An optimization problem is considered where the objective function f (x) is black-box and multiextremal and the information about its gradient ∇ f (x) is available during the search. It is supposed that ∇ f (x) satisfies the Lipschitz condition over the admissible hyperinterval with an unknown Lipschitz constant K. Some numerical Lipschitz global optimization methods based on geometric ideas with the usage of different estimates of the Lipschitz constant K are presented. Results of their systematic experimental investigation are reported and commented on.
REFERENCES
1.
Y. G.
Evtushenko
, and M. A.
Posypkin
, Optim. Lett.
7
, 819
–829
(2013
).2.
C. A.
Floudas
, and P. M.
Pardalos
, editors, Encyclopedia of Optimization
(6
Volumes), Springer
, 2009
, 2nd edn.3.
J. M.
Fowkes
, N. I. M.
Gould
, and C. L.
Farmer
, J. Global Optim.
56
, 1791
–1815
(2013
).4.
V. P.
Gergel
, J. Global Optim.
10
, 257
–281
(1997
).5.
D. E.
Kvasov
, and Y. D.
Sergeyev
, J. Comput. Appl. Math.
236
, 4042
–4054
(2012
).6.
R.
Paulavičius
, and J.
Žilinskas
, Simplicial Global Optimization
, Springer
, New York
, 2014
.7.
Y. D.
Sergeyev
, and D. E.
Kvasov
, Diagonal Global Optimization Methods
, FizMatLit
, Moscow
, 2008
, in Russian.8.
Y. D.
Sergeyev
, and D. E.
Kvasov
, “Lipschitz global optimization,” in Wiley Encyclopedia of Operations Research and Management Science
, Wiley
, New York
, 2011
, vol. 4
, pp. 2812
–2828
.9.
Y. D.
Sergeyev
, and D. E.
Kvasov
, Commun. Nonlinear Sci. Numer. Simulat.
21
, 99
–111
(2015
).10.
Y. D.
Sergeyev
, R. G.
Strongin
, and D.
Lera
, Introduction to Global Optimization Exploiting Space-Filling Curves
, Springer
, New York
, 2013
.11.
R. G.
Strongin
, and Y. D.
Sergeyev
, Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms
, Kluwer Academic Publishers
, Dordrecht
, 2000
.12.
W.
Baritompa
, and A.
Cutler
, J. Global Optim.
4
, 329
–341
(1994
).13.
L.
Breiman
, and A.
Cutler
, Math. Program.
58
, 179
–199
(1993
).14.
S. Y.
Gorodetsky
, Vestnik of Lobachevsky State University of Nizhni Novgorod
1
, 144
–155
(2012
), in Russian.15.
D. E.
Kvasov
, and Y. D.
Sergeyev
, Numer. Algebra Contr. Optim.
2
, 69
–90
(2012
).16.
D.
Lera
, and Y. D.
Sergeyev
, SIAM J. Optim.
23
, 508
–529
(2013
).17.
Y. D.
Sergeyev
, Math. Program.
81
, 127
–146
(1998
).18.
D. E.
Kvasov
, and Y. D.
Sergeyev
, Optim. Lett.
3
, 303
–318
(2009
).19.
D.
Lera
, and Y. D.
Sergeyev
, Commun. Nonlinear Sci. Numer. Simulat.
23
, 328
–342
(2015
).20.
J. D.
Pintér
, Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications)
, Kluwer Academic Publishers
, Dordrecht
, 1996
.21.
Y. D.
Sergeyev
, and D. E.
Kvasov
, SIAM J. Optim.
16
, 910
–937
(2006
).22.
Y. D.
Sergeyev
, J. Optim. Theory Appl.
107
, 145
–168
(2000
).23.
D. E.
Kvasov
, and Y. D.
Sergeyev
, Automat. Remote Contr.
74
, 1435
–1448
(2013
).24.
D.
Di Serafino
, G.
Liuzzi
, V.
Piccialli
, F.
Riccio
, and G.
Toraldo
, J. Optim. Theory Appl.
151
, 175
–190
(2011
).25.
R.
Paulavičius
, J.
Žilinskas
, and A.
Grothey
, Optim. Methods Softw.
26
, 487
–498
(2011
).26.
R.
Battiti
, M.
Brunato
, and F.
Mascia
, Reactive Search and Intelligent Optimization
, Springer
, New York
, 2009
.27.
K.
Barkalov
, A.
Polovinkin
, I.
Meyerov
, S.
Sidorov
, and N.
Zolotykh
, “SVM regression parameters optimization using parallel global search algorithm,” in Parallel Computing Technologies
, Springer
, 2013
, vol. 7979
of LNCS, pp. 154
–166
.28.
R.
Paulavičius
, Y. D.
Sergeyev
, D. E.
Kvasov
, and J.
Žilinskas
, J. Global Optim.
59
, 545
–567
(2014
).29.
A. A.
Zhigljavsky
, and A.
Žilinskas
, Stochastic Global Optimization
, Springer
, New York
, 2008
.30.
J. W.
Gillard
, and A. A.
Zhigljavsky
, J. Global Optim.
57
, 733
–751
(2013
).31.
A.
Žilinskas
, J. Global Optim.
48
, 173
–182
(2010
).32.
Y.
Sergeyev
, and D.
Kvasov
, “On deterministic diagonal methods for solving global optimization problems with Lipschitz gradients,” in Optimization, Control, and Applications in the Information Age
, Springer
, 2015
, vol. 130
of PROMS, pp. 319
–337
.33.
D. E.
Kvasov
, C.
Pizzuti
, and Y. D.
Sergeyev
, Numer. Math.
94
, 93
–106
(2003
).34.
35.
Y. D.
Sergeyev
, Comput. Math. Math. Phys.
35
, 705
–717
(1995
).36.
D. E.
Kvasov
, 4OR – Quart. J. Oper. Res.
6
, 403
–406
(2008
).37.
D. E.
Kvasov
, and Y. D.
Sergeyev
, Adv. Eng. Softw.
80
, 58
–66
(2015
).38.
J. M.
Calvin
, and A.
Žilinskas
, Comp. Math. Appl.
50
, 157
–169
(2005
).39.
V. P.
Gergel
, Comput. Math. Math. Phys.
36
, 729
–742
(1996
).40.
V. P.
Gergel
, and Y. D.
Sergeyev
, Comput. Math. Appl.
37
, 163
–179
(1999
).41.
Y. D.
Sergeyev
, and V. A.
Grishagin
, J. Optim. Theory Appl.
80
, 513
–536
(1994
).42.
Y. D.
Sergeyev
, and V. A.
Grishagin
, J. Comput. Anal. Appl.
3
, 123
–145
(2001
).43.
Y. D.
Sergeyev
, D.
Famularo
, and P.
Pugliese
, J. Global Optim.
21
, 317
–341
(2001
).44.
R. G.
Strongin
, and Y. D.
Sergeyev
, Parallel Comput.
18
, 1259
–1273
(1992
).45.
V. A.
Grishagin
, and R. G.
Strongin
, Eng. Cybernetics
22
, 117
–122
(1984
).46.
M.
Gaviano
, D.
Lera
, D. E.
Kvasov
, and Y. D.
Sergeyev
, ACM Trans. Math. Software
29
, 469
–480
(2003
).
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