Vladimir I. Norkin

Biography

Vladimir Norkin, Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine (Ukraine), male, born 04.01.1951.

• Vladimir Norkin is an expert in Operations Research, Dr. of Sciences, and part-time Professor of the Chair of Applied Mathematics (since 2015) at the Faculty of Applied Mathematics of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, a leading researcher (since 2001) at V.M.Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine.

• He graduated from Kolmogorov’s mathematical school (1966-1968) and from Moscow Institute of Physics and Technology, Faculty of Control Theory and Applied Mathematics (1974), and since 1974 is employed with V.M.Glushkov Institute of Cybernetics (Kyiv, Ukraine); obtained Candidate (1984) and Doctor (1998) degree in Operations Research from this Institute.

• He published more than 100 research papers in the field of operations research and its applications and a monograph “Methods of Nonconvex Optimization”, Nauka, Moscow, 1987 (with co-authors). His research interests include optimization theory, stochastic programming, mathematical methods of systems analysis, financial and actuarial mathematics, mathematical economics, mathematical statistics, machine learning.

• He visited and collaborated with International Institute for Applied Systems Analysis (Laxenburg, Austria, 1994-2002, part-time contracts), Free University of Amsterdam (the Netherlands, 2005-2008, visitor), University of California (Davis, USA, 2010.10-2011.09, full time), University of Vienna (2022.04-2022.06), Chemnitz University of Technology (Chemnitz, Germany, 2022.07-2023.06).

• He was a member of the International Committee on Stochastic Programming (1996-2002), a member of Editorial Board of “SIAM Journal of Optimization” (2008-2016). Currently, he is a member of Committee on Systems Analysis at the National Academy of Sciences of Ukraine (2022-pres.).

• He is a laureate of “Kjell Gunnarson's Risk Management Prize” (1997) of the Swedish Insurance Society, “V.M. Glushkov Prize in the field of informatics” (2007) of the National Academy of Sciences of Ukraine, a holder of Fulbright Scholarship (2010.10-2011.09), a year grand from Volkswagen Foundation (2022.07-2023.06).

• He was the leader of project G3127 financed by Science and Technology Center in Ukraine (STCU, 2005-2006), the leader of project 2020.02/0121 financed by National Research Fund of Ukraine (2020-2023), and participated in many international projects.

• He participated in the international conferences: VI, VII, IX, XI, XIII, XV international conferences on stochastic programming (1992 - 2019), International Mathematical Congress (1998, Berlin), International Symposium on Mathematical Programming (2003, Copenhagen), European Conference on Operations Research (EURO-2012, Vilnius), Ukrainian Mathematical Congress (2001, 2009, Kyiv), The World Congress on Global Optimization (2023, Athens, Greece), 4th International Conference "Numerical Computations: Theory and Algorithms" (2023 , Calabria, Italy) and others.

• Ratings in international scientific data bases (01.01.2024):

Scopus: h–index (Hirsch index) – 12; WoS: h–index – 11;

Goоgle Scholar: citations – 2164, h-index – 21.

https://orcid.org/0000-0003-3255-0405 , https://www.scopus.com/authid/detail.uri?authorId=6602822409

https://www.webofscience.com/wos/author/record/335225

https://scholar.google.com.ua/citations?user=s0c4hIEAAAAJ&hl=en

Main scientific results

The concept of generalized-differentiable functions was introduced (Norkin, 1978, 1980, 1983).

• Norkin V.I. Nonlocal minimization algorithms of nondifferentiable functions. Cybernetics, 1978. Vol. 14, No. 5, P. 704-707. https://doi.org/10.1007/BF01069307

• Norkin V.I. Generalized differentiable functions. 1980. Cybernetics, Vol. 16, No. 1, P. 10-12. https://doi.org/10.1007/BF01099354

• Norkin V.I. Generalized gradient method in nonconvex nonsmooth optimization. Abstract of Candidate (Ph.D.) Thesis, V.M. Glushkov Institute of Cybernetics, Kiev, 1983, 16 p. (In Russian).

The concept of mollified (smoothed) subdifferential was introduced (Ermoliev, Norkin, Wets, 1992, 1995).

• Ermoliev Yu.M., Norkin V.I., and Wets R.J-B. The minimization of semicontinuous functions: Mollifier subgradients, SIAM Journal on Control and Optimization, 1995, No. 1, P. 149-167. https://doi.org/10.1137/S0363012992238369

• Ermoliev Y.M., Norkin V.I. On Constrained Discontinuous Optimization. Proceedings of 3rd GAMM/IFIP Workshop (Neubiberg/Munchen, 1996), Stochastic optimization: Numerical methods and technical applications, Lecture Notes in Economics and Mathematical Systems 458, Berlin, Springer, 1998, P. 128-142. https://doi.org/10.1007/978-3-642-45767-8_7

The concept of normalized convergence of random variables was introduced (Norkin, 1989, 1992, Ermoliev, Norkin, 1990, 1991).

• Ermoliev Yu.M., Norkin V.I. Normalized convergence of random variables and its applications. Cybernetics, 1990, Vol. 26, No. 6, P. 912-918.  https://doi.org/10.1007/BF01069497

• Ermoliev Yu.M., Norkin.V.I. Normalized convergence in stochastic optimization. Annals of Operations Research, 1991, No. 1, P.187-198. https://doi.org/10.1007/BF02204816

• Norkin V.I. Normalized convergence of random variables. Cybernetics and Systems Analysis, 1992, Vol. 28, No. 3, P. 396-402. https://doi.org/10.1007/BF01125420

• Norkin V.I. Stability of stochastic optimization models and statistical methods of stochastic programming. Preprint 89-53, Glushkov Institute of Cybernetics, Kiev, 1989 (In Russian).

The theory of generalized differentiable functions and corresponding methods of non-smooth non-convex deterministic and stochastic optimization have been developed. (Norkin, 1978-1986; Mikhalevich, Gupal, Norkin, 1987; Ermoliev, Norkin, 1998, 2003).

• Mikhalevich V.S., Gupal A.M., Norkin V.I. Methods of Nonconvex Optimization. Moscow, Nauka, 1987. (In Russian).

• Norkin V.I. A method of minimizing an undifferentiable function with generalized-gradient averaging. Cybernetics, 1980, Vol. 16, No. 6, P. 890-892. https://doi.org/10.1007/BF01069064

• Norkin V.I. Method of generalized gradient descent. Cybernetics, 1985, Vol. 21, No. 4, P. 495-505. https://doi.org/10.1007/BF01070609

• Norkin V.I. Stochastic Lipschitz functions. Cybernetics, 1986. Vol. 22, No. 2, P. 226-233. https://doi.org/10.1007/BF01074785

• Norkin V.I. Stochastic generalized-differentiable functions in the problem of nonconvex nonsmooth stochastic optimization. Cybernetics, 1986. Vol. 22, No. 6, P. 804-809. https://doi.org/10.1007/BF01068698

• Norkin V.I. Stochastic methods for solving nonconvex stochastic programming problems and their applications. Abstract of Doctor Thesis, Glushkov Institute of Cybernetics of the Ukrainian Academy of Sciences, Kiev, 1998, 32 p. (In Ukrainian).

• Ermoliev Y.M., Norkin V.I. Solution of nonconvex nonsmooth stochastic optimization problems. Cybernetics and systems analysis, 2003, Vol. 39, No.5, P. 701-715. https://doi.org/10.1023/B:CASA.0000012091.84864.65 (Q4)

• Ermoliev Yu.M., Norkin V.I. Stochastic generalized gradient method for solving nonconvex nonsmooth stochastic optimization problems. Cybernetics and systems analysis, 1998, Vol. 34, No. 2, P. 196–215. https://doi.org/10.1007/BF02742069

The theory of α-concave functions and measures was developed (Norkin, 1989, 1983; Norkin, Royenko, 1989, 1991).

• Norkin V.I., Roenko N.V. a-concave functions and measures and their applications. Cybernetics and Systems Analysis, 1991, Vol. 27, No. 6, P. 860-869. https://doi.org/10.1007/BF01246517

• Norkin V.I. The Analysis and Optimization of Probability Functions, Working paper WP-93-6, Int. Inst. for Appl. Syst. Anal., Laxenburg, Austria, 1993.

• Norkin V.I. Optimization of probabilities. Preprint 89-6, Glushkov Institute of Cybernetics, Kiev, 1989 (In Russian).

A stochastic branch-and-bound method was developed for solving non-convex stochastic optimization problems based on the method of interchainge relaxation (Norkin, 1993; Norkin, Ermoliev, Ruszczynski, 1994, 1998; Norkin, Pflug, Ruszczynski, 1996, 1998; Norkin, Onischenko, 2008).

• Norkin V.I. Global Stochastic Optimization: Branch and Probabilistic Bound Method. In: Methods of Control and Decision-Making under Risk and Uncertainty, Ed. Yu.M.Ermoliev, Glushkov Institute of Cybernetics, Kiev, 1993, P. 3-12 (In Russian).

• Norkin V.I., Pflug G.Ch. and Ruszczynski A. A branch and bound method for stochastic global optimization. Math. Progr., 1998, Vol. 83, P. 425-450. https://doi.org/10.1007/BF02680569

• Norkin V.I., Ermoliev Yu.M., Ruszczynski A. On optimal allocation of indivisibles under uncertainty. Operations Research, 1998, Vol. 46, No. 3, P.381-395. https://doi.org/10.1287/opre.46.3.381

• Norkin V.I. Global Optimization of Probabilities by the Stochastic Branch and Bound Method, Proceedings of 3rd GAMM/IFIP Workshop (Neubiberg/Munchen, 1996), Stochastic optimization: Numerical methods and technical applications, Lecture Notes in Economics and Mathematical Systems 458, Berlin, Springer, 1998, P. 186-201. https://doi.org/10.1007/978-3-642-45767-8_11

• Norkin V.I., Onishchenko B.O. Reliability optimization of a complex system by the stochastic branch and bound method. Cybernetics and systems analysis, 2008, Vol. 44, No. 3, P. 418-428. DOI:10.1007/s10559-008-9000-5  (Q3)

A stochastic smoothing method has been developed for solving non-smooth and discontinuous optimization problems (Gupal,  Norkin, 1977;  Ermoliev,  Norkin,  Wets, 1992, 1995; Ermoliev, Norkin, 1997, 2004; Norkin, 2020).

• Gupal A.M., Norkin V.I. Algorithm for the minimization of discontinuous functions. Cybernetics, 1977. Vol. 13, No. 2, P. 220-223. https://doi.org/10.1007/BF01073313

• Norkin V.I. A stochastic smoothing method for nonsmooth global optimization. Cybernetics and Computer technologies. 2020, Issue 1, 5-14. https://doi.org/10.34229/2707-451X.20.1.1

• Ermoliev Yu.M., Norkin V.I., and Wets R.J-B. The minimization of semicontinuous functions: Mollifier subgradients, SIAM Journal on Control and Optimization, 1995, No. 1, P. 149-167. https://doi.org/10.1137/S0363012992238369

• Ermoliev Yu.M., Norkin.V.I. Om nonsmooth and discontinuous problems of stochastic systems optimization. European J. of Operational Research, 1997, Vol. 101, P. 230-244. https://doi.org/10.1016/S0377-2217(96)00395-5

• Ermoliev Y.M., Norkin V.I. Stochastic Optimization of Risk Functions via Parametric Smoothing, In Dynamic Stochastic Optimization, Eds. K.Marti, Y.Ermoliev and G.Pflug, Lecture Notes in Economics and Mathematical Systems 532, Springer, 2004, P. 225-247.

• https://doi.org/10.1007/978-3-642-55884-9_11 (Q4)

A method of exact projective penalty functions has been developed for the equivalent reduction of general conditional optimization problems to unconditional non-smooth optimization problems, which do not need to determine the penalty coefficient and do not use the value of the objective function outside the feasible set (Norkin 2020, 2023).

• Norkin V.I. A stochastic smoothing method for nonsmooth global optimization. Cybernetics and Computer technologies. 2020, Issue 1, P. 5-14. https://doi.org/10.34229/2707-451X.20.1.1

• Norkin V.I. The exact projective penalty method for constrained optimization. Journal of Global Optimization. 2023. https://doi.org/10.1007/s10898-023-01350-4 (Q1)

A vector Branch and Bound method was developed for solving optimization problems in partially ordered spaces (Norkin, 2017, 2019).

• Norkin V.I. B&B method for discrete partial order optimization. Computational Management Science. Comput Manag Sci, 2019, Vol. 16, P. 577–592. https://doi.org/10.1007/s10287-019-00346-4 (Q1)

• Norkin V.I. B&B method for discrete partial order and quasiorder optimizations. Dopovidi NANU (Reports of the National Academy of Sciences of Ukraine), 2019, No. 1, P. 16-22. https://doi.org/10.15407/dopovidi2019.01.016

• Norkin V.I. B&B Solution Technique for Multicriteria Stochastic Optimization Problems. In: Optimization Methods and Applications, S. Butenko et al. (eds.). Springer Optimization and its Applications 130, pp. 345-378. – Springer International Publishing, 2017. https://doi.org/10.1007/978-3-319-68640-0_17 (Q4)

A method of reducing complex probabilistic optimization problems with a discrete distribution of random data to mixed-integer programming problems has been developed (Norkin, 2010; Norkin, Kibzun, Naumov, 2013, 2014). 

• Kibzun A.I., Naumov A.V., Norkin V.I. On reducing a quantile optimization problem with discrete distribution to a mixed integer programming problem. Automation and Remote Control, 2013, Vol. 74, Issue 6, P. 951-967. https://doi.org/10.1134/S0005117913060064 (Q2)

• Norkin V.I., Kibzun A.I., Naumov A.V. Reducing two-stage probabilistic optimization problems with discrete distribution of random data to mixed-integer programming problems. Cybernetics and Systems Analysis, 2014, Vol. 50, No. 5, P. 679-692. https://doi.org/10.1007/s10559-014-9658-9  (Q2)

• Norkin V.I., Kibzun A.I., Naumov A.V. Reduction of two-stage probabilistic optimization problems with discrete distribution of random data to mixed integer programming problems, in Stochastic programming and its applications, P.S.Knopov and V.I.Zorkaltsev (Eds.), Irkutsk, Melentiev Energy Systems Institute of Siberian Branch of the Russian Academy of Sciences, 2012, pp. 76-103.

• Norkin V.I. On mixed integer reformulations of monotonic probabilistic programming problems with discrete distributions, 05/17/2010, http://www.optimization-online.org/DB_HTML/2010/05/2619.html

The tangent minorant method was developed for solving deterministic and stochastic global optimization problems (Norkin, 1992; Norkin, Onishchenko, 2003-2005)

• Norkin V.I. On Pijavski’s method for solving general global optimization problem. Comp. Maths and Math. Phys., 1992, No. 7, P. 873-886.

• Norkin V.I., Onischenko B.O. Minorant methods of stochastic global optimization. Cybernetics and systems analysis, 2005, Vol. 41, No.2, P. 203-214. https://doi.org/10.1007/s10559-005-0053-4 (Q3)

• Norkin V.I., Onischenko B.O. Minorant methods for stochastic global optimization. Dagstuhl Seminar Proceedings 05031 – Algorithms for Optimization with Incomplete Information, S. Albers, R. H. Möhring, G. C. Pflug, R. Schultz (Eds.), 2005, Paper211. http://drops.dagstuhl.de/opus/volltexte/2005/211/pdf/05031.NorkinVladimir2.Paper.211.pdf

• Norkin V.I., Onischenko B.O. On the global minimization of minimum functions by the minorant method. Teoria optimalnyh risheniy (Theory of optimal decisions), Ed. N.Z.Shor, Glushkov Institute of Cybernetics, Kiev, 2004, No. 3, P. 56-63. (in Russian).

• Norkin V.I., Onischenko B.O. A branch and bound method with minorant estimates used to solve stochastic global optimization problems. Komputernaya matematika (Computer mathematics), Institute of Cybernetics, Kiev, 2004, No. 1, P. 91-101. (in Russian).

• Norkin V.I., Onischenko B.O. On a stochastic analogue of Piyavskii’s global optimization method. Teoriya optimalnyh rischen (Theory of optimal decisions), Institute of Cybernetics, Kiev, 2003, N 2, P. 61-67 (in Russian).

The error backpropagation method for learning deep non-smooth non-convex neural networks based on the Hamilton-Pontryagin formalism is substantiated (Norkin, 2019-2021).

• Norkin V.I. Generalized gradients in dynamic optimization, optimal control, and machine learning problems. Cybernetics and Systems Analysis, 2020, Vol. 56, No. 2, P.243-258. DOI 10.1007/s10559-020-00240-x (Q3)

• Norkin V.I. Stochastic generalized gradient methods for training nonconvex nonsmooth neural networks. Cybernetics and Systems Analysis, 57 (2021), pp. 714-729, https://doi.org/10.9841007/s10559-021-00397-z (Q2)

• Norkin V.I. Substantiation of the backpropagation technique via the Hamilton—Pontryagin formalism for training nonconvex nonsmooth neural networks. Dopovidi NANU (Reports of the National Academy of Sciences of Ukraine), 2019. No. 12. P. 19-26. https://doi.org/10.15407/dopovidi2019.12.019

The method of maximizing aggregated utility for finding economic equilibrium is justified (Norkin, Ermoliev, Fischer, 1997; Norkin, 1999).

• Ermoliev Y.M., Fischer G., Norkin V.I. On convergence of one method for searching economic equilibrium. Cybernetics and Systems Analysis, 1997, Vol. 33, No. 6, P. 854–866. https://doi.org/10.1007/BF02733225

• Norkin V.I. On a possibility to reduce a general equilibrium model to optimization problems. Cybernetics and Systems Analysis, 1999, Vol. 35, No. 5, P. 743–753. https://doi.org/10.1007/BF02733408 (Q3)

The method of successive approximations for solving the Wiener-Hopf equations with a probability kernel was substantiated (Norkin, 2006)

• Norkin V.I. On solution of Wiener-Hopf equation with probabilistic kernel. Cybernetics and systems analysis, 2006, Vol. 42, No.2, P. 195-201. https://doi.org/10.1007/s10559-006-0054-y (Q3)

The graphic law of large numbers for random multivalued mappings was substantiated. (Norkin, Wets, 2013)

• Norkin V.I., Wets R.J.-B. On a strong graphical law of large numbers for random semicontinuous mappings. Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 2013, No. 3, P. 102–111. https://www.mathnet.ru/rus/vspui/y2013/i3/p102

• Norkin V.I., Wets R.J-B. Law of small numbers as concentration inequalities for sums of independent random sets and random set valued mappings. Proceedings of the International workshop “Stochastic programming for implementation and advanced applications” (STOPROG-2012, July 3-6, 2012, Neringa, Lithuania), L.Sakalauskas, A.Tomasgard, S.W.Wallase (Eds.), Vilnius Gediminas Technical University, Vilnius, 2012, pp. 94-99. DOI:10.5200/stoprog.2012.17

Strong uniform convergence of the Vapnyk’s kernel support vectors machine in the Reproducing Kernel Hilbert Spaces was proved (Norkin, Keyzer, 2009).

• Norkin V.I., Keyzer M.A. On convergence of kernel learning estimators. SIAM J. on Optimization, 2009, Vol. 20, No. 3, pp. 1205–1223. https://doi.org/10.1137/070696817 (Q1)

• Norkin V.I., Keyzer M.A. Efficiency of classification methods based on empirical risk minimization. Cybernetics and Systems Analysis, 2009, Vol. 45, No. 5, P.750-761. DOI 10.1007/s10559-009-9153-x  (Q2)

• Norkin V.I., Keyzer M.A. Asymptotic efficiency of kernel support vector machines (SVM). Cybernetics and Systems Analysis, 2009, Vol. 45, No. 4, P.575-588. https://doi.org/10.1007/s10559-009-9125-1  (Q2)

• Norkin V.I., Keyzer M.A. On stochastic optimization and statistical learning in Reproducing Kernel Hilbert Spaces by SVM. Informatica (Vilnius), 2009, Vol. 20, No. 2, 273–292. https://doi.org/10.15388/Informatica.2009.250 (Q2)

Finite convergence of nearest neighbor learning method on mistakes in the case of non-intersecting compact classes (generalization of Novikoff's theorem) was proved (Norkin, 2022).

• Norkin V.I. On the finite convergence of the NN classification learning on mistakes. Dopov. Nac. akad. nauk Ukr. 2022. № 1. С. 34-38. https://doi.org/10.15407/dopovidi2022.01.034

Roy's method for choosing the optimal safe financial portfolio is generalized (Norkin, Boyko, 2012).

• Norkin V.I., Boyko S.V. Safety-first portfolio selection. Cybernetics and Systems Analysis, Vol. 48, No. 2, P. 180-191. DOI: 10.1007/s10559-012-9396-9 (Q3)

• Norkin V.I. On mixed integer reformulations of monotonic probabilistic programming problems with discrete distributions, 05/17/2010, http://www.optimization-online.org/DB_HTML/2010/05/2619.html