March  2018, 8(1): 47-62. doi: 10.3934/naco.2018003

Globalizer: A novel supercomputer software system for solving time-consuming global optimization problems

Lobachevsky State University of Nizhni Novgorod, 23, Gagarin Avenue, Nizhni Novgorod, Russia

* Corresponding author: Konstantin Barkalov

Received  December 2016 Revised  September 2017 Published  March 2018

Fund Project: The study was supported by the Russian Science Foundation, project No 16-11-10150.

In this paper, we describe the Globalizer software system for solving the global optimization problems. The system is designed to maximize the use of computational potential of the modern high-performance computational systems in order to solve the most time-consuming optimization problems. The highly parallel computations are facilitated using various distinctive computational schemes: processing several optimization iterations simultaneously, reducing multidimensional optimization problems using multiple Peano space-filling curves, and multi-stage computing based on the nested block reduction schemes. These novelties provide for the use of the supercomputer system capabilities with shared and distributed memory and with large numbers of processors to solve the global optimization problems efficiently.

Citation: Victor Gergel, Konstantin Barkalov, Alexander Sysoyev. Globalizer: A novel supercomputer software system for solving time-consuming global optimization problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 47-62. doi: 10.3934/naco.2018003
References:
[1]

K. Barkalov and V. Gergel, Multilevel scheme of dimensionality reduction for parallel global search algorithms, in Proceedings of the 1st International Conference on Engineering and Applied Sciences Optimization, (2014), 2111-2124. Google Scholar

[2]

K. Barkalov and V. Gergel, Parallel global optimization on GPU, J. Glob. Optim., 66 (2016), 3-20.  doi: 10.1007/s10898-016-0411-y.  Google Scholar

[3]

K. BarkalovV. Gergel and I. Lebedev, Use of Xeon Phi coprocessor for solving global optimization problems, LNCS, 9251 (2015), 307-318.  doi: 10.1007/978-3-319-21909-7_31.  Google Scholar

[4]

K. BarkalovV. GergelI. Lebedev and A. Sysoev, Solving the global optimization problems on heterogeneous cluster systems, CEUR Workshop Proceedings, 1482 (2015), 411-419.   Google Scholar

[5]

K. BarkalovA. PolovinkinI. MeyerovS. Sidorov and N. Zolotykh, SVM regression parameters optimization using parallel global search algorithm, LNCS, 7979 (2013), 154-166.  doi: 10.1007/978-3-642-39958-9_14.  Google Scholar

[6]

M. R. Bussieck and A. Meeraus, General algebraic modeling system (GAMS), in Modeling Languages in Mathematical Optimization (ed. J. Kallrath), Springer, (2004), 137-157. doi: 10.1007/978-1-4613-0215-5_8.  Google Scholar

[7]

Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, 1998.  Google Scholar

[8]

R. Čiegis, D. Henty, B. Kågström and J. Žilinskas, Parallel Scientific Computing and Optimization: Advances and Applications, Springer, 2009. doi: 10.1007/978-0-387-09707-7.  Google Scholar

[9]

I. N. Egorov, G. V. Kretinin, I. A. Leshchenko and S. V. Kuptzov, IOSO optimization toolkit — novel software to create better design, in 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2002. Available from http://www.iosotech.com/text/2002_4329.pdf. Google Scholar

[10]

D. FamularoP. Pugliese and Y. D. Sergeyev, A global optimization technique for checking parametric robustness, Automatica, 35 (1999), 1605-1611.  doi: 10.1016/S0005-1098(99)00058-8.  Google Scholar

[11]

G. Fasano and J. D. Pintér, Modeling and Optimization in Space Engineering, Springer, 2013. doi: 10.1007/978-1-4614-4469-5.  Google Scholar

[12]

C. A. Floudas and M. P. Pardalos, State of the Art in Global Optimization: Computational Methods and Applications, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-1-4613-3437-8.  Google Scholar

[13]

C. A. Floudas and M. P. Pardalos, Recent Advances in Global Optimization, Princeton University Press, 2016. doi: 10.1007/s10898-008-9332-8.  Google Scholar

[14]

J. M. Gablonsky and C. T. Kelley, A locally-biased form of the DIRECT algorithm, J. Glob. Optim., 21 (2001), 27-37.  doi: 10.1023/A:1017930332101.  Google Scholar

[15]

M. GavianoD. E. KvasovD. Lera and Y. D. Sergeev, Software for generation of classes of test functions with known local and global minima for global optimization, ACM Trans. Math. Software, 29 (2003), 469-480.  doi: 10.1145/962437.962444.  Google Scholar

[16]

V. Gergel and I. Lebedev, Heterogeneous parallel computations for solving global optimization problems, Procedia Comput. Sci., 66 (2015), 53-62.  doi: 10.1016/j.procs.2015.11.008.  Google Scholar

[17]

V. Gergel, A software system for multi-extremal optimization, Eur. J. Oper. Res., 65 (1993), 305-313.   Google Scholar

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V. Gergel, A method for using derivatives in the minimization of multiextremum functions, Comput. Math. Math. Phys., 36 (1996), 729-742.   Google Scholar

[19]

V. Gergel, A global optimization algorithm for multivariate functions with Lipschitzian first derivatives, J. Glob. Optim., 10 (1997), 257-281.  doi: 10.1023/A:1008290629896.  Google Scholar

[20]

V. Gergel, et al., High performance computing in biomedical applications, Procedia Computer Science, 18 (2013), 10-19. doi: 10.1016/j.procs.2013.05.164.  Google Scholar

[21]

V. Gergel, et al., Recognition of surface defects of cold-rolling sheets based on method of localities, International Review of Automatic Control, 8 (2015), 51-55. doi: 10.15866/ireaco.v8i1.4935.  Google Scholar

[22]

V. Gergel and S. Sidorov, A two-level parallel global search algorithm for solving computationally intensive multi-extremal optimization problems, LNCS, 9251 (2015), 505-515.  doi: 10.1007/978-3-319-21909-7_49.  Google Scholar

[23]

V. A. Grishagin and R. G. Strongin, Optimization of multi-extremal functions subject to monotonically unimodal constraints, Engineering Cybernetics, 5 (1984), 117-122.   Google Scholar

[24]

K. Holmstrm and M. M. Edvall, The TOMLAB optimization environment, Modeling Languages in Mathematical Optimization, Springer, (2004), 369-376. doi: 10.1007/978-1-4613-0215-5_19.  Google Scholar

[25]

R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-03199-5.  Google Scholar

[26]

D. R. JonesC. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant, J. Optim. Theory Appl., 79 (1993), 157-181.  doi: 10.1007/BF00941892.  Google Scholar

[27]

R. B. Kearfott, GlobSol user guide, Optim. Methods Softw., 24 (2009), 687-708.  doi: 10.1080/10556780802614051.  Google Scholar

[28]

D. E. Kvasov and Y. D. Sergeyev, Deterministic approaches for solving practical black-box global optimization problems, Adv. Eng. Softw., 80 (2015), 58-66.  doi: 10.1016/j.advengsoft.2014.09.014.  Google Scholar

[29]

D. E. KvasovD. MennitiA. PinnarelliY. D. Sergeyev and N. Sorrentino, Tuning fuzzy power-system stabilizers in multi-machine systems by global optimization algorithms based on efficient domain partitions, Electric Power Systems Research, 78 (2008), 1217-1229.  doi: 10.1016/j.epsr.2007.10.009.  Google Scholar

[30]

D. E. KvasovC. Pizzuti and Y. D. Sergeyev, Local tuning and partition strategies for diagonal GO methods, Numerische Mathematik, 94 (2003), 93-106.  doi: 10.1007/s00211-002-0419-8.  Google Scholar

[31]

L. Liberti, Writing global optimization software, in Nonconvex Optimization and Its Applications, Springer, 84 (2006), 211-262. doi: 10.1007/0-387-30528-9_8.  Google Scholar

[32]

Y. Lin and L. Schrage, The global solver in the LINDO API, Optim. Methods Softw., 24 (2009), 657-668.  doi: 10.1080/10556780902753221.  Google Scholar

[33]

M. Locatelli and F. Schoen, Global Optimization: Theory, Algorithms and Applications, SIAM, 2013. doi: 10.1137/1.9781611972672.  Google Scholar

[34]

G. Luque and E. Alba, Parallel Genetic Algorithms. Theory and Real World Applications, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-22084-5.  Google Scholar

[35]

M. MongeauH. KarsentyV. Rouzé and J. B. Hiriart-Urruty, Comparison of public-domain software for black box global optimization, Optim. Methods Softw., 13 (2000), 203-226.  doi: 10.1080/10556780008805783.  Google Scholar

[36]

K. M. Mullen, Continuous global optimization in R, J. Stat. Softw. , 60 (2014). doi: 10.18637/jss.v060.i06.  Google Scholar

[37]

M. P. Pardalos, A. A. Zhigljavsky and J. Žilinskas, Advances in Stochastic and Deterministic Global Optimization, Springer, 2016. doi: 10.1007/978-3-319-29975-4.  Google Scholar

[38]

J. D. Pintér, Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications), Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-1-4757-2502-5.  Google Scholar

[39]

J. D. Pintér, Software development for global optimization, Lectures on Global Optimization. Fields Institute Communications, 55 (2009), 183-204.   Google Scholar

[40]

L. M. Rios and N. V. Sahinidis, Derivative-free optimization: a review of algorithms and comparison of software implementations, J. Glob. Optim., 56 (2013), 1247-1293.  doi: 10.1007/s10898-012-9951-y.  Google Scholar

[41]

N. V. Sahinidis, BARON: A general purpose global optimization software package, J. Glob. Optim., 8 (1996), 201-205.  doi: 10.1007/BF00138693.  Google Scholar

[42]

Y. D. Sergeyev, An information global optimization algorithm with local tuning, SIAM J. Optim., 5 (1995), 858-870.  doi: 10.1137/0805041.  Google Scholar

[43]

Y. D. Sergeyev, Multidimensional global optimization using the first derivatives, Comput. Math. Math. Phys., 39 (1999), 743-752.   Google Scholar

[44]

Y. D. Sergeyev and D. E. Kvasov, Global search based on efficient diagonal partitions and a set of Lipschitz constants, SIAM Journal on Optimization, 16 (2006), 910-937.  doi: 10.1137/040621132.  Google Scholar

[45]

Y. D. Sergeyev and V. A. Grishagin, Parallel asynchronous global search and the nested optimization scheme, J. Comput. Anal. Appl., 3 (2001), 123-145.  doi: 10.1023/A:1010185125012.  Google Scholar

[46]

Y. D. Sergeyev, R. G. Strongin and D. Lera, Introduction to Global Optimization Exploiting Space-filling Curves, Springer, 2013. doi: 10.1007/978-1-4614-8042-6.  Google Scholar

[47]

Y. D. SergeyevD. Famularo and P. Pugliese, Index branch-and-bound algorithm for Lipschitz univariate global optimization with multiextremal constraints, J. Glob. Optim., 21 (2001), 317-341.  doi: 10.1023/A:1012391611462.  Google Scholar

[48]

R. G. Strongin, Numerical Methods in Multi-Extremal Problems (Information-Statistical Algorithms), Moscow: Nauka, In Russian, 1978.  Google Scholar

[49]

R. G. Strongin, Algorithms for multi-extremal mathematical programming problems employing a set of joint space-filling curves, J. Glob. Optim., 2 (1992), 357-378.  doi: 10.1007/BF00122428.  Google Scholar

[50]

R. G. Strongin, V. P. Gergel, V. A. Grishagin and K. A. Barkalov, Parallel Computations for Global Optimization Problems, Moscow State University (In Russian), Moscow, 2013. Google Scholar

[51]

R. G. Strongin and Y. D. Sergeyev, Global Optimization with Non-convex Constraints. Sequential and Parallel Algorithms, Kluwer Academic Publishers, Dordrecht (2000, 2nd ed. 2013, 3rd ed. 2014). doi: 10.1007/978-1-4615-4677-1.  Google Scholar

[52]

A. Törn and A. Žilinskas, Global Optimization, Springer, 1989. doi: 10.1007/3-540-50871-6.  Google Scholar

[53]

P. Venkataraman, Applied Optimization with MATLAB Programming, John Wiley & Sons, 2009. Google Scholar

[54]

A. A. Zhigljavsky, Theory of Global Random Search, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3436-1.  Google Scholar

show all references

References:
[1]

K. Barkalov and V. Gergel, Multilevel scheme of dimensionality reduction for parallel global search algorithms, in Proceedings of the 1st International Conference on Engineering and Applied Sciences Optimization, (2014), 2111-2124. Google Scholar

[2]

K. Barkalov and V. Gergel, Parallel global optimization on GPU, J. Glob. Optim., 66 (2016), 3-20.  doi: 10.1007/s10898-016-0411-y.  Google Scholar

[3]

K. BarkalovV. Gergel and I. Lebedev, Use of Xeon Phi coprocessor for solving global optimization problems, LNCS, 9251 (2015), 307-318.  doi: 10.1007/978-3-319-21909-7_31.  Google Scholar

[4]

K. BarkalovV. GergelI. Lebedev and A. Sysoev, Solving the global optimization problems on heterogeneous cluster systems, CEUR Workshop Proceedings, 1482 (2015), 411-419.   Google Scholar

[5]

K. BarkalovA. PolovinkinI. MeyerovS. Sidorov and N. Zolotykh, SVM regression parameters optimization using parallel global search algorithm, LNCS, 7979 (2013), 154-166.  doi: 10.1007/978-3-642-39958-9_14.  Google Scholar

[6]

M. R. Bussieck and A. Meeraus, General algebraic modeling system (GAMS), in Modeling Languages in Mathematical Optimization (ed. J. Kallrath), Springer, (2004), 137-157. doi: 10.1007/978-1-4613-0215-5_8.  Google Scholar

[7]

Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, 1998.  Google Scholar

[8]

R. Čiegis, D. Henty, B. Kågström and J. Žilinskas, Parallel Scientific Computing and Optimization: Advances and Applications, Springer, 2009. doi: 10.1007/978-0-387-09707-7.  Google Scholar

[9]

I. N. Egorov, G. V. Kretinin, I. A. Leshchenko and S. V. Kuptzov, IOSO optimization toolkit — novel software to create better design, in 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2002. Available from http://www.iosotech.com/text/2002_4329.pdf. Google Scholar

[10]

D. FamularoP. Pugliese and Y. D. Sergeyev, A global optimization technique for checking parametric robustness, Automatica, 35 (1999), 1605-1611.  doi: 10.1016/S0005-1098(99)00058-8.  Google Scholar

[11]

G. Fasano and J. D. Pintér, Modeling and Optimization in Space Engineering, Springer, 2013. doi: 10.1007/978-1-4614-4469-5.  Google Scholar

[12]

C. A. Floudas and M. P. Pardalos, State of the Art in Global Optimization: Computational Methods and Applications, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-1-4613-3437-8.  Google Scholar

[13]

C. A. Floudas and M. P. Pardalos, Recent Advances in Global Optimization, Princeton University Press, 2016. doi: 10.1007/s10898-008-9332-8.  Google Scholar

[14]

J. M. Gablonsky and C. T. Kelley, A locally-biased form of the DIRECT algorithm, J. Glob. Optim., 21 (2001), 27-37.  doi: 10.1023/A:1017930332101.  Google Scholar

[15]

M. GavianoD. E. KvasovD. Lera and Y. D. Sergeev, Software for generation of classes of test functions with known local and global minima for global optimization, ACM Trans. Math. Software, 29 (2003), 469-480.  doi: 10.1145/962437.962444.  Google Scholar

[16]

V. Gergel and I. Lebedev, Heterogeneous parallel computations for solving global optimization problems, Procedia Comput. Sci., 66 (2015), 53-62.  doi: 10.1016/j.procs.2015.11.008.  Google Scholar

[17]

V. Gergel, A software system for multi-extremal optimization, Eur. J. Oper. Res., 65 (1993), 305-313.   Google Scholar

[18]

V. Gergel, A method for using derivatives in the minimization of multiextremum functions, Comput. Math. Math. Phys., 36 (1996), 729-742.   Google Scholar

[19]

V. Gergel, A global optimization algorithm for multivariate functions with Lipschitzian first derivatives, J. Glob. Optim., 10 (1997), 257-281.  doi: 10.1023/A:1008290629896.  Google Scholar

[20]

V. Gergel, et al., High performance computing in biomedical applications, Procedia Computer Science, 18 (2013), 10-19. doi: 10.1016/j.procs.2013.05.164.  Google Scholar

[21]

V. Gergel, et al., Recognition of surface defects of cold-rolling sheets based on method of localities, International Review of Automatic Control, 8 (2015), 51-55. doi: 10.15866/ireaco.v8i1.4935.  Google Scholar

[22]

V. Gergel and S. Sidorov, A two-level parallel global search algorithm for solving computationally intensive multi-extremal optimization problems, LNCS, 9251 (2015), 505-515.  doi: 10.1007/978-3-319-21909-7_49.  Google Scholar

[23]

V. A. Grishagin and R. G. Strongin, Optimization of multi-extremal functions subject to monotonically unimodal constraints, Engineering Cybernetics, 5 (1984), 117-122.   Google Scholar

[24]

K. Holmstrm and M. M. Edvall, The TOMLAB optimization environment, Modeling Languages in Mathematical Optimization, Springer, (2004), 369-376. doi: 10.1007/978-1-4613-0215-5_19.  Google Scholar

[25]

R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-03199-5.  Google Scholar

[26]

D. R. JonesC. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant, J. Optim. Theory Appl., 79 (1993), 157-181.  doi: 10.1007/BF00941892.  Google Scholar

[27]

R. B. Kearfott, GlobSol user guide, Optim. Methods Softw., 24 (2009), 687-708.  doi: 10.1080/10556780802614051.  Google Scholar

[28]

D. E. Kvasov and Y. D. Sergeyev, Deterministic approaches for solving practical black-box global optimization problems, Adv. Eng. Softw., 80 (2015), 58-66.  doi: 10.1016/j.advengsoft.2014.09.014.  Google Scholar

[29]

D. E. KvasovD. MennitiA. PinnarelliY. D. Sergeyev and N. Sorrentino, Tuning fuzzy power-system stabilizers in multi-machine systems by global optimization algorithms based on efficient domain partitions, Electric Power Systems Research, 78 (2008), 1217-1229.  doi: 10.1016/j.epsr.2007.10.009.  Google Scholar

[30]

D. E. KvasovC. Pizzuti and Y. D. Sergeyev, Local tuning and partition strategies for diagonal GO methods, Numerische Mathematik, 94 (2003), 93-106.  doi: 10.1007/s00211-002-0419-8.  Google Scholar

[31]

L. Liberti, Writing global optimization software, in Nonconvex Optimization and Its Applications, Springer, 84 (2006), 211-262. doi: 10.1007/0-387-30528-9_8.  Google Scholar

[32]

Y. Lin and L. Schrage, The global solver in the LINDO API, Optim. Methods Softw., 24 (2009), 657-668.  doi: 10.1080/10556780902753221.  Google Scholar

[33]

M. Locatelli and F. Schoen, Global Optimization: Theory, Algorithms and Applications, SIAM, 2013. doi: 10.1137/1.9781611972672.  Google Scholar

[34]

G. Luque and E. Alba, Parallel Genetic Algorithms. Theory and Real World Applications, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-22084-5.  Google Scholar

[35]

M. MongeauH. KarsentyV. Rouzé and J. B. Hiriart-Urruty, Comparison of public-domain software for black box global optimization, Optim. Methods Softw., 13 (2000), 203-226.  doi: 10.1080/10556780008805783.  Google Scholar

[36]

K. M. Mullen, Continuous global optimization in R, J. Stat. Softw. , 60 (2014). doi: 10.18637/jss.v060.i06.  Google Scholar

[37]

M. P. Pardalos, A. A. Zhigljavsky and J. Žilinskas, Advances in Stochastic and Deterministic Global Optimization, Springer, 2016. doi: 10.1007/978-3-319-29975-4.  Google Scholar

[38]

J. D. Pintér, Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications), Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-1-4757-2502-5.  Google Scholar

[39]

J. D. Pintér, Software development for global optimization, Lectures on Global Optimization. Fields Institute Communications, 55 (2009), 183-204.   Google Scholar

[40]

L. M. Rios and N. V. Sahinidis, Derivative-free optimization: a review of algorithms and comparison of software implementations, J. Glob. Optim., 56 (2013), 1247-1293.  doi: 10.1007/s10898-012-9951-y.  Google Scholar

[41]

N. V. Sahinidis, BARON: A general purpose global optimization software package, J. Glob. Optim., 8 (1996), 201-205.  doi: 10.1007/BF00138693.  Google Scholar

[42]

Y. D. Sergeyev, An information global optimization algorithm with local tuning, SIAM J. Optim., 5 (1995), 858-870.  doi: 10.1137/0805041.  Google Scholar

[43]

Y. D. Sergeyev, Multidimensional global optimization using the first derivatives, Comput. Math. Math. Phys., 39 (1999), 743-752.   Google Scholar

[44]

Y. D. Sergeyev and D. E. Kvasov, Global search based on efficient diagonal partitions and a set of Lipschitz constants, SIAM Journal on Optimization, 16 (2006), 910-937.  doi: 10.1137/040621132.  Google Scholar

[45]

Y. D. Sergeyev and V. A. Grishagin, Parallel asynchronous global search and the nested optimization scheme, J. Comput. Anal. Appl., 3 (2001), 123-145.  doi: 10.1023/A:1010185125012.  Google Scholar

[46]

Y. D. Sergeyev, R. G. Strongin and D. Lera, Introduction to Global Optimization Exploiting Space-filling Curves, Springer, 2013. doi: 10.1007/978-1-4614-8042-6.  Google Scholar

[47]

Y. D. SergeyevD. Famularo and P. Pugliese, Index branch-and-bound algorithm for Lipschitz univariate global optimization with multiextremal constraints, J. Glob. Optim., 21 (2001), 317-341.  doi: 10.1023/A:1012391611462.  Google Scholar

[48]

R. G. Strongin, Numerical Methods in Multi-Extremal Problems (Information-Statistical Algorithms), Moscow: Nauka, In Russian, 1978.  Google Scholar

[49]

R. G. Strongin, Algorithms for multi-extremal mathematical programming problems employing a set of joint space-filling curves, J. Glob. Optim., 2 (1992), 357-378.  doi: 10.1007/BF00122428.  Google Scholar

[50]

R. G. Strongin, V. P. Gergel, V. A. Grishagin and K. A. Barkalov, Parallel Computations for Global Optimization Problems, Moscow State University (In Russian), Moscow, 2013. Google Scholar

[51]

R. G. Strongin and Y. D. Sergeyev, Global Optimization with Non-convex Constraints. Sequential and Parallel Algorithms, Kluwer Academic Publishers, Dordrecht (2000, 2nd ed. 2013, 3rd ed. 2014). doi: 10.1007/978-1-4615-4677-1.  Google Scholar

[52]

A. Törn and A. Žilinskas, Global Optimization, Springer, 1989. doi: 10.1007/3-540-50871-6.  Google Scholar

[53]

P. Venkataraman, Applied Optimization with MATLAB Programming, John Wiley & Sons, 2009. Google Scholar

[54]

A. A. Zhigljavsky, Theory of Global Random Search, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3436-1.  Google Scholar

Figure 1.  A Peano curve approximation for the third density level
Figure 2.  The computational scheme for obtaining the value of the reduced one-dimensional function $\varphi(y(x))$
Figure 3.  The architecture of the Globalizer system
Table 1.  Averaged number of executed iterations of the compared global optimization methods
$N$Problem classDIRECTDIRECTlMAGS
4 Simple$>$47282(4)1898311953
Hard $>$95708(7)6875425263
5 Simple $>$16057(1)1675815920
Hard $>$217215(16) $>$269064(4) $>$148342(4)
$N$Problem classDIRECTDIRECTlMAGS
4 Simple$>$47282(4)1898311953
Hard $>$95708(7)6875425263
5 Simple $>$16057(1)1675815920
Hard $>$217215(16) $>$269064(4) $>$148342(4)
Table 2.  Averaged numbers of iterations executed by GPMAGS for solving the test optimization problems
p$N=4$$N=5$
Simple Hard Simple Hard
Serial trial computations1119532526315920 $>$148342 (4)
Parallel computations247621117813378109075
on CPU423725972520351868
813932874377351868
Parallel computations601713933823452
on Xeon Phi120851822491306
2404210397381
p$N=4$$N=5$
Simple Hard Simple Hard
Serial trial computations1119532526315920 $>$148342 (4)
Parallel computations247621117813378109075
on CPU423725972520351868
813932874377351868
Parallel computations601713933823452
on Xeon Phi120851822491306
2404210397381
Table 3.  Speedup of parallel computations executed by GPMAGS
p $N=4$ $N=5$
Simple Hard Simple Hard
Serial trial computations. 1 11953 25263 15920 > 148342 (4)
Average number of iterations
Parallel computations 2 2.52 2.32 1.21 1.41
of CPU. 4 5.05 4.24 3.13 2.92
Speedup 8 8.68 8.88 4.24 6.66
Parallel computations 60 8.18 7.37 9.99 6.66
of Xeon Phi. 120 16.316 15.815 15.215 17.317
Speedup 240 33.133 27.827 38.838 59.359
p $N=4$ $N=5$
Simple Hard Simple Hard
Serial trial computations. 1 11953 25263 15920 > 148342 (4)
Average number of iterations
Parallel computations 2 2.52 2.32 1.21 1.41
of CPU. 4 5.05 4.24 3.13 2.92
Speedup 8 8.68 8.88 4.24 6.66
Parallel computations 60 8.18 7.37 9.99 6.66
of Xeon Phi. 120 16.316 15.815 15.215 17.317
Speedup 240 33.133 27.827 38.838 59.359
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