Vladimir
NORKIN’s research
Current research interests:
Research directions:
Summary of main results
Research history
1974 – 1988.
On graduation from Kolmogorov’s
Mathematical School (1968) and Moscow
Institute of Physics and Technology (1974)
I started to work at the Institute of Cybernetics of the Ukrainian
Academy of Sciences, Kiev, Ukraine, and thus fell under influence of Kiev optimization
school (Mikhalevich V.S., Ermoliev
Y.M., Pschenichny B.N., Shor
N.Z. et al.). My very first paper (1975, joint with Pilyugin
N.N.) on hypersonic gas dynamics represents my master thesis and was submitted
in 1974 when I was a student of Moscow Institute of Physics and Technology. My first paper (1977, joint with Gupal A.M.) at Institute of Cybernetics was devoted to a
rather exotic at that time problem of optimization of discontinuous functions.
In the second paper (1978) I generalized a classical differentiability concept
for the analysis of nonsmooth nonconvex
functions. My main result for this period was extension of subgradient
methods by Snor N.Z. and stochastic qusigradient methods by Ermoliev
Y.M. from convex to nonconvex deterministic and
stochastic optimization problems. Another noticeable result concerns to the
generalization of Pijavski’s global optimization
method (published at Proceedings of the
1989 -2002.
Since 1989 I have started to collaborate with International Institute for
Applied Systems Analysis (IIASA),
Current research. IIASA’s
land use models inspired consideration of spatial general equilibrium models
with continuum of agents and application of stochastic optimization technique
to these models and to nonparametric spatial statistics problems (joint works
with Y.Ermoliev and M.Keyzer). IIASA’s research on insurance and financial
mechanisms of mitigation to catastrophic risks attracted my attention to
problems of actuarial and financial mathematics. Now jointly with PhD students I develop effective
numerical methods for solution of integral equations arising in risk theory and
methods for optimization of insurance and optional contracts.
In Norkin (1983, 1986a,b),
Mikhalevich, Gupal and Norkin (1987), Ermoliev and Norkin (1997, 1998) theory (and methods) of local nonconvex nonsmooth stochastic
optimization is developed (see a review paper by Ermoliev
and Norkin (2003)). The considered optimization
problems consist in minimization of a nonconvex
mathematical expectation function under constraints. The consideration is based
on a concept of the so-called generalized differentiable function, Norkin (1986b). The class of generalized differentiable
functions includes convex and concave functions and is closed with respect to
maximum, minimum, superposition and mathematical expectation operations. Such
functions appear, for example, in areas of optimization of servicing systems,
communication networks and other discrete event systems, Ermoliev
and Norkin (1997). There is a calculus of
(stochastic) subgradients as measurable selections of
multivalued subdifferential
mappings for compound generalized differentiable functions. In Mikhalevich, Gupal and Norkin (1987) a number of methods of (stochastic) subgradient type (with projection on the constraint set,
stochastic linearization, stochastic reduced subgradient,
averaged stochastic subgradient and others) for solving
nonconvex nonsmooth
stochastic programming problems were developed and investigated. These methods
do not require exact calculation of the expectation objective function, but use
realizations of random function’s values and (sub)gradients
obtained through Monte Carlo simulations.
In Norkin
(1993a) the properties of (nonsmooth nonconvex) probability functions and stochastic subgradient methods for their optimization are studied. A
probability function represents the probability that some real random variable
depending on control parameters exceeds the prescribed threshold. Conditions of
(some kind) quasi-concavity are established. Convergence of stochastic subgradient method with additional averaging of stochastic
trajectories is established.
Selected references
Ermoliev Y.M., Norkin V.I. (2003). Methods for solution of nonconvex nonsmooth stochastic optimization
problems, Kibernetika i
sistemnyi analiz (in Russian, English translation in Cybernetics and Systems
Analysis), 2003, N 5, 60-81.
Norkin V.I. (1983).Generalized gradient method in nonconvex nonsmooth optimization, Abstract of Candidate (Ph.D.)
Thesis, Glushkov Institute of Cybernetics,
Kiev, 1983, 16p.
(In Russian).
Norkin V.I. (1986a). On random Lipschitz
functions, Kibernetika,
1986, No.2, 66-71,76 (In Russian, English translation
in Cybernetics,
V. 22, No.2)
Norkin V.I. (1986b). Random generalized differentiable functions in nonconvex nonsmooth stochastic
programming problems, Kibernetika,
1986, No.6, 98-102 (In Russian, English translation in Cybernetics, V. 22, No.6,
804-809).
Mikhalevich V.S., Gupal A.M. and Norkin V.I. (1987). Methods of Nonconvex Optimization (Abstract), Nauka,
Ermoliev Y.M., Norkin V.I. (1998). Stochastic generalized gradient method for
solving nonconvex nonsmooth
stochastic optimization problems, Kibernetika i sistemny analiz,
1998, N 2, 50-71 (see also IIASA
Interim Report IR-97-021 , Abstract; English
translation in Cybernetics and systems analysis, V. 34, N 2).
Ermoliev Y.M., Norkin V.I. (1997). Om nonsmooth
and discontinuous problems of stochastic systems optimization, European
J. of Operational Research, 1997, Vol. 101, 230-244.
Norkin V.I. (1993a). The Analysis and Optimization of Probability
Functions, Working paper WP-93-6, Int. Inst. for Appl. Syst.
Anal., Laxenburg, Austria, 1993.
In Norkin, Pflug and Ruszczynski (1998), Norkin, Ermoliev and Ruszczynski (1998) a Stochastic Branch and Bound method for
solving stochastic integer and stochastic global optimization problems is
developed. The method is based on specific (stochastic lower and upper) bounds
of optimal values of stochastic programming problems. For different classes of
problems such bounds can be obtained by means of interchanging minimization and
expectation (or probability) operators (the so-called interchange relaxation, Norkin (1993b)). For many linear stochastic programming
problems such bounds can be calculated explicitly and for the others one has to
solve some deterministic estimation optimization problems with randomly taken
data. Convergence conditions a.s.
and with probability 1-e are established. In Norkin (1998) the Stochastic Branch and Bound method is
extended for a global optimization of probability functions.
In Norkin
(1992) a minorant approach to global optimization is
developed. As a source of global information on a problem function general
tangent minorants to the function graph are used. In
particular minorant tangent cones and paraboloids can be employed. A calculus of tangent minorants is developed. Pijavski’s
global optimization method is generalized to general constrained global
optimization problems.
In Norkin
(1993), Norkin and Onischenko
(2003, 2004) a concept of stochastic tangent minorants
is introduced and on this basis the minorant method
is extended to multiextremal stochastic optimization
problems with expectation/probability type functions. As a stochastic tangent minorant of an expectation function one can use a tangent minorant of the corresponding random function. In this case
exact calculation of minorants for the expectation
function may be problematic but the calculation of stochastic minorants is possible in many cases.
Selected references
Norkin V.I., Pflug
G.Ch. and Ruszczynski A. (1998). A branch and bound method for
stochastic global optimization, Math. Progr., 1998, V. 83, 425-450 (see also IIASA
Working Paper WP-96-065, Abstract ).
Norkin V.I., Ermoliev
Y.M. and Ruszczynski A. (1998). On optimal allocation of
indivisibles under uncertainty, Operations Research, 1998, Vol. 46,
N 3, 381-395 (see also IIASA
Working paper WP-94-21, Abstract ).
Norkin V.I. (1992b).On Pijavski’s
method for solving general global optimization problem, Zhurnal Vychislitel'noj Matematiki
i Matematicheskoj Fiziki, 1992, N 7, 992-1006 (in Russian, English
translation in Comp. Maths and Math. Phys., 1992, N
7, 873-886).
Norkin V.I. (1998). 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, 186-201.
Norkin V.I. (1993b). 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, 3-12 (In Russian).
Norkin V.I., Onischenko B.O. (2003). On stochastic
analogue of Piyavski’s global optimization method, Teoria optimalnyh risheniy (Theory of
optimal decisions), Ed. N.Z.Shor, Glushkov Institute of Cybernetics,
Norkin
V.I., Onischenko B.O. (2004). 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,
Norkin
V.I., Onischenko B.O. (2004).
A branch and bound
method with minorant estimates used to solve
stochastic global optimization problems,
Komputernaya matematika (Computer mathematics),
In Norkin (1978, 1980a) the so-called
generalized differentiable functions were introduced and in Norkin
(1978, 1980b), Mikhalevich, Gupal
and Norkin (1987), Ermoliev
and Norkin (1998) theory and methods of their local
optimization was developed. Generalized differentiable functions enjoy many
nice properties that make them a convenient model of nonconvexity
and nonsmoothness in optimization problems. While a
differentiable function f(y) admits a
linear expansion at a given point x,
i.e. f(y)=f(x)+<g(x),y-x>+o(x,|y-x|), a generalized differentiable function f(y) admits a qusilinear
expansion, f(y)=f(x)+<g(y),y-x>+o(x,y,g) with (sub)gradients g taken not at x but at
an evaluation point y. The class of
generalized differentiable functions includes continuously differentiable,
convex, concave, semismooth functions and is closed
with respect to maximum, minimum, superposition and mathematical expectation
operations. Generalized differentiable functions are locally Lipschitzian and may have no directional derivatives. Such kind of functions spring into being, for example, in areas of
optimization of servicing systems, communication networks and other discrete
event systems. They are used for construction of Newton’s type methods
to solve generalized equations. There is a calculus of subgradients
for compound generalized differentiable functions. In Mikhalevich,
Gupal and Norkin (1987), Ermoliev and Norkin (1998) many
methods of subgradient type (with projection on the
constraint set, stochastic linearization, stochastic reduced subgradient, averaged subgradient
and others) for solving nonconvex nonsmooth
constrained optimization problems were developed and investigated.
In Gupal and Norkin (1977), Ermoliev , Norkin and Wets (1995), Ermoliev
and Norkin (1998, 2004) we develop a stochastic
smoothing approach to optimization of generally discontinuous functions by
means of their approximation by smooth mollified (averaged) functions. The idea
of the approach consists in approximation of the original problem by a family
of nonconvex stochastic optimization problems and
application to approximate problems all the technique developed in stochastic
programming (methods of calculation of stochastic gradients without the exact
calculation of multidimensional integrals, the technique of the nonconvex stochastic second Lyapunov
method for the proof of convergence a.s. of
stochastic approximation algorithms, acceleration technique for stochastic
optimization methods, nonstationary stochastic
optimization technique). Besides, in Ermoliev , Norkin and Wets (1995), Ermoliev and Norkin (1998, 2004)
mollified (averaged) functions are employed to construct (mollifier)
subdifferentials of discontinuous functions. The last
concept has appeared as a result of application of ideas of distribution theory
by L.Schwartz (or generalized function by S.Sobolev) to the range of nonsmooth
optimization where quite different approaches have been used before. Optimality
conditions in terms of mollifier subdifferentials
of discontinuous functions are established. Convergence of the approximation
scheme is proved, i.e. the convergence of minimums of approximation problems to
minimums of the original problem. Convergence a.s. of stochastic finite-difference procedures for
optimization of discontinuous functions is proved.
Selected references
Mikhalevich V.S., Gupal A.M. and Norkin V.I. Methods of Nonconvex
Optimization (Abstract), Nauka,
Ermoliev Y.M., Norkin V.I. (2003). Methods
for solution of nonconvex nonsmooth stochastic optimization problems, Kibernetika i sistemnyi analiz, 2003, N 5, 60-81 (in Russian, English translation in Cybernetics and Systems
Analysis, 2003, Vol. 39, Issue 5, pp. 701-715).
Ermoliev Y.M., Norkin V.I. (1998). Stochastic generalized gradient method for
solving nonconvex nonsmooth
stochastic optimization problems, Kibernetika i sistemny analiz,
1998, N 2, 50-71 (see also IIASA
Interim Report IR-97-021 , Abstract; English
translation in Cybernetics and systems analysis, V. 34, N 2).
Norkin V.I. (1978). Nonlocal
algorithms for minimization of nondifferentiable
functions, Kibernetika,
1978, No.5, 44-48 (In Russian, English translation in Cybernetics, V. 14,
No.5).
Norkin V.I. (1980a).Generalized
differentiable functions, Kibernetika, 1980, No.1, 9-11 (In Russian, English
translation in Cybernetics, V. 16, No.1)
Norkin V.I. (1980b). Nondifferentiable
function minimization method with averaging of generalized gradients, Kibernetika,
1980, No.6, 88-89 (In Russian, English translation in Cybernetics, 1980, V. 16,
No.6)
Gupal A.M. and Norkin
V.I. (1977).
Minimization algorithms for discontinues functions, Kibernetika, 1977, No.2, 101-107
(In Russian, English translation in Cybernetics, V. 13, No.2).
Ermoliev Yu.M., Norkin V.I. and Wets R.J-B. (1995). The
minimization of semicontinuous functions: Mollifier subgradients, SIAM
Journal on Control and Optimization, 1995, N 1, 149-167.
Ermoliev Y.M. and Norkin
V.I. (1998). 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, 128-142.
Ermoliev Y.M., Norkin V.I. (2004). 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, pp. 225-247; see also IIIASA
Interim Report IR-03-033).
In Norkin (1989, 1992c), Ermoliev
and Norkin (1990, 1991) a new concept of normalized
convergence of random variables is introduced and investigated. This concept is
close to convergence in probability (with some rate) but has some advantages,
for instance, it allows to construct confidence
intervals for the converging random variables. The following properties of the
normalized convergence are established. From a normalized convergence a
convergence in probability (with some less rate)
follows. Conversely, formally a convergence in probability is a normalized
convergence with a degenerated distribution function. From a convergence in mean it
follows not only a convergence in probability (that is well known) but also a
normalized convergence of the corresponding random variables. Normalized
convergence is preserved under Hölder
transformations of the corresponding random variables. It is also persisted
under summation, product, division and direct product of normalized convergent
sequences of random variables. Every probabilistic limit theorem implies a
normalized convergence of the corresponding random variables. For normalized
convergence Cauchy convergence criterion takes place. In Norkin
(1989, 1992a), Ermoliev and Norkin
(1990, 1991) the concept of normalized convergence is applied to the analysis
of convergence of the so-called empirical mean (statistical) method in
stochastic optimization.
In Norkin
and Roenko (1991), Norkin
(1993) certain concavity properties of functions and probability measures are
systematically studied, and in Norkin (1993) a
stochastic subgradient approach to optimization of
probability functions is developed.
In Ermoliev
and Norkin (1998) a general stochastic version of the
second Lyapunov’s method is developed and applied to
establishing some nonstationary laws of large number
(with moving expectations) and to proving almost sure convergence of some Monte
Carlo optimization and estimation procedures.
In Keyzer, Ermoliev
and Norkin (2002)
a risk minimization approach to estimate a flexible form that meets a priori
restrictions on slope and curvature by means of constraints on both the
estimated parameters and the function values is presented. The resulting
constrained risk minimization combines parametric and nonparametric estimation
and contains integrals and implicit constraints. Within econometrics,
simulation has become a common tool to solve problems of this kind. However, it
appears that in our case, the simulation approach only applies when the model
is linear in parameters, has simple constraints on parameters and a quadratic
risk function. To deal with other cases, we use a stochastic optimization
technique known as the stochastic quasi-gradient method for stationary and nonstationary problems with Cesàro
averaging. This method is also applicable to an expanding series of random
observations, and produces asymptotically (weakly) convergent estimates.
Selected references
Norkin V.I. (1992c). On Normalized
Convergence of Random Variables, Kibernetika i Sistemnyi Analiz,
1992, N 2, 107-120 (in Russian, English translation in Cybernetics and Systems Analysis,
1992, V. 28, N 2).
Norkin V.I. (1989). Stability
of stochastic optimization models and statistical methods of stochastic
programming, Preprint 89-53, Glushkov Institute of
Cybernetics, Kiev, 1989 (In Russian).
Ermoliev Y.M. and Norkin
V.I. (1990). Normalized convergence of random variables and its applications, Kibernetika,
1990, N 6, 85-89. (In Russian, English translation in Cybernetics,
1990, V. 26, N 6).
Ermoliev Y.M. and Norkin
V.I. (1991). Normalized convergence in stochastic optimization, Annals
of Operations Research, 1991, N 1, 187-198.
Norkin V.I. (1992a). On Convergence and Rate of Convergence of
Empirical Mean Value Method in Mathematical Statistics and Stochastic
Programming, Kibernetika i Sistemnyi Analiz, 1992, N 3,
84-92 (in Russian, English translation in Cybernetics and Systems Analysis,
1992, V. 28, N 3).
Ermoliev Y.M. and Norkin
V.I. (1998). On nonstationary law of large numbers for dependent random
variables and its application to stochastic optimization, Kibernetika i sistemny analiz, 1998, N 4, 94-106 (In Russian, English
translation in Cybernetics and systems analysis, Vol. 34, N4; see also IIASA
Interim Report IR-98-009, Abstract).
Norkin V.I. and Roenko
N.V. (1991). a-concave
functions and measures and their applications, Kibernatica i Systemnyi Analiz, 1991, N 6, 84-94. (In
Russian, English translation in Cybernetics and Systems Analysis,
1991, V. 27, N 6, 860-869).
Keyzer M.A., Ermoliev
Y.M., Norkin V.I. (2002). Estimation
of econometric models by risk minimization: Stochastic quasi-gradient
approach (Abstract), IIASA
Interim Report IR-02-021, Int. Inst. for Appl. Syst. Anal., 2002, 32p.
In a series of papers jointly with staff
members of the International Institute of Applied Systems Analysis (IIASA,
Austria) Amendola A., Ermoliev
Y.M., MacDonald G., Ermolieva T.Y.) we develop a number of models for insurance and mitigation
against rare large claims caused by catastrophic events (earthquakes,
hurricanes, storms, floods, droughts, fires, financial crises and etc.). These
are portfolio (also welfare, equilibrium, risk process) type models which have
a form of stochastic (two-stage, dynamic) programming problems, describing
trade-off between expected profit and risk of bankruptcy of an insurance
company. Decision variables are fractions of insured damages. The essential
part of the approach is a Monte Carlo simulation of catastrophes and subsequent
dependent claims. Mathematical results (my main contribution) include reduction
of chance constrained stochastic optimization problem to optimization of
penalized expected profit, the development and investigation of an adaptive
Monte Carlo optimization method, estimation of the probability of ruin for the
case of rare dependent claims.
In
Ermoliev et al. (1998, 2000) it was shown that
optimal values of a chance constrained problem and a simple recourse one become
close if a reliability level tends to one and recourse (penalty) prices tend to
infinity.
Selected references
Ermoliev Y.M., Ermolieva
T.Y., MacDonald G. and Norkin V.I. (2000). Stochastic Optimization of Insurance Portfolios
for Managing Exposure to Catastrophic Risks, Annals of Operations Research
99, 2000, 207-225.(see also IIASA
Interim Report IR-98-056, Abstract).
Amendola A., Ermoliev
Y.M., Ermolieva T.Y., MacDonald G.J. and Norkin V.I. (2000). A system approach to
management of catastrophic risks (Abstract), European
J. of Operational Research,
2000, V.122, P.452-460.
Ermoliev Y.M., Ermolieva
T.Y., MacDonald G.J. and Norkin V.I. (2000). Insurability of catastrophic risks:
the stochastic optimization model (Abstract), Optimization,
2000, Vol. 47, 251-265.
Ermoliev Y.M., Ermolieva
T.Y., MacDonald G. and Norkin V.I. (2001). Problems of catastrophic risks
insurance, Kibernetika i sistemnyi analiz, 2001, N 2,
99-110 (in Russian, English
translation in Cybernetics and Systems
Analysis, 2001, Vol. 37, Issue 2, pp. 220-234).
Norkin V.I. (1999). On modeling spatial,
temporal and magnitude aspects in insuring cataclysmic events, Proceedings of the Second International
School on Actuarial and Finantial Mathematics, Kyiv, Ukraine, 8-12 June, 1999, Theory
of Stochastic Processes(Kiev), 1999, ¹1-2, 98-110.
Ermoliev Y.M. and Norkin V.I. (2003). Risk and
Extended Expected Utility Functions: Optimization Approaches, Interim
Report IR-03-033, Int. Inst. for Appl. Syst. Anal., 2003, 23p..
One more group of results concerns methods for
searching economic equilibrium. We develop two approaches, the reduction of the
equilibrium problem to a sequence of optimization problem (Norkin,
Ermoliev and Fischer (1997)) and decomposition of a
general equilibrium problem
by means of stochastic quasi-gradient method (Ermoliev,
Norkin and Keyzer (2000)).
It is shown that under certain
conditions (fixed consumer budgets and etc., see Norkin
(1999)) general equilibrium model is reduced to convex (joint/aggregate
utility) optimization problem and thus can be effectively solved. General case
is reduced to iterative application of utility aggregation and optimization. It
was discovered that convergence of this method for searching economic
equilibrium is closely related to the ergodicity of
certain Markov chains.
The second approach consists in
randomization of a classical Walrasian tâtonement process such that price adaptation is made
not in the direction aggregate excess demand but in a random direction
reflecting excess demand of a small number randomly sampled market agents. Some
sufficient convergence conditions and rate of convergence to equilibrium are
established.
Paper by Ermoliev , Norkin and Keyzer (2001) develops a practical modeling
framework for land use planning and presents the associated stochastic
algorithms for numerical implementation. We focus on the case in which transfers
among social group adjust to support social welfare optimization. It appears that the problem becomes more
tractable if it is treated as the minimization of a dual welfare function, that solely depends on prices but is evaluated
after integration over space. Next, we allow for (nonrival)
demand that simultaneoulsy benefits several agents,
in order to represent general informational infrastructure as well as
investments with uncertain outcomes. This leads to a minimax
problem, with a dual welfare function to be minimized with respect to prices
and maximized with respect to nonrival demand.
Selected
references
Norkin V.I., Ermoliev
Y.M. and Fischer G. (1997). On convergence of one method for searching economic equilibrium, Kibernetika i sistemnyi analiz, 1997, N 6
(In Russian, English translation in Cybernetics and Systems Analysis, see
also IIASA
Working Paper WP-96-118).
Norkin V.I. (1999). On a possibility to
reduce a general equilibrium model to optimization problems, Kibernetika i sistemnyi analiz (in Russian,
English translation in Cybernetics and Systems Analysis),
1999, N 5, 75-86.
Ermoliev Yu., Norkin
and Keyzer M.A. (2000). Global convergence of the stochastic tatonement process, J.of
Mathematical Economics, 2000,
V.34, P.173-190.
Ermoliev Yu., Norkin and Keyzer M.A. (2001). General
equilibrium and welfare modeling in spatial continuum: a practical
framework for land use planning (Abstract),
Interim
Report IR-01-033, Int. Inst. for Appl. Syst. Anal., August 2001, 28p.
In Norkin (2003,
1990) to discover and identify a (linear) dependence in experimental data, a
principal component method in its geometric form is applied: data are
approximated by a linear manifold formed by some first principal components of
a data space. If the dimension of the manifold is less than a dimension of the
space, then a linear dependence in data is identified. Problems of selection of
dependent, independent and neutral variables and problem of multicollinearity
of observations are resolved in a natural way in this approach.
In Kirilyuk, Norkin
and Domrachev (2000-2002) an approach of
nonparametric indexes (data envelope analysis) is applied to actors of
Ukrainian financial and agricultural markets to estimate their position
relative to market efficiency frontier.
Selected references
Norkin V.I. (2003). On
application of principle components method in multi-dimensional regression analysis, Komputernaya matematika (Computer
mathematics),
Norkin V.I. (1990). Construction of linear
models on the basis of limited experimental data, in: Mathematical Methods of Decision-Making under uncertainty, Eds. Yu.M.Ermoliev and I.N.Kovalenko, Glushkov Institute of Cybernetics, Kiev, 1990, 77-82 (In
Russian).
Kirilyuk V.S., Norkin V.I. and Domrachev V.N.
(2002). A Nonparametric Index
Approach for Estimating Subjects of Financial Market by Profitability-Risk
Criterion by Example of Commercial Banks, Problemy upravleniya i informatiki, 2002, ¹ 6,
120-131 (in Russian, English
translation in J. of Automation and
Information Sciences, 2002, Volume34, Issue 12).
Kirilyuk V.S., Norkin V.I.
and Domrachev V.N. (2002). On the use of a return-risk frontier
for the evaluation of banks functioning, Finansovye riski
(Financial risks), 2002, No. 1-2(29), pp.75-77. (In
Russian).
Kirilyuk V.S., Norkin V.I.
and Domrachev V.N. (2001). Method of
nonparametric indexes for the analysis of productivity, technical growth and
efficiency chainge on the example of Ukrainian
agriculture in 1996-1999, Finansovye riski (Financial
risks), 2001, No. 3(27), pp.77-84. (In Russian).