43–57.

23. Shor N.Z., Solomon D.I. Decomposition methods in fractional-linear programming. Chisinau: Shtiintsa, 1989. 204 p.

24. Mikhalevich V.S., Sergienko I.V., Trubin V.A., Shor N.Z. et al. Package of applied programs for solving large-scale production and transport planning problems (PLANER). Kibernetika. 1983. No. 3. pp. 57–71.

25. Mikhalevich V.S., Sergienko I.V., Shor N.Z. et al. Software package DISPRO-3: purpose, classes of problems to be solved, system and algorithmic support. Kibernetika. 1985. No. 1. pp. 56–71.

26. Mikhalevich V.S., Sergienko I.V., Shor N.Z. et al. Software package for solving discrete and nonlinear optimization problems (DISNEL package). Kibernetika. 1991. No. 3. pp. 36–45.

27. Shor N.Z., Sergienko I.V., Shylo V.P. et al. Problems of optimal design of reliable networks. Kyiv: Nauk. dumka, 2005. 230 p.

28. Kiseleva E.M., Shor N.Z. Continuous problems of optimal partitioning of sets: Theory, algorithms, applications. Kyiv: Nauk. dumka. 2005. 564 p.

29. Sergienko I.V., Mikhalevich M.V., Stetsyuk P.I., Koshlai L.B. Models and information technologies for decision support during structural and technological transformations. Kibernetika i sistemnyj analiz. 2009. No. 2. P. 26–49.

30. Stetsyuk P.I. Ellipsoid Methods and -Algorithms. Chisinau: Eureka. 2014. 488 p.

31. Shor N.Z., Davydov A.S. On a method for obtaining estimates in quadratic extremal problems with Boolean variables. Kibernetika. 1985. No. 2. P. 48–50.

32. Shor N.Z. On an approach to obtaining global extrema in polynomial problems of mathematical programming. Kibernetika. 1987. No. 5. P. 102–106.

33. Shor N.Z. On a class of estimates of the global minimum of polynomial functions. Kibernetika. 1987. No. 6. pp. 9–11.

34. Shor N.Z., Stetsenko S.I. Quadratic extremal problems and nondifferentiable optimization. Kyiv: Nauk. Dumka, 1989. 208 p.

35. Shor N.Z. Minimization of matrix functions and nondifferentiable optimization. In: Review of Applied and Industrial Optimization. Vol. 2. Kyiv, 1995. pp. 113–138.

36. Shor N.Z., Stetsyuk P.I. Lagrangian bounds in multiextremal polynomial and discrete optimization problems. J. of Global Optimization. 2002. Vol. 23. P. 1–41.

37. Stetsyuk P.I. Algorithms and Ellipsoids. Kibernetika i sistemnyj analiz. 1996. No. 1. P. 113–134.