Abstract
Many practical engineering design problems need constrained optimization. The literature reports several meta-heuristic algorithms have been applied to solve constrained optimization problems. In many cases, the algorithms fail due to violation of constraints. Recently in 2014, a new meta-heuristic algorithm known as symbiotic organism search (SOS) is reported by Cheng and Prayogo. It is inspired by the natural phenomenon of interaction between organisms in an ecosystem which help them to survive and grow. In this paper, the SOS algorithm is combined with augmented Lagrange multiplier (ALM) method to solve the constrained optimization problems. The ALM is accurate and effective as the constraints in this case do not have the power to restrict the search space or search direction. The orthogonal array strategies have gained popularity among the meta-heuristic researchers due to its potentiality to enhance the exploitation process of the algorithms. Simultaneously, researchers are also looking at designing parallel version of the meta-heuristics to reduce the computational burden. In order to enhance the performance, an Orthogonal Parallel SOS (OPSOS) is developed. The OPSOS along with ALM method is a suitable combination which is used here to solve twelve benchmark nonlinear constrained problems and four engineering design problems. Simulation study reveals that the proposed approach has almost similar accuracy with lower run time than ALM with Orthogonal SOS. Comparative analysis also establish superior performance over ALM with orthogonal colliding bodies optimization, modified artificial bee colony, augmented Lagrangian-based particle swarm optimization and Penalty function-based genetic algorithm.
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Abdullahi M, Ngadi MA (2015) Symbiotic organism search optimization based task scheduling in cloud computing environment. Future Gener Comput Syst 56:640–650
Afonso MV, Bioucas-Dias JM, Figueiredo MA (2011) An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans Image Process 20(3):681–695
Baghel V, Nanda SJ, Panda G (2011) New GOPSO and its application to robust identification. In: Proceedings of IEEE international conference on energy, automation, and signal, pp 1–6
Cagnina LC, Esquivel SC, Coello CAC (2011) Solving constrained optimization problems with a hybrid particle swarm optimization algorithm. Eng Optim 43(8):843–866
Cheng MY, Prayogo D (2014) Symbiotic organisms search: a new metaheuristic optimization algorithm. Comput Struct 139:98–112
Cheng MY, Prayogo D, Tran DH (2015) Optimizing multiple-resources leveling in multiple projects using discrete symbiotic organisms search. J Comput Civil Eng 30(3):04015036
Coello CAC (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Methods Appl Mech Eng 191(11):1245–1287
Cung VD, Martins SL, Ribeiro CC, Roucairol C (2002) Strategies for the parallel implementation of metaheuristics. Essays and surveys in metaheuristics. Springer, US
Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186(2–4):311–338
Fogarty TC, Huang R (1990) Implementing the genetic algorithm on transputer based parallel processing systems. In: International conference on parallel problem solving from nature, Springer, Berlin, pp 145–149
Ghasemishabankareh B, Li X, Ozlen M (2016) Cooperative coevolutionary differential evolution with improved augmented Lagrangian to solve constrained optimisation problems. Inf Sci 369:441–456
Gong W, Cai Z, Ling CX (2006) ODE: a fast and robust differential evolution based on orthogonal design. In: Proceedings of 19th Australian joint conference on artificial intelligence. Advances in artificial intelligence, Springer, Berlin, pp 709–718
Gong W, Cai Z, Jiang L (2008) Enhancing the performance of differential evolution using orthogonal design method. Appl Math Comput 206(1):56–69
He Q, Wang L (2007) An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng Appl Artif Intell 20(1):89–99
He Q, Wang L (2007) A hybrid particle swarm optimization with a feasibility-based rule for constrained optimization. Appl Math Comput 186(2):1407–1422
Ho SY, Lin HS, Liauh WH, Ho SJ (2008) OPSO: orthogonal particle swarm optimization and its application to task assignment problems. IEEE Trans Syst Man Cybern A Syst Hum 38(2):288–298
Huang F, Wang L, He Q (2007) An effective co-evolutionary differential evolution for constrained optimization. Appl Math Comput 186(1):340–356
Jansen PW, Perez RE (2011) Constrained structural design optimization via a parallel augmented Lagrangian particle swarm optimization approach. Comput Struct 89(13):1352–1366
Joines JA, Houck CR (1994) On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GA’s. In: IEEE congress on evolutionary computation, pp 579–584
Kamankesh H, Agelidis VG, Kavousi-Fard A (2016) Optimal scheduling of renewable micro-grids considering plug-in hybrid electric vehicle charging demand. Energy 100:285–297
Karaboga D, Akay B (2011) A modified artificial bee colony (ABC) algorithm for constrained optimization problems. Appl Soft Comput 11(3):3021–3031
Kong H, Li N, Shen Y (2015) Adaptive double chain quantum genetic algorithm for constrained optimization problems. Chin J Aeronaut 28(1):214–228
Lemonge ACC, Barbosa HJC (2004) An adaptive penalty scheme for genetic algorithms in structural optimization. Int J Numer Methods Eng 59:703–736
Leung YW, Wang Y (2001) An orthogonal genetic algorithm with quantization for global numerical optimization. IEEE Trans Evol Comput 5(1):41–53
Lin CH (2013) A rough penalty genetic algorithm for constrained optimization. Inf Sci 241:119–137
Liu H, Li P, Wen Y (2006) Parallel ant colony optimization algorithm. In: 6th IEEE world congress on intelligent control and automation, vol 1, pp 3222–3226
Liu H, Cai Z, Wang Y (2010) Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization. Appl Soft Comput 10(2):629–640
Long W, Liang X, Huang Y, Chen Y (2014) An effective hybrid cuckoo search algorithm for constrained global optimization. Neural Comput Appl 25(3–4):911–926
MATLAB Box plot, available at Mathworks online: http://in.mathworks.com/help/stats/boxplot.html?refresh=true (2016)
Nanda SJ, Panda G (2013) Automatic clustering algorithm based on multi-objective immunized PSO to classify actions of 3D human models. Eng Appl Artif Intell 26(5):1429–1441
Panda A, Pani S (2016) A symbiotic organisms search algorithm with adaptive penalty function to solve multi-objective constrained optimization problems. Appl Soft Comput 46:344–360
Panda A, Pani S (2016) A WNN model trained with orthogonal colliding bodies optimization for accurate identification of hammerstein plant. In: Proceedings of IEEE congress on evolutionary computation (CEC-2016), Vancouver
Panda A, Pani S (2016) Improved Identification of hammerstein plant using a non-linear model trained with symbiotic organisms search. In: Proceedings of IEEE region 10 conference (TENCON 2016), pp 247–250
Pattnaik SS, Bakwad KM, Devi S, Panigrahi BK, Das S (2011) Parallel bacterial foraging optimization. In: Handbook of swarm intelligence. Springer, Berlin, pp 487–502
Prayogo D (2015) An innovative parameter-free symbiotic organisms search (SOS) for solving construction-engineering problems. Ph.D. Thesis, Department of Construction Engineering, National Taiwan University of Science and Technology
Ray T, Liew KM (2003) Society and civilization: an optimization algorithm based on the simulation of social behavior. IEEE Trans Evol Comput 7(4):386–396
Rocha AMA, Martins TF, Fernandes EM (2011) An augmented Lagrangian fish swarm based method for global optimization. J Comput Appl Math 235(16):4611–4620
Runarsson TP, Yao X (2000) Stochastic ranking for constrained evolutionary optimization. IEEE Trans Evol Comput 4:284–294
Runarsson TP, Yao X (2005) Search biases in constrained evolutionary optimization. IEEE Trans Syst Man Cybern C Appl Rev 35:233–243
Schutte JF, Reinbolt JA, Fregly BJ, Haftka RT, George AD (2004) Parallel global optimization with the particle swarm algorithm. Int J Numer Methods Eng 61(13):2296–2315
Sedlaczek K, Eberhard P (2006) Using augmented Lagrangian particle swarm optimization for constrained problems in engineering. Struct Multidiscip Optim 32(4):277–286
Shukla UP, Nanda SJ (2016) Parallel social spider clustering algorithm for high dimensional datasets. Eng Appl Artif Intell 56:75–90
Tahk MJ, Sun BC (2000) Coevolutionary augmented Lagrangian methods for constrained optimization. IEEE Trans Evol Comput 4(2):114–124
Tejani GG, Savsani VJ, Patel VK (2016) Adaptive symbiotic organisms search (SOS) algorithm for structural design optimization. J Comput Des Eng 3(3):226–249
Tran DH, Cheng MY, Prayogo D (2016) A novel multiple objective symbiotic organisms search (MOSOS) for timecostlabor utilization tradeoff problem. Knowl Based Syst 94:132–145
Vincent FY, Redi AP, Yang CL, Ruskartina E, Santosa B (2017) Symbiotic organisms search and two solution representations for solving the capacitated vehicle routing problem. Appl Soft Comput 52:657–672
Yang J, Bouzerdoum A, Phung SL (2010) A particle swarm optimization algorithm based on orthogonal design. In: Proceedings of IEEE evolutionary computation (CEC 10), pp 1–7
Yeniay O (2005) Penalty function methods for constrained optimization with genetic algorithms. Math Comput Appl 10(1):45–56
Zahara E, Kao Y (2009) Hybrid NelderMead simplex search and particle swarm optimization for constrained engineering design problems. Expert Syst Appl 36(2):3880–3886
Zhan ZH, Zhang J, Li Y, Shi YH (2011) Orthogonal learning particle swarm optimization. IEEE Trans Evol Comput 15(6):832–847
Zhang C, Lin Q, Gao L, Li X (2015) Backtracking search algorithm with three constraint handling methods for constrained optimization problems. Expert Syst Appl 42(21):7831–7845
Zhang Q, Leung YW (1999) An orthogonal genetic algorithm for multimedia multicast routing. IEEE Trans Evol Comput 3: 53–62
Acknowledgements
Ms. Arnapurna Panda received research grants in from of institute research scholar fellowship from Ministry of HRD, Govt. of India to to carry out her Ph.D work at Indian Institute of Technology Bhubaneswar.
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Appendix
Appendix
Test Function-1: Quadratic Problem
subject to
where the bounds are
The global optimum is at \(z^*=(1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1)\) and \(f(z^*)=-15\).
Constraints \(g_{1}-g_{6}\) are active.
Test Function-2: Nonlinear Problem
subject to
where \(n=20\) and \(0\le z_i \le 10\) for \(i=1, 2,\ldots ,n\).
The global maximum is at \(z_i^*= 1 \sqrt{n}\) for \(i=1,2 \ldots ,n\) and \(f(z_i^*)=0.803619.\) Constraint \(g_1\) is active.
Test Function-3: Nonlinear Problem
subject to
where \(n=10\) and \(0\le z_i\le 1\) for \(i=1, 2, \ldots ,n\).
The global maximum is at \(z_i^*=1 \sqrt{n}\) for \(i=1,2 \ldots ,n\) and \(f(z^*)=1\).
Test Function-4: Quadratic Problem
subject to
where \(78\le z_1 \le 102\), \(33 \le z_2 \le 45\), \(27 \le z_i \le 45\) for \(i=3,4,5\). The optimal solution is \(z^*=(78, 33, 29.995256025682, 45, 36.775812905788)\) and \(f(z^*) =-30{,}665.539.\) Constraints \( g_1\) and \(g_6\) are active.
Test Function-5: Nonlinear Problem
subject to
where \(0 \le z_1\le 1200, 0 \le z_2 \le 1200, -0.55 \le z_3 \le 0.55\) and \(-0.55 \le z_4 \le 0.55. \) The best known solution is \(z^*=(679.9453,1026.067,0.1188764,-0.3962336)\) and \(f(z^*)=5126.4981\).
Test Function-6: Nonlinear Problem
subject to
where \(13\le z_1 \le 100\) and \(0\le z_2\le 100\). The optimum solution is \(z^*=(14.095,0.84296)\) and \(f(z_i^*)=-6961.81388\). Both constraints \(g_1\) and \(g_2\) are active.
Test Function-7: Quadratic Problem
subject to
where \(-10\le z_i \le 10\) for \(i=1,2,\ldots ,10\).The global optimum is \(z^*=(2.171996, 2.363683, 8.773926, 5.095984, 0.9906548, 1.430574, 1.321644, 9.828726, 8.280092, 8.375927)\), where \(f(z^*)=24.3062091\). Constraints \(g_1-g_6\) are active.
Test Function-8: Nonlinear Problem
subject to
where \(0\le z_i \le 10\) for \(i= 1, 2\). The optimal solution is located at \(z^*=(1.2279713, 4.2453733)\) and \(f(z^*)=0.0095825.\)
Test Function-9: Nonlinear Problem
subject to
where \(-10\le z_i \le 10\) for \(i=1,2,\ldots 7\). The global optimum is \(z^*=(2.330499, 1.951372, -0.4775414, 4.365726, -0.6244870, 1.038131, 1.594227)\) and \(f(z^*)=680. 6300573\). The constraints \(g_1-g_2\) are active.
Test Function-10: Linear Problem
f10: Minimize \( f(\mathop {z}\limits ^{\rightarrow })= z_1+z_2+z_3\) subject to
where \(100\le z_1 \le 10{,}000\), \(1000 \le z_i \le 10{,}000\) for \(i=2,3\); \(10 \le z_i \le 1000\) for \(i=4,\ldots 8\). The global optimum is \(z^*=(579.19, 1360.13, 5109.92, 182.0174, 295.5985, 217.9799, 286.40, 395.5979)\), where \(f(z^*)=7049.25\). The constraints \(g_1-g_3\) are active.
Test Function-11: Quadratic Problem
subject to
where \(-1\le z_1 \le 1;-1\le z_2\le 1\).
The optimum solution is \(z^*=(+-1 \sqrt{2},12)\), where \(f(z^*)=0.75\).
Test Function-12: Quadratic Problem
subject to
where \(0\le z_i \le 10\), \(i=1, 2, 3\) and \(p, q, r=1,2,\ldots ,9\). The global optimum solution is at \(z^*=(5, 5, 5)\), and \(f(z^*)=1.\)
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Panda, A., Pani, S. An orthogonal parallel symbiotic organism search algorithm embodied with augmented Lagrange multiplier for solving constrained optimization problems. Soft Comput 22, 2429–2447 (2018). https://doi.org/10.1007/s00500-017-2693-5
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DOI: https://doi.org/10.1007/s00500-017-2693-5