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Papers by Vladimir Norkin

Research paper thumbnail of Catastrophic Risk Management and Economic Growth

Analysis, Schlossplatz 1, A-2361 Laxenburg, Austria, Telephone: (+43 2236) 807 402 Vladimir Norki... more Analysis, Schlossplatz 1, A-2361 Laxenburg, Austria, Telephone: (+43 2236) 807 402 Vladimir Norkin, norkin@dept130.cyber.kiev.ua, Glushkov's Institute of Cybernetics, Kiev, Ukraine

Research paper thumbnail of A Nonparametric Index Approach for Estimating Subjects of Financial Market by Profitability-Risk Criterion by Example of Commercial Banks

Journal of Automation and Information Sciences, 2002

Research paper thumbnail of Polyhedral Coherent Risk Measures and their Application to Investment Decisions Support under Catastrophic Flood Risks

In this chapter we review some results of the PCRM theory, consider applications to decision maki... more In this chapter we review some results of the PCRM theory, consider applications to decision making support in conditions of risk, and develop numerical methods for searching optimal decisions. An investment decisions making under catastrophic flood risks is considered as a particular application.

Research paper thumbnail of On a strong graphical law of large numbers for random semicontinuous mappings

In the paper we establish a strong graphical law of large numbers (LLN) for random outer semicont... more In the paper we establish a strong graphical law of large numbers (LLN) for random outer semicontinuous mappings, providing conditions when graphs of sample average mappings converge to the graph of the expectation mapping with probability one. This result extends a known LLN for compact valued random sets to random uniformly bounded (by an integrable function) set valued mappings. We give also an equivalent formulation for the graphical LLN by means of some fattened mappings. The study is motivated by applications of the set convergence and the graphical LLN in stochastic variational analysis, including approximation and solution of stochastic generalized equations, stochastic variational inequalities and stochastic optimization problems. The nature of these applications consists in sample average approximation of the inclusion mappings, application of the graphical LLN and obtaining from here a graphical approximation of the set of solutions. Bibliogr. 23.

Research paper thumbnail of On Stochastic Optimization and Statistical Learning in Reproducing Kernel Hilbert Spaces by Support Vector Machines (SVM)

The paper studies stochastic optimization problems in Reproducing Kernel Hilbert Spaces (RKHS). T... more The paper studies stochastic optimization problems in Reproducing Kernel Hilbert Spaces (RKHS). The objective function of such problems is a mathematical expectation functional depending on decision rules (or strategies), i.e. on functions of observed random parameters. Fea- sible rules are restricted to belong to a RKHS. This kind of problems arises in on-line decision making and in statistical learning theory. We solve the problem by sample average approximation combined with Tihonov's regularization and establish sufficient conditions for uniform conver- gence of approximate solutions with probability one, jointly with a rule for downward adjustment of the regularization factor with increasing sample size.

Research paper thumbnail of Erratum to: Mathematical Modeling of Distributed Catastrophic and Terrorist Risks

Cybernetics and Systems Analysis, 2015

ABSTRACT The paper shows how the mathematical tools of the theory of controlled Markov fields can... more ABSTRACT The paper shows how the mathematical tools of the theory of controlled Markov fields can be applied to model catastrophic risks caused by natural events or terrorist threats. The examples of problem statements of long-term investment in security are given. A survey of solution methods for stochastic optimal control problems is proposed. It is shown that these problems can be reduced to finite-dimensional stochastic programming problems and can be solved by the stochastic quasigradient method. (Erratum: author misspelling, DOI 10.1007/s10559-015-9724-y).

Research paper thumbnail of Stochastic Optimization of Risk Functions via Parametric Smoothing

Lecture Notes in Economics and Mathematical Systems, 2004

Research paper thumbnail of On Constrained Discontinuous Optimization

Lecture Notes in Economics and Mathematical Systems, 1998

Research paper thumbnail of THE MINIMIZATION OF DISCONTINUOUS FUNCTIONS: MOLLIFIER SUBGRADIENTS

To minimize discontinuous functions, that arise in the context of sys- tems with jumps for exampl... more To minimize discontinuous functions, that arise in the context of sys- tems with jumps for example, we propose a new approach based on approximation via averaged functions (obtained by convolution with mollifiers). The properties of averaged functions are studied, after it is shown that they can be used in an ap- proximation scheme consistent with minimization. A new notion of

Research paper thumbnail of General equilibrium and welfare modelling in spatial continuum

Research paper thumbnail of Estimation of econometric models by risk minimization: a stochastic quasigradient approach

Research paper thumbnail of Mathematical Modeling of Distributed Catastrophic and Terrorist Risks1

ABSTRACT The paper shows how the mathematical tools of the theory of controlled Markov fields can... more ABSTRACT The paper shows how the mathematical tools of the theory of controlled Markov fields can be applied to model catastrophic risks caused by natural events or terrorist threats. The examples of problem statements of long-term investment in security are given. A survey of solution methods for stochastic optimal control problems is proposed. It is shown that these problems can be reduced to finite-dimensional stochastic programming problems and can be solved by the stochastic quasigradient method.

Research paper thumbnail of General equilibrium models with discrete choices in a spatial continuum

The treatment of spatial characteristics through probability distributions makes it possible to u... more The treatment of spatial characteristics through probability distributions makes it possible to use stochastic optimization methods and to obtain efficiency results and competitive equilibrium prices for general equilibrium models with discrete choices in spatial continuum. Along these lines, and combining results from stochastic optimization with principles established by Aumann and Hildenbrand for economies with continuum of traders the paper develops

Research paper thumbnail of Reducing Two-Stage Probabilistic Optimization Problems with Discrete Distribution of Random Data to Mixed-Integer Programming Problems*

Research paper thumbnail of Constrained optimization of discontinuous systems

Research paper thumbnail of On nonsmooth problems of stochastic systems optimization

A class of stochastic optimization problems is analyzed that cannot be solved by deterministic an... more A class of stochastic optimization problems is analyzed that cannot be solved by deterministic and standard stochastic approximation methods. We consider risk control problems, optimization of stochastic networks and discrete event systems, screening irreversible changes, pollution control. The results of Ermoliev, Norkin, Wets [ll] are extended to the case of problems involving random variables and general constraints. It is shown that the concept of mollifier subgradient leads to easily implementable computational procedures for stochastic systems with Lipschitz and discontinuous expectation functions. New optimality conditions are formulated enabling to design stochastic search procedures for constrained optimization of discontinuous systems.

Research paper thumbnail of Risk and extended expected utility functions: optimization approaches

Research paper thumbnail of Metody nevypukloi optimizatsii

Research paper thumbnail of Stochastic generalized gradient method with application to insurance risk management

a We would like to thank Gordon MacDonald and Joanne Linnerooth-Bayer for their helpful comments.

Research paper thumbnail of Methods of nonconvex optimization

Research paper thumbnail of Catastrophic Risk Management and Economic Growth

Analysis, Schlossplatz 1, A-2361 Laxenburg, Austria, Telephone: (+43 2236) 807 402 Vladimir Norki... more Analysis, Schlossplatz 1, A-2361 Laxenburg, Austria, Telephone: (+43 2236) 807 402 Vladimir Norkin, norkin@dept130.cyber.kiev.ua, Glushkov's Institute of Cybernetics, Kiev, Ukraine

Research paper thumbnail of A Nonparametric Index Approach for Estimating Subjects of Financial Market by Profitability-Risk Criterion by Example of Commercial Banks

Journal of Automation and Information Sciences, 2002

Research paper thumbnail of Polyhedral Coherent Risk Measures and their Application to Investment Decisions Support under Catastrophic Flood Risks

In this chapter we review some results of the PCRM theory, consider applications to decision maki... more In this chapter we review some results of the PCRM theory, consider applications to decision making support in conditions of risk, and develop numerical methods for searching optimal decisions. An investment decisions making under catastrophic flood risks is considered as a particular application.

Research paper thumbnail of On a strong graphical law of large numbers for random semicontinuous mappings

In the paper we establish a strong graphical law of large numbers (LLN) for random outer semicont... more In the paper we establish a strong graphical law of large numbers (LLN) for random outer semicontinuous mappings, providing conditions when graphs of sample average mappings converge to the graph of the expectation mapping with probability one. This result extends a known LLN for compact valued random sets to random uniformly bounded (by an integrable function) set valued mappings. We give also an equivalent formulation for the graphical LLN by means of some fattened mappings. The study is motivated by applications of the set convergence and the graphical LLN in stochastic variational analysis, including approximation and solution of stochastic generalized equations, stochastic variational inequalities and stochastic optimization problems. The nature of these applications consists in sample average approximation of the inclusion mappings, application of the graphical LLN and obtaining from here a graphical approximation of the set of solutions. Bibliogr. 23.

Research paper thumbnail of On Stochastic Optimization and Statistical Learning in Reproducing Kernel Hilbert Spaces by Support Vector Machines (SVM)

The paper studies stochastic optimization problems in Reproducing Kernel Hilbert Spaces (RKHS). T... more The paper studies stochastic optimization problems in Reproducing Kernel Hilbert Spaces (RKHS). The objective function of such problems is a mathematical expectation functional depending on decision rules (or strategies), i.e. on functions of observed random parameters. Fea- sible rules are restricted to belong to a RKHS. This kind of problems arises in on-line decision making and in statistical learning theory. We solve the problem by sample average approximation combined with Tihonov's regularization and establish sufficient conditions for uniform conver- gence of approximate solutions with probability one, jointly with a rule for downward adjustment of the regularization factor with increasing sample size.

Research paper thumbnail of Erratum to: Mathematical Modeling of Distributed Catastrophic and Terrorist Risks

Cybernetics and Systems Analysis, 2015

ABSTRACT The paper shows how the mathematical tools of the theory of controlled Markov fields can... more ABSTRACT The paper shows how the mathematical tools of the theory of controlled Markov fields can be applied to model catastrophic risks caused by natural events or terrorist threats. The examples of problem statements of long-term investment in security are given. A survey of solution methods for stochastic optimal control problems is proposed. It is shown that these problems can be reduced to finite-dimensional stochastic programming problems and can be solved by the stochastic quasigradient method. (Erratum: author misspelling, DOI 10.1007/s10559-015-9724-y).

Research paper thumbnail of Stochastic Optimization of Risk Functions via Parametric Smoothing

Lecture Notes in Economics and Mathematical Systems, 2004

Research paper thumbnail of On Constrained Discontinuous Optimization

Lecture Notes in Economics and Mathematical Systems, 1998

Research paper thumbnail of THE MINIMIZATION OF DISCONTINUOUS FUNCTIONS: MOLLIFIER SUBGRADIENTS

To minimize discontinuous functions, that arise in the context of sys- tems with jumps for exampl... more To minimize discontinuous functions, that arise in the context of sys- tems with jumps for example, we propose a new approach based on approximation via averaged functions (obtained by convolution with mollifiers). The properties of averaged functions are studied, after it is shown that they can be used in an ap- proximation scheme consistent with minimization. A new notion of

Research paper thumbnail of General equilibrium and welfare modelling in spatial continuum

Research paper thumbnail of Estimation of econometric models by risk minimization: a stochastic quasigradient approach

Research paper thumbnail of Mathematical Modeling of Distributed Catastrophic and Terrorist Risks1

ABSTRACT The paper shows how the mathematical tools of the theory of controlled Markov fields can... more ABSTRACT The paper shows how the mathematical tools of the theory of controlled Markov fields can be applied to model catastrophic risks caused by natural events or terrorist threats. The examples of problem statements of long-term investment in security are given. A survey of solution methods for stochastic optimal control problems is proposed. It is shown that these problems can be reduced to finite-dimensional stochastic programming problems and can be solved by the stochastic quasigradient method.

Research paper thumbnail of General equilibrium models with discrete choices in a spatial continuum

The treatment of spatial characteristics through probability distributions makes it possible to u... more The treatment of spatial characteristics through probability distributions makes it possible to use stochastic optimization methods and to obtain efficiency results and competitive equilibrium prices for general equilibrium models with discrete choices in spatial continuum. Along these lines, and combining results from stochastic optimization with principles established by Aumann and Hildenbrand for economies with continuum of traders the paper develops

Research paper thumbnail of Reducing Two-Stage Probabilistic Optimization Problems with Discrete Distribution of Random Data to Mixed-Integer Programming Problems*

Research paper thumbnail of Constrained optimization of discontinuous systems

Research paper thumbnail of On nonsmooth problems of stochastic systems optimization

A class of stochastic optimization problems is analyzed that cannot be solved by deterministic an... more A class of stochastic optimization problems is analyzed that cannot be solved by deterministic and standard stochastic approximation methods. We consider risk control problems, optimization of stochastic networks and discrete event systems, screening irreversible changes, pollution control. The results of Ermoliev, Norkin, Wets [ll] are extended to the case of problems involving random variables and general constraints. It is shown that the concept of mollifier subgradient leads to easily implementable computational procedures for stochastic systems with Lipschitz and discontinuous expectation functions. New optimality conditions are formulated enabling to design stochastic search procedures for constrained optimization of discontinuous systems.

Research paper thumbnail of Risk and extended expected utility functions: optimization approaches

Research paper thumbnail of Metody nevypukloi optimizatsii

Research paper thumbnail of Stochastic generalized gradient method with application to insurance risk management

a We would like to thank Gordon MacDonald and Joanne Linnerooth-Bayer for their helpful comments.

Research paper thumbnail of Methods of nonconvex optimization