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Page 184

172 STOCHASTIC APPROXIMATION AND ITS APPLICATIONS

Remark 4.2.2 Results in Sections 4.1 and 4.2 are proved for the case,

where the two-sided randomized differences are
used where and are given by (4.1.3) and (4.1.4), respectively.
But, all results presented in Sections 4.1 and 4.2 are also valid for the
case where the one-sided randomized differences

are used, where and are given by (4.1.3) and (4.1.6), respec-
tively.

In this case, in (4.1.27), (4.1.28) and in the expression of should
be replaced by 1, and (4.1.29)–(4.1.32) disappear. Accordingly, (4.1.36)
changes to

Theorems 4.1.1-4.1.4 and 4.2.1 remain unchanged. The conclusion of

Theorem 4.2.2 remains valid too, if in Condition iv)

changes to

4.3. Global Optimization
As pointed out at the beginning of the chapter, the KW algorithm may

lead to a local minimizer of Before the 1980s, the random search
or its combination with a local search method was the main stochastic
approach to achieve the global minimum when the values of L can exactly
be observed without noise. When the structural property of L is used
for local search, a rather rapid convergence rate can be derived, but it
is hard to escape a local attraction domain. The random search has
a chance to fall into any attraction domain, but its convergence rate
decreases exponentially as the dimension of the problem increases.

Simulating annealing is an attractive method for global optimization,
but it provides only convergence in probability rather than path-wise
convergence. Moreover, simulation shows that for functions with a few
local minima, simulated annealing is not efficient. This motivates one
to combine KW-type method with random search. However, a simple
combination of SA and random search does not work: in order to reach
the global minimum one has to reduce the noise effect as time goes on.

A hybrid algorithm composed of a search method and the KW algo-
rithm is presented in the sequel with main effort devoted to design eas-

Page 185

Optimization by Stochastic Approximation 173

ily realizable switching rules and to provide an effective noise-reducing
method.

We define a global optimization algorithm, which consists of three
parts: search, selection, and optimization. To be fixed, let us discuss
the global minimization problem. In the search part, we choose an ini-
tial value and make the local search by use of the KW algorithm with
randomized differences and expanding truncations described in Section
4.1 to approach the bottom of the local attraction domain. At the same
time, the average of the observations for L is used to serve as an estimate
of the local minimum of L in this attraction domain. In the selection
part, the estimates obtained for the local minima of L are compared with
each other, and the smallest one among them together with the corre-
sponding minimizer given by the KW algorithm are selected. Then, the
optimization part takes place, where again the local search is carried out,
i.e., the KW algorithm without any truncations is applied to improve
the estimate for the minimizer. At the same time, the corresponding
minimum of L is reestimated by averaging the noisy observations. After
this, the algorithm goes back to the search part again.

For the local search, we use observations (4.1.3) and (4.1.4), or (4.1.5)
and (4.1.6). To be fixed, let us use (4.1.5) and (4.1.6).

In the sequel, by KW algorithm with expanding truncations we mean
the algorithm defined by (4.1.11) and (4.1.12) with

where and are given by (4.1.5) and (4.1.6), respectively. Sim-
ilar to (4.1.9) and (4.1.10) we have

where

By KW algorithm we mean

with defined by (4.3.2).
It is worth noting that unlike (4.1.8), is used in (4.3.1).

Roughly speaking, this is because in the neighborhood of a miminizer

of is increasing, and in (4.1.11) should be an
observation on

Page 368

Nonconvex Optimization and Its Applications

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R. Reemtsen and J.�J. Rückmann (eds.): Semi�Infinite Programming. 1998

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B. Ricceri and S. Simons (eds.): Minimax Theory and Applications. 1998

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J.�P. Crouzeix, J.�E. Martinez�Legaz and M. Volle (eds.): Generalized Convexitiy,
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J. Outrata, M. Kočvara and J. Zowe: Nonsmooth Approach to Optimization Problems

with Equilibrium Constraints. 1998 ISBN 0�7923�5170�3

D. Motreanu and P.D. Panagiotopoulos: Minimax Theorems and Qualitative Proper�

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J.F. Bard: Practical Bilevel Optimization. Algorithms and Applications. 1999

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H.D. Sherali and W.P. Adams: A Reformulation�Linearization Technique for Solving

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F. Forgó, J. Szép and F. Szidarovszky: Introduction to the Theory of Games. Concepts,
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C.A. Floudas and P.M. Pardalos (eds.): Handbook of Test Problems in Local and

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T. Stoilov and K. Stoilova: Noniterative Coordination in Multilevel Systems. 1999

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J. Haslinger, M. Miettinen and P.D. Panagiotopoulos: Finite Element Method for

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V. Korotkich: A Mathematical Structure of Emergent Computation. 1999

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C.A. Floudas: Deterministic Global Optimization: Theory, Methods and Applications.

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F. Giannessi (ed.): Vector Variational Inequalities and Vector Equilibria. Mathemat�

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D. Y. Gao: Duality Principles in Nonconvex Systems. Theory, Methods and Applica�

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C.A. Floudas and P.M. Pardalos (eds.): Optimization in Computational Chemistry

and Molecular Biology. Local and Global Approaches. 2000 ISBN 0�7923�6155�5

G. Isac: Topological Methods in Complementarity Theory. 2000 ISBN 0�7923�6274�8

P.M. Pardalos (ed.): Approximation and Complexity in Numerical Optimization: Con�

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V. Demyanov and A. Rubinov (eds.): Quasidifferentiability and Related Topics. 2000

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Page 369

Nonconvex Optimization and Its Applications

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R.G. Strongin and Y.D. Sergeyev: Global Optimization with Non-Convex Constraints.
2000 ISBN 0-7923-6490-2
X.-S. Zhang: Neural Networks in Optimization. 2000 ISBN 0-7923-6515-1
H. Jongen, P. Jonker and F. Twilt: Nonlinear Optimization in Finite Dimen-
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R. Horst, P.M. Pardalos and N.V. Thoai: Introduction to Global Optimization. 2nd
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S.P. Uryasev (ed.): Probabilistic Constrained Optimization. Methodology and
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D.Y. Gao, R.W. Ogden and G.E. Stavroulakis (eds.): Nonsmooth/Nonconvex Mech-
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M. do Rosário Grossinho and S.A. Tersian: An Introduction to Minimax Theorems
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A. Migdalas, P.M. Pardalos and P. Värbrand (eds.): From Local to Global Optimiza-
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N. Hadjisavvas and P.M. Pardalos (eds.): Advances in Convex Analysis and Global
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R.P. Gilbert, P.D. Panagiotopoulos† and P.M. Pardalos (eds.): From Convexity to
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D.-Z. Du, P.M. Pardalos and W. Wu: Mathematical Theory of Optimization. 2001

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M.A. Goberna and M.A. López (eds.): Semi-Infinite Programming. Recent Advances.
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F. Giannessi, A. Maugeri and P.M. Pardalos (eds.): Equilibrium Problems: Nonsmooth
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G. Dzemyda, V. Šaltenis and A. Žilinskas (eds.): Stochastic and Global Optimization.
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D. Klatte and B. Kummer: Nonsmooth Equations in Optimization. Regularity, Cal-
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G. Isac, V.A. Bulavsky and V.V. Kalashnikov: Complementarity, Equilibrium, Effi-
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