September  2020, 10(3): 317-330. doi: 10.3934/naco.2020004

On a new smoothing technique for non-smooth, non-convex optimization

Department of Mathematics, Suleyman Demirel University, Isparta, Turkey

* Corresponding author: Ahmet Sahiner

Received  May 2019 Revised  October 2019 Published  February 2020

Fund Project: This study has been supported by the Teaching Staff Training Project Units of Suleyman Demirel University (OYP-05545-DR-13)

In many global optimization techniques, the local search methods are used for different issues such as to obtain a new initial point and to find the local solution rapidly. Most of these local search methods base on the smoothness of the problem. In this study, we propose a new smoothing approach in order to smooth out non-smooth and non-Lipschitz functions playing a very important role in global optimization problems. We illustrate our smoothing approach on well-known test problems in the literature. The numerical results show the efficiency of our method.

Citation: Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004
References:
[1]

A. M. BagirovA. Al Nuaimat and N. Sultanova, Hyperbolic smoothing function method for minimax problems, Optimization, 62 (2013), 759-782.  doi: 10.1080/02331934.2012.675335.  Google Scholar

[2]

D. Bertsekas, Nondifferentiable optimization via approximation, Mathematical Programming Study, 3 (1975), 1-25.  doi: 10.1007/BFb0120696.  Google Scholar

[3]

B. C. CetinJ. Barhen and J. W. Burdick, Terminal repeller unconstrained subenergy tunneling (TRUST) for fast global optimization, J. Optimiz. Theory Appl., 77 (1993), 97-126.  doi: 10.1007/BF00940781.  Google Scholar

[4]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problem, Comput. Optim. Appl., 5 (1996), 91-138.  doi: 10.1007/BF00249052.  Google Scholar

[5]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program. Ser. B, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[6]

X. ChenM. K. Ng and C. Zhang, Non-lipschitz $l_p$-regularization and box constrained model for image processing, IEEE Trans. Image. Process., 21 (2012), 4709-4721.  doi: 10.1109/TIP.2012.2214051.  Google Scholar

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.  Google Scholar

[8]

V. F. Demyanov and V. N. Malozemov, Introduction to Minimax, 2$^{nd}$ edition, Dover, New York 1990.  Google Scholar

[9]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt, 1995.  Google Scholar

[10]

R. P. Ge, A filled function method for finding global minimizer of a function of several variables, Math. Program., 46 (1990), 191-204.  doi: 10.1007/BF01585737.  Google Scholar

[11]

R. P. Ge, The theory of filled function method for finding global minimizer of a nonlinearly constrained minimization problem, J. Comput. Math., 5 (1987), 1-9.   Google Scholar

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[13]

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

[14]

A. Ketfi-Cherif and A. Ziadi, Global descent method for constrained continuous global optimization, Appl. Math. Comput., 244 (2014), 209-221.  doi: 10.1016/j.amc.2014.06.089.  Google Scholar

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S. KirkpatrickJr. C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680.   Google Scholar

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D. E. Kvasov and M. S. Mukhametzhanov, Metaheuristic vs. deterministic global optimization algorithms: The univariate case, Appl. Math. Comput., 318 (2018), 245-259.  doi: 10.1016/j.amc.2017.05.014.  Google Scholar

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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

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J. Lee and D. Skipper, Virtuous smoothing for global optimization, J. Glob. Optim., 69 (2017), 677-697.  doi: 10.1007/s10898-017-0533-x.  Google Scholar

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A. V. Levy and A. Montalvo, The tunneling algorithm for the global minimization of functions, SIAM J. Sci. Stat. Comput., 6 (1985), 15-29.  doi: 10.1137/0906002.  Google Scholar

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S. J. Lian, Smoothing approximation to $l_1$ exact penalty for inequality constrained optimization, Appl. Math. Comput., 219 (2012), 3113-3121.  doi: 10.1016/j.amc.2012.09.042.  Google Scholar

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H. LinY. WangY. Gao and X. Wang, A filled function method for global optimization with inequality constraints, Comput. Appl. Math., 37 (2018), 1524-1536.  doi: 10.1007/s40314-016-0407-8.  Google Scholar

[22]

S. MaY. Yang and H. Liu, A parameter free filled function for unconstrained global optimization, Appl. Math. Comput., 215 (2010), 3610-3619.  doi: 10.1016/j.amc.2009.10.057.  Google Scholar

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C. K. NgD. Li and L. S. Zhang, Global descent method for global optimization, SIAM J. Optim., 20 (2010), 3161-3184.  doi: 10.1137/090749815.  Google Scholar

[24]

M. NikolovaC. K. Ng and C. P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE T. Image Process., 19 (2010), 3073-3088.  doi: 10.1109/TIP.2010.2052275.  Google Scholar

[25]

A. OzmenE. Kropat and G.-W. Weber, Spline regression models for complex multi-modal regulatory networks, Optim. Method Softw., 29 (2014), 515-534.  doi: 10.1080/10556788.2013.821611.  Google Scholar

[26]

A. Ozmen and G. W. Weber, RMARS: Robustification of multivariate adaptive regression spline under polyhedral uncertainty, J. Comput. Appl. Math., 259 (2014), 914-924.  doi: 10.1016/j.cam.2013.09.055.  Google Scholar

[27]

R. PaulaviciusJ. Zilinskas and A. Grothey, Parallel branch and bound for global optimization with combination of Lipschitz bounds, Optim. Method Softw., 26 (2011), 487-498.  doi: 10.1080/10556788.2010.551537.  Google Scholar

[28]

R. PaulaviciusY. D. SergeyevD. E. Kvasov and J. Zilinskas, Globally-biased DISIMPL algorithm for expensive global optimization, J. Glob. Optim., 59 (2014), 545-567.  doi: 10.1007/s10898-014-0180-4.  Google Scholar

[29]

R. PaulaviciusC. Lakhdar and J. Zilinskas, Global optimization based on bisection of rectangles, function values at diagonals, and a set of Lipschitz constants, J. Glob. Optim., 79 (2018), 5-20.  doi: 10.1007/s10898-016-0485-6.  Google Scholar

[30]

S. S. RaoK. SundararajuB. G. Prakash and C. Balakrishna, A fuzzy goal programming approach for structural optimization, AIAA Journal, 30 (1992), 1425-1432.  doi: 10.2514/3.11079.  Google Scholar

[31]

M. ResenerS. HaffnerL. A. Pereira and P. M. Pardalos, Optimization techniques applied to planning of electric power distribution systems: A bibliographic survey, Energy Syst., 9 (2018), 473-509.  doi: 10.1007/s12667-018-0276-x.  Google Scholar

[32]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998.  Google Scholar

[33]

A. SahinerN. Yilmaz and O. Demirozer, Mathematical modeling and application of filled function method in entomology, Int. Journal Pest. Manage., 60 (2014), 232-237.  doi: 10.1080/09670874.2014.958879.  Google Scholar

[34]

A. SahinerG. Kapusuz and N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Cont. Optim., 6 (2016), 161-173.  doi: 10.3934/naco.2016006.  Google Scholar

[35]

Y. D. Sergeyev, R. G. Strongin and D. Lera, Introduction to Global Optimization Exploiting Space-Filling Curves, Springer, New York, 2013.  Google Scholar

[36]

Y. D. SergeyevD. E. Kvasov and M. S. Mukhametzhanov, On the efficiency of nature-inspired metaheuristics in expensive global optimization with limited budget, Sci. Rep., 8 (2018), 1-9.  doi: 10.1038/s41598-017-18940-4.  Google Scholar

[37]

Y. Shi and R. Eberhart, A modified particle swarm optimizer, In Proceedings of IEEE International Conference on Evolutionary Computation, (1998), 69–73. Google Scholar

[38]

R. G. Strogin and Y. D. Sergeyev, Global Optimization with Nonconvex Constraints: Sequential and Paralel Algorithms, Kluwer Academic Publishers, Dordecht, 2000. Google Scholar

[39]

H. Tuy, Cutting plane methods for global optimization, In: Encyclopedia of Optimization (eds. C. Floudas and P. Pardalos), 2$^{nd}$ Edition, Springer, New York, (2009), 590–594. Google Scholar

[40]

Y. WangW. Fang and T. Wu, A cut-peak function method for global optimization, J. Comput. Appl. Math., 230 (2009), 135-142.  doi: 10.1016/j.cam.2008.10.069.  Google Scholar

[41]

F. WeiY. Wang and H. Lin, A new filled function method with two parameters for global optimization, J. Optim. Theory Appl., 163 (2014), 510-527.  doi: 10.1007/s10957-013-0515-1.  Google Scholar

[42]

Z. Y. WuF. S. BaiH. W. J. Lee and Y. J. Yang, A filled function method for constrained global optimization, J. Glob. Optim., 39 (2007), 495-507.  doi: 10.1007/s10898-007-9152-2.  Google Scholar

[43]

Z. Y. WuD. Li and S. Zhang, Global descent methods for unconstrained global optimization, J. Glob. Optim., 50 (2011), 379-396.  doi: 10.1007/s10898-010-9587-8.  Google Scholar

[44]

A. E. Xavier, Penalizacao Hiperbolica: Um Novo Metodo para Resolutato de Problemas de Otimizao, M. Sc. Thesis, COPPE–Federal University of Rio de Janeiro, 1982. Google Scholar

[45]

A. E. Xavier, The hyperbolic smoothing clustering method, Patt. Recog., 43 (2010), 731-737.  doi: 10.1016/j.patcog.2009.06.018.  Google Scholar

[46]

A. E. Xavier and A. A. F. Oliveira, Optimal covering of plane domains by circles via hyperbolic smoothing, J. Glob. Optim., 31 (2005), 493-504.  doi: 10.1007/s10898-004-0737-8.  Google Scholar

[47]

L. Xu, J. Lee and D. Skipper, More virtuous smoothing, SIAM J. Optim., 29 (2019) 1240–1259. doi: 10.1137/18M1172831.  Google Scholar

[48]

Y. T. XuY. Zhang and S. G. Wang, A modified tunneling function method for non-smooth global optimization and its application in artificial neural network, Appl. Math. Model., 39 (2015), 6438-6450.  doi: 10.1016/j.apm.2015.01.059.  Google Scholar

[49]

Y. Yang and Y. Shang, A new filled function method for unconstrained global optimization, Appl. Math. Comput., 173 (2006), 501-512.  doi: 10.1016/j.amc.2005.04.046.  Google Scholar

[50]

H. Yin, An adaptive smoothing method for continuous minimax problems, Asia-Pac. J. Oper. Res., 32 (2015), 1-19.  doi: 10.1142/S0217595915400011.  Google Scholar

[51]

I. Zang, A smoothing out technique for min-max optimization, Math. Program., 19 (1980), 61-77.  doi: 10.1007/BF01581628.  Google Scholar

[52]

L. S. ZhangC. K. NgD. Li and W. W. Tian, A new filled function method for global optimization, J. Glob. Optim., 28 (2004), 17-43.  doi: 10.1023/B:JOGO.0000006653.60256.f6.  Google Scholar

[53]

A. Zhigljavsky and A. Zilinskas, Stochastic Global Optimization, Springer, New York, 2008.  Google Scholar

[54]

J. Zilinskas, Branch and bound with simplicial partitions for global optimization, Math. Model. Anal., 13 (2008), 145-159.  doi: 10.3846/1392-6292.2008.13.145-159.  Google Scholar

show all references

References:
[1]

A. M. BagirovA. Al Nuaimat and N. Sultanova, Hyperbolic smoothing function method for minimax problems, Optimization, 62 (2013), 759-782.  doi: 10.1080/02331934.2012.675335.  Google Scholar

[2]

D. Bertsekas, Nondifferentiable optimization via approximation, Mathematical Programming Study, 3 (1975), 1-25.  doi: 10.1007/BFb0120696.  Google Scholar

[3]

B. C. CetinJ. Barhen and J. W. Burdick, Terminal repeller unconstrained subenergy tunneling (TRUST) for fast global optimization, J. Optimiz. Theory Appl., 77 (1993), 97-126.  doi: 10.1007/BF00940781.  Google Scholar

[4]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problem, Comput. Optim. Appl., 5 (1996), 91-138.  doi: 10.1007/BF00249052.  Google Scholar

[5]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program. Ser. B, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[6]

X. ChenM. K. Ng and C. Zhang, Non-lipschitz $l_p$-regularization and box constrained model for image processing, IEEE Trans. Image. Process., 21 (2012), 4709-4721.  doi: 10.1109/TIP.2012.2214051.  Google Scholar

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.  Google Scholar

[8]

V. F. Demyanov and V. N. Malozemov, Introduction to Minimax, 2$^{nd}$ edition, Dover, New York 1990.  Google Scholar

[9]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt, 1995.  Google Scholar

[10]

R. P. Ge, A filled function method for finding global minimizer of a function of several variables, Math. Program., 46 (1990), 191-204.  doi: 10.1007/BF01585737.  Google Scholar

[11]

R. P. Ge, The theory of filled function method for finding global minimizer of a nonlinearly constrained minimization problem, J. Comput. Math., 5 (1987), 1-9.   Google Scholar

[12] J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975.   Google Scholar
[13]

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

[14]

A. Ketfi-Cherif and A. Ziadi, Global descent method for constrained continuous global optimization, Appl. Math. Comput., 244 (2014), 209-221.  doi: 10.1016/j.amc.2014.06.089.  Google Scholar

[15]

S. KirkpatrickJr. C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680.   Google Scholar

[16]

D. E. Kvasov and M. S. Mukhametzhanov, Metaheuristic vs. deterministic global optimization algorithms: The univariate case, Appl. Math. Comput., 318 (2018), 245-259.  doi: 10.1016/j.amc.2017.05.014.  Google Scholar

[17]

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

[18]

J. Lee and D. Skipper, Virtuous smoothing for global optimization, J. Glob. Optim., 69 (2017), 677-697.  doi: 10.1007/s10898-017-0533-x.  Google Scholar

[19]

A. V. Levy and A. Montalvo, The tunneling algorithm for the global minimization of functions, SIAM J. Sci. Stat. Comput., 6 (1985), 15-29.  doi: 10.1137/0906002.  Google Scholar

[20]

S. J. Lian, Smoothing approximation to $l_1$ exact penalty for inequality constrained optimization, Appl. Math. Comput., 219 (2012), 3113-3121.  doi: 10.1016/j.amc.2012.09.042.  Google Scholar

[21]

H. LinY. WangY. Gao and X. Wang, A filled function method for global optimization with inequality constraints, Comput. Appl. Math., 37 (2018), 1524-1536.  doi: 10.1007/s40314-016-0407-8.  Google Scholar

[22]

S. MaY. Yang and H. Liu, A parameter free filled function for unconstrained global optimization, Appl. Math. Comput., 215 (2010), 3610-3619.  doi: 10.1016/j.amc.2009.10.057.  Google Scholar

[23]

C. K. NgD. Li and L. S. Zhang, Global descent method for global optimization, SIAM J. Optim., 20 (2010), 3161-3184.  doi: 10.1137/090749815.  Google Scholar

[24]

M. NikolovaC. K. Ng and C. P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE T. Image Process., 19 (2010), 3073-3088.  doi: 10.1109/TIP.2010.2052275.  Google Scholar

[25]

A. OzmenE. Kropat and G.-W. Weber, Spline regression models for complex multi-modal regulatory networks, Optim. Method Softw., 29 (2014), 515-534.  doi: 10.1080/10556788.2013.821611.  Google Scholar

[26]

A. Ozmen and G. W. Weber, RMARS: Robustification of multivariate adaptive regression spline under polyhedral uncertainty, J. Comput. Appl. Math., 259 (2014), 914-924.  doi: 10.1016/j.cam.2013.09.055.  Google Scholar

[27]

R. PaulaviciusJ. Zilinskas and A. Grothey, Parallel branch and bound for global optimization with combination of Lipschitz bounds, Optim. Method Softw., 26 (2011), 487-498.  doi: 10.1080/10556788.2010.551537.  Google Scholar

[28]

R. PaulaviciusY. D. SergeyevD. E. Kvasov and J. Zilinskas, Globally-biased DISIMPL algorithm for expensive global optimization, J. Glob. Optim., 59 (2014), 545-567.  doi: 10.1007/s10898-014-0180-4.  Google Scholar

[29]

R. PaulaviciusC. Lakhdar and J. Zilinskas, Global optimization based on bisection of rectangles, function values at diagonals, and a set of Lipschitz constants, J. Glob. Optim., 79 (2018), 5-20.  doi: 10.1007/s10898-016-0485-6.  Google Scholar

[30]

S. S. RaoK. SundararajuB. G. Prakash and C. Balakrishna, A fuzzy goal programming approach for structural optimization, AIAA Journal, 30 (1992), 1425-1432.  doi: 10.2514/3.11079.  Google Scholar

[31]

M. ResenerS. HaffnerL. A. Pereira and P. M. Pardalos, Optimization techniques applied to planning of electric power distribution systems: A bibliographic survey, Energy Syst., 9 (2018), 473-509.  doi: 10.1007/s12667-018-0276-x.  Google Scholar

[32]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998.  Google Scholar

[33]

A. SahinerN. Yilmaz and O. Demirozer, Mathematical modeling and application of filled function method in entomology, Int. Journal Pest. Manage., 60 (2014), 232-237.  doi: 10.1080/09670874.2014.958879.  Google Scholar

[34]

A. SahinerG. Kapusuz and N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Cont. Optim., 6 (2016), 161-173.  doi: 10.3934/naco.2016006.  Google Scholar

[35]

Y. D. Sergeyev, R. G. Strongin and D. Lera, Introduction to Global Optimization Exploiting Space-Filling Curves, Springer, New York, 2013.  Google Scholar

[36]

Y. D. SergeyevD. E. Kvasov and M. S. Mukhametzhanov, On the efficiency of nature-inspired metaheuristics in expensive global optimization with limited budget, Sci. Rep., 8 (2018), 1-9.  doi: 10.1038/s41598-017-18940-4.  Google Scholar

[37]

Y. Shi and R. Eberhart, A modified particle swarm optimizer, In Proceedings of IEEE International Conference on Evolutionary Computation, (1998), 69–73. Google Scholar

[38]

R. G. Strogin and Y. D. Sergeyev, Global Optimization with Nonconvex Constraints: Sequential and Paralel Algorithms, Kluwer Academic Publishers, Dordecht, 2000. Google Scholar

[39]

H. Tuy, Cutting plane methods for global optimization, In: Encyclopedia of Optimization (eds. C. Floudas and P. Pardalos), 2$^{nd}$ Edition, Springer, New York, (2009), 590–594. Google Scholar

[40]

Y. WangW. Fang and T. Wu, A cut-peak function method for global optimization, J. Comput. Appl. Math., 230 (2009), 135-142.  doi: 10.1016/j.cam.2008.10.069.  Google Scholar

[41]

F. WeiY. Wang and H. Lin, A new filled function method with two parameters for global optimization, J. Optim. Theory Appl., 163 (2014), 510-527.  doi: 10.1007/s10957-013-0515-1.  Google Scholar

[42]

Z. Y. WuF. S. BaiH. W. J. Lee and Y. J. Yang, A filled function method for constrained global optimization, J. Glob. Optim., 39 (2007), 495-507.  doi: 10.1007/s10898-007-9152-2.  Google Scholar

[43]

Z. Y. WuD. Li and S. Zhang, Global descent methods for unconstrained global optimization, J. Glob. Optim., 50 (2011), 379-396.  doi: 10.1007/s10898-010-9587-8.  Google Scholar

[44]

A. E. Xavier, Penalizacao Hiperbolica: Um Novo Metodo para Resolutato de Problemas de Otimizao, M. Sc. Thesis, COPPE–Federal University of Rio de Janeiro, 1982. Google Scholar

[45]

A. E. Xavier, The hyperbolic smoothing clustering method, Patt. Recog., 43 (2010), 731-737.  doi: 10.1016/j.patcog.2009.06.018.  Google Scholar

[46]

A. E. Xavier and A. A. F. Oliveira, Optimal covering of plane domains by circles via hyperbolic smoothing, J. Glob. Optim., 31 (2005), 493-504.  doi: 10.1007/s10898-004-0737-8.  Google Scholar

[47]

L. Xu, J. Lee and D. Skipper, More virtuous smoothing, SIAM J. Optim., 29 (2019) 1240–1259. doi: 10.1137/18M1172831.  Google Scholar

[48]

Y. T. XuY. Zhang and S. G. Wang, A modified tunneling function method for non-smooth global optimization and its application in artificial neural network, Appl. Math. Model., 39 (2015), 6438-6450.  doi: 10.1016/j.apm.2015.01.059.  Google Scholar

[49]

Y. Yang and Y. Shang, A new filled function method for unconstrained global optimization, Appl. Math. Comput., 173 (2006), 501-512.  doi: 10.1016/j.amc.2005.04.046.  Google Scholar

[50]

H. Yin, An adaptive smoothing method for continuous minimax problems, Asia-Pac. J. Oper. Res., 32 (2015), 1-19.  doi: 10.1142/S0217595915400011.  Google Scholar

[51]

I. Zang, A smoothing out technique for min-max optimization, Math. Program., 19 (1980), 61-77.  doi: 10.1007/BF01581628.  Google Scholar

[52]

L. S. ZhangC. K. NgD. Li and W. W. Tian, A new filled function method for global optimization, J. Glob. Optim., 28 (2004), 17-43.  doi: 10.1023/B:JOGO.0000006653.60256.f6.  Google Scholar

[53]

A. Zhigljavsky and A. Zilinskas, Stochastic Global Optimization, Springer, New York, 2008.  Google Scholar

[54]

J. Zilinskas, Branch and bound with simplicial partitions for global optimization, Math. Model. Anal., 13 (2008), 145-159.  doi: 10.3846/1392-6292.2008.13.145-159.  Google Scholar

Figure 1.  (a) The green and solid graph is the graph of $ q_1(t) $, the blue and dashed one is the graph of $ \tilde{q}_{1}(t,0.5) $, the red and dotted one is the graph of $ \tilde{q}_{1}(t,1) $. (b) The green and solid graph is the graph of $ q_{1/2}(t) $, the blue and dashed one is the graph of $ \tilde{q}_{1/2}(t,0.5) $, the red and dotted one is the graph of $ \tilde{q}_{1/2}(t,1) $
Figure 2.  (a) The graph of the function $ f(x,y) $, (b) The graph of $ \tilde{f}(x,\beta) $
Figure 3.  (a) The graph of the function $ F_{\lambda}(x) $, (b) the graph of the function $ \tilde{F}_{\lambda}(x,\beta) $
Table 1.  The values of $ a_j $, $ b_j $ and $ p_j $ in Example 1
$ j $ $ 1 $ $ 2 $ $ 3 $ $ 4 $ $ 5 $
$ a_j $ $ 3 $ $ 2 $ $ 3 $ $ 3.5 $ $ 5 $
$ b^j $ $ (4,3) $ $ (-2,-3) $ $ (-3,3) $ $ (4,-4) $ $ (0,0) $
$ p_j $ $ 2 $ $ 2 $ $ 2 $ $ 1 $ $ \frac{2}{3} $
$ j $ $ 1 $ $ 2 $ $ 3 $ $ 4 $ $ 5 $
$ a_j $ $ 3 $ $ 2 $ $ 3 $ $ 3.5 $ $ 5 $
$ b^j $ $ (4,3) $ $ (-2,-3) $ $ (-3,3) $ $ (4,-4) $ $ (0,0) $
$ p_j $ $ 2 $ $ 2 $ $ 2 $ $ 1 $ $ \frac{2}{3} $
Table 2.  Table of minimization process of the Example 1
$ \beta $ $ k $ $ \bar{x}_k $ $ \tilde{f}(\bar{x}_k,\beta) $ $ f(\bar{x}_k) $ $ x_k^* $ $ f(x_k^*) $
$ 10^{-4} $ $ 1 $ $ (3.9999, 2.9998) $ $ -4.4178 $ $ -4.4178 $ $ (3.9944,2.9777) $ $ -4.4193 $
$ 2 $ $ (4.0000, -4.0000) $ $ -4.4704 $ $ -4.4708 $ $ (4.0000,-3.9808) $ $ -4.4064 $
$ 3 $ $ (-0.0000, -0.0000) $ $ -5.7907 $ $ -5.8048 $ $ (-0.0000, 0.0000) $ $ -5.8048 $
$ 10^{-6} $ $ 1 $ $ (3.9829,2.9846) $ $ -4.2006 $ $ -4.4188 $ $ (3.9944, 2.9777) $ $ -4.4193 $
$ 2 $ $ (4.0000,-4.0000) $ $ -4.4673 $ $ -4.4706 $ $ (4.0000,-4.0000) $ $ -4.4706 $
$ 3 $ $ (-0.0000, 0.0000) $ $ -5.8040 $ $ -5.8042 $ $ (0.0000,-0.0000) $ $ -5.8047 $
$ \beta $ $ k $ $ \bar{x}_k $ $ \tilde{f}(\bar{x}_k,\beta) $ $ f(\bar{x}_k) $ $ x_k^* $ $ f(x_k^*) $
$ 10^{-4} $ $ 1 $ $ (3.9999, 2.9998) $ $ -4.4178 $ $ -4.4178 $ $ (3.9944,2.9777) $ $ -4.4193 $
$ 2 $ $ (4.0000, -4.0000) $ $ -4.4704 $ $ -4.4708 $ $ (4.0000,-3.9808) $ $ -4.4064 $
$ 3 $ $ (-0.0000, -0.0000) $ $ -5.7907 $ $ -5.8048 $ $ (-0.0000, 0.0000) $ $ -5.8048 $
$ 10^{-6} $ $ 1 $ $ (3.9829,2.9846) $ $ -4.2006 $ $ -4.4188 $ $ (3.9944, 2.9777) $ $ -4.4193 $
$ 2 $ $ (4.0000,-4.0000) $ $ -4.4673 $ $ -4.4706 $ $ (4.0000,-4.0000) $ $ -4.4706 $
$ 3 $ $ (-0.0000, 0.0000) $ $ -5.8040 $ $ -5.8042 $ $ (0.0000,-0.0000) $ $ -5.8047 $
Table 3.  The values of $ a_j $, $ b_j $ and $ p_j $ in Example 2
$ j $ $ 1 $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $
$ a_j $ $ 1.8 $ $ 1.2 $ $ 4.8 $ $ 3.5 $ $ 2.8 $ $ 2.1 $
$ b^j $ $ -7 $ $ 2 $ $ -3.5 $ $ 6 $ $ -1 $ $ 4 $
$ p_j $ $ 2 $ $ 2 $ $ 1 $ $ 1 $ $ \frac{2}{5} $ $ \frac{1}{2} $
$ j $ $ 1 $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $
$ a_j $ $ 1.8 $ $ 1.2 $ $ 4.8 $ $ 3.5 $ $ 2.8 $ $ 2.1 $
$ b^j $ $ -7 $ $ 2 $ $ -3.5 $ $ 6 $ $ -1 $ $ 4 $
$ p_j $ $ 2 $ $ 2 $ $ 1 $ $ 1 $ $ \frac{2}{5} $ $ \frac{1}{2} $
Table 4.  Table of minimization process of the Example 2
$ \beta $ $ k $ $ \bar{x}_k $ $ \tilde{F}_{\lambda}(\bar{x}_k,\beta) $ $ F_{\lambda}(\bar{x}_k) $ $ f(\bar{x}_k) $ $ x_k^* $ $ f(x_k^*) $
$ 10^{-4} $ $ 1 $ $ 4.0001 $ $ -5.0017 $ $ -5.0320 $ $ -5.0320 $ $ 4.0000 $ $ -5.0504 $
$ 2 $ $ 6.0000 $ $ -5.7891 $ $ -5.7893 $ $ -5.7893 $ $ 6.0000 $ $ -5.7893 $
$ 3 $ $ -3.5000 $ $ -7.0153 $ $ -7.0156 $ $ -7.0156 $ $ -3.5000 $ $ -7.0156 $
$ 10^{-6} $ $ 1 $ $ -0.9998 $ $ -5.2906 $ $ -5.3361 $ $ -5.3361 $ $ -1.0000 $ $ -5.4260 $
$ 2 $ $ 3.9999 $ $ -2.5950 $ $ -2.5966 $ $ -2.5966 $ $ 4.0000 $ $ -2.5967 $
$ 3 $ $ -3.5000 $ $ -7.0132 $ $ -7.0156 $ $ -7.0156 $ $ -3.5000 $ $ -7.0156 $
$ \beta $ $ k $ $ \bar{x}_k $ $ \tilde{F}_{\lambda}(\bar{x}_k,\beta) $ $ F_{\lambda}(\bar{x}_k) $ $ f(\bar{x}_k) $ $ x_k^* $ $ f(x_k^*) $
$ 10^{-4} $ $ 1 $ $ 4.0001 $ $ -5.0017 $ $ -5.0320 $ $ -5.0320 $ $ 4.0000 $ $ -5.0504 $
$ 2 $ $ 6.0000 $ $ -5.7891 $ $ -5.7893 $ $ -5.7893 $ $ 6.0000 $ $ -5.7893 $
$ 3 $ $ -3.5000 $ $ -7.0153 $ $ -7.0156 $ $ -7.0156 $ $ -3.5000 $ $ -7.0156 $
$ 10^{-6} $ $ 1 $ $ -0.9998 $ $ -5.2906 $ $ -5.3361 $ $ -5.3361 $ $ -1.0000 $ $ -5.4260 $
$ 2 $ $ 3.9999 $ $ -2.5950 $ $ -2.5966 $ $ -2.5966 $ $ 4.0000 $ $ -2.5967 $
$ 3 $ $ -3.5000 $ $ -7.0132 $ $ -7.0156 $ $ -7.0156 $ $ -3.5000 $ $ -7.0156 $
Table 5.  The list of non-smooth test problems
Problem No. Function Name Dimension $ n $ Region Global minimum
2 Problem 1 in [14] $ 1 $ $ [-10,10] $ $ 1 $
3 Problem 2 in [14] $ 1 $ $ [-10,10] $ $ 1 $
4 Problem 3 in [14] $ 2 $ $ [-10,10] $ $ 2 $
5, 6, 7, 8, 9, 10, 11 Problem 7 in [14] $ 2,3,4,5,10,15,20 $ $ [-10,10] $ $ 0 $
Problem No. Function Name Dimension $ n $ Region Global minimum
2 Problem 1 in [14] $ 1 $ $ [-10,10] $ $ 1 $
3 Problem 2 in [14] $ 1 $ $ [-10,10] $ $ 1 $
4 Problem 3 in [14] $ 2 $ $ [-10,10] $ $ 2 $
5, 6, 7, 8, 9, 10, 11 Problem 7 in [14] $ 2,3,4,5,10,15,20 $ $ [-10,10] $ $ 0 $
Table 6.  The numerical results for the non-smooth test problems
Problem AFA GDA
No. $ x_0 $ $ f^* $ f.eval Time(sec) $ f^* $ f.eval Time(sec)
$ 2 $ $ 10 $ $ 1.0000 $ $ 106 $ $ 0.0155 $ $ 1.0000 $ $ 362 $ $ 0.1552 $
$ 3 $ $ 10 $ $ 1.0000 $ $ 219 $ $ 0.0178 $ $ 1.0000 $ $ 524 $ $ 0.3779 $
$ 4 $ $ (10,10) $ $ 2.0000 $ $ 539 $ $ 0.2676 $ $ 2.0000 $ $ 1136 $ $ 2.4738 $
$ 5 $ $ (5,5) $ $ 1.0000 $ $ 367 $ $ 0.2432 $ $ 0.0000 $ $ 1212 $ $ 2.9559 $
$ 6 $ $ (5,5,5) $ $ 2.9806e-005 $ $ 471 $ $ 0.2458 $ $ 0.0000 $ $ 1678 $ $ 5.1616 $
$ 7 $ $ (5,\dots,5) $ $ 2.9986e-005 $ $ 614 $ $ 0.2771 $ $ 0.0000 $ $ 2614 $ $ 7.5685 $
$ 8 $ $ (5,\dots,5) $ $ 4.9992e-005 $ $ 733 $ $ 0.3065 $ $ 2e-004 $ $ 3484 $ $ 10.0016 $
$ 9 $ $ (5,\dots,5) $ $ 9.9991e-005 $ $ 2270 $ $ 0.4037 $ $ 6e-004 $ $ 10344 $ $ 37.5289 $
$ 10 $ $ (5,\dots,5) $ $ 1.4982e-006 $ $ 3278 $ $ 0.4986 $ $ 8e-004 $ $ 14390 $ $ 55.3796 $
$ 11 $ $ (5,\dots,5) $ $ 1.6520e-013 $ $ 8262 $ $ 0.7262 $ $ 5e-004 $ $ 42446 $ $ 118.4403 $
Problem AFA GDA
No. $ x_0 $ $ f^* $ f.eval Time(sec) $ f^* $ f.eval Time(sec)
$ 2 $ $ 10 $ $ 1.0000 $ $ 106 $ $ 0.0155 $ $ 1.0000 $ $ 362 $ $ 0.1552 $
$ 3 $ $ 10 $ $ 1.0000 $ $ 219 $ $ 0.0178 $ $ 1.0000 $ $ 524 $ $ 0.3779 $
$ 4 $ $ (10,10) $ $ 2.0000 $ $ 539 $ $ 0.2676 $ $ 2.0000 $ $ 1136 $ $ 2.4738 $
$ 5 $ $ (5,5) $ $ 1.0000 $ $ 367 $ $ 0.2432 $ $ 0.0000 $ $ 1212 $ $ 2.9559 $
$ 6 $ $ (5,5,5) $ $ 2.9806e-005 $ $ 471 $ $ 0.2458 $ $ 0.0000 $ $ 1678 $ $ 5.1616 $
$ 7 $ $ (5,\dots,5) $ $ 2.9986e-005 $ $ 614 $ $ 0.2771 $ $ 0.0000 $ $ 2614 $ $ 7.5685 $
$ 8 $ $ (5,\dots,5) $ $ 4.9992e-005 $ $ 733 $ $ 0.3065 $ $ 2e-004 $ $ 3484 $ $ 10.0016 $
$ 9 $ $ (5,\dots,5) $ $ 9.9991e-005 $ $ 2270 $ $ 0.4037 $ $ 6e-004 $ $ 10344 $ $ 37.5289 $
$ 10 $ $ (5,\dots,5) $ $ 1.4982e-006 $ $ 3278 $ $ 0.4986 $ $ 8e-004 $ $ 14390 $ $ 55.3796 $
$ 11 $ $ (5,\dots,5) $ $ 1.6520e-013 $ $ 8262 $ $ 0.7262 $ $ 5e-004 $ $ 42446 $ $ 118.4403 $
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