Vladimir
NORKIN’s research
Research directions:
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 - 2005) a concept of stochastic tangent minorants
is introduced and on this basis the 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. (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. (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. (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., 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),
Norkin
V.I., Onischenko B.O. (2005).
Minorant methods of
stochastic global optimization // Cybernetics
and systems analysis. – Vol. 41 (2005). – No.2. – P. 203-214 (Translated
from Kibernetika i Sistemnyi Analiz, No. 2, pp. 56–70, 2005).