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A novel two-stage constraints handling framework for real-world multi-constrained multi-objective optimization problem based on evolutionary algorithm

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Abstract

Multi-constrained multi-objective optimization is a challenging topic, which is very common in dealing with real-world problems. This paper proposes a novel two-stage ρg / μg framework based on multi-objective evolutionary algorithm (MOEA) to solve the multi-constrained multi-objective optimization problems (MCMOPs), which dynamically balances the diversity and convergence of solutions. During the multi-constraints handling process, ρg / μg -MOEA makes the reduction of violated constraints as its primary goal, and converges to feasible regions by a proposed ρg -criterion based constraints relaxation method. Moreover, in the late stage of evolution, by introducing the improved dynamic stochastic ranking (DSR) strategy, the “potential” infeasible individuals are utilized to find more feasible regions, which would guarantee a good distribution of the obtained Pareto frontiers. Thereafter, the proposed framework combined with non-dominated sorting genetic algorithm II (NSGAII) is applied to ten benchmark functions and a series of real-world MCMOPs, and the performances are compared with those obtained by some state-of-the-art constraints handling methods. Experimental results indicate that the proposed ρg / μg framework outperforms the current efficient methods in dealing with test CMOPs, and can achieve satisfactory results when solving real-world MCMOPs.

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Abbreviations

x :

decision vector/ solution.

f m(x):

m-th objective function.

g i(x):

i-th inequality constraint.

h j(x):

j-th equality constraint.

qp :

total amount of inequality and equality constraints, respectively.

l i u i :

lower and upper bound of i-th variable, respectively.

X :

feasible region.

G(x):

overall constraints violation.

G i(x),w i :

violation and weight values of i-th type of constraints, respectively.

K :

amount of constraints types.

N F :

amount of violated constraints.

\( {\overline{N}}_F \) :

normalization value ofNF.

n F, k :

amount of violated k-th type of constraints.

N F, curmin :

amount of minimum violated constraints of the individuals in the current population.

N F, curmax :

amount of maximum violated constraints of the individuals in the current population.

w F, j :

weight values of j-th type of constraint in MOP F.

g F, j,\( {\overline{g}}_{F,j,i} \) :

violation value of j-th type of constraint in MOP F and its normalized form, respectively.

g F, j, curmax :

maximum violation value of j-th type of constraint in current population.

g F, j, wholemin :

record minimum violation value of j-th type of constraint population.

L F,\( {\overline{L}}_F \) :

comprehensive evaluation index of infeasibility and its normalized form, respectively.

L F, curmin :

minimum value of LF in the current population.

L F, curmax :

maximum value of LF in the current population.

M F, total :

total amount of constraints in MOP F.

ρ :

proportion of violated constraints in all the constraints.

ρ g :

threshold of acceptable solutions.

gen cur :

number of current generation.

gen set :

number of preset generation.

s :

evaluation parameter.

μg :

probability that s equals to 1.

P :

population size.

MOP:

multi-objective optimization problem.

CMOP:

constrained multi-objective optimization problem.

MCMOP:

multi-constrained multi-objective optimization problem.

MOEA :

multi-objective evolutionary algorithm.

NSGAII:

non-dominated sorting genetic algorithm II.

PFM :

penalty function method.

CDP:

constraint domination principle.

ACDP:

angle-based ACDP

IEpsilon:

improved epsilon

SR:

stochastic ranking.

DSR:

dynamic stochastic ranking.

GD:

generational distance.

MS :

maximum spread.

TSC:

two set coverage.

CNSGAII:

constrained NSGAII.

DMCS :

dynamic multi-constraints handling strategy.

MEM-MG:

multi-objective energy management problem of microgrid.

References

  1. Tanabe R, Ishibuchi H (2019) A review of evolutionary multi-modal multi-objective optimization. IEEE Trans Evol Comput 24(1):193–200

    Article  Google Scholar 

  2. Arjmandzadeh Z, Nazemi A, Safi M (2019) Solving multiobjective random interval programming problems by a capable neural network framework. Appl Intell 49(3):1566–1579

    Article  Google Scholar 

  3. Moradi H, Ebrahimpourkomleh H (2018) Development of a multi-objective optimization evolutionary algorithm based on educational systems. Appl Intell 48(9):2954–2966

    Article  Google Scholar 

  4. Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip O 26(6):369–395

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhang Y, Gong DW, Cheng J (2017) Multi-objective particle swarm optimization approach for cost-based feature selection in classification. IEEE Acm T Comput Bi 14(1):64–75

    Google Scholar 

  6. Zhang Y, Gong DW, Gao XZ, Tian T, Sun XY (2020) Binary differential evolution with self-learning for multi-objective feature selection. Inform Sciences 507:67–85

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang Y, Cheng S, Shi YH, Gong DW, Zhao XC (2019) Cost-sensitive feature selection using two-archive multi-objective artificial bee colony algorithm. Expert Syst Appl 137:46–58

    Article  Google Scholar 

  8. Zhang Y, Gong DW, Sun JY, Qu BY (2018) A decomposition-based archiving approach for multi-objective evolutionary optimization. Inform Sciences 430-431:397–413

    Article  Google Scholar 

  9. Mac TT, Copot C, Tran DT, De Keyser R (2017) A hierarchical global path planning approach for mobile robots based on multi-objective particle swarm optimization. Appl Soft Comput 59:68–76

    Article  Google Scholar 

  10. Li K, Chen R, Fu G, Yao X (2019) Two-archive evolutionary algorithm for constrained multiobjective optimization. IEEE Trans Evol Comput 23(2):303–315

    Article  Google Scholar 

  11. Bastani M, Damgacioglu H, Celik N (2018) A δ-constraint multi-objective optimization framework for operation planning of smart grids. Sustain Cities Soc 38:21–30

    Article  Google Scholar 

  12. Fonseca CM, Fleming PJ (1998) Multiobjective optimization and multiple constraint handling with evolutionary algorithms. I A unified formulation IEEE Trans Syst Man Cybern Syst 28(1):26–37

    Article  Google Scholar 

  13. Cai X, Mei Z, Fan Z, Zhang Q (2018) A constrained decomposition approach with grids for evolutionary multiobjective optimization. IEEE Trans Evol Comput 22(4):564–577

    Article  Google Scholar 

  14. Ma X, Zhang Q, Tian G, Yang J, Zhu Z (2018) On Tchebycheff decomposition approaches for multiobjective evolutionary optimization. IEEE Trans Evol Comput 22(2):226–244

    Article  Google Scholar 

  15. Jiang X, Yu Y, Zhao L, Liu H (2017) Constrained nondominated neighbor immune multiobjective optimization algorithm for multimedia delivery. Multimed Tools Appl 76(16):17297–17317

    Article  Google Scholar 

  16. Trivedi A, Srinivasan D, Sanyal K, Ghosh A (2017) A survey of multiobjective evolutionary algorithms based on decomposition. IEEE Trans Evol Comput 21(3):440–462

    Google Scholar 

  17. Woldesenbet YG, Yen GG, Tessema B (2009) Constraint handling in multiobjective evolutionary optimization. IEEE Trans Evol Comput 13(3):514–525

    Article  Google Scholar 

  18. Li B, Li J, Tang K, Yao X (2015) Many-objective evolutionary algorithms: a survey. ACM Comput Surv 48(1):1–13

    Article  Google Scholar 

  19. Li X, Lai J, Tang R (2017) A hybrid constraints handling strategy for multiconstrained multiobjective optimization problem of microgrid economical/environmental dispatch. Complexity 2017:1–12

    Article  MATH  Google Scholar 

  20. Sanseverino ER, Silvestre ML, Ippolito MG, De Paola A, Re GL (2011) An execution, monitoring and replanning approach for optimal energy management in microgrids. Energy 36(5):3429–3436

    Article  Google Scholar 

  21. Zhou H, Qiao J (2019) Multiobjective optimal control for wastewater treatment process using adaptive MOEA/D. Appl Intell 49(3):1098–1126

    Article  Google Scholar 

  22. Li X, Xia R (2019) A dynamic multi-constraints handling strategy for multi-objective energy management of microgrid based on MOEA. IEEE Access 7:138732–138744

    Article  Google Scholar 

  23. Deb K (2001) Multi-objective optimization using evolutionary algorithms. John Wiley and Sons, Chichester

    MATH  Google Scholar 

  24. 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

    Article  Google Scholar 

  25. Babu BV, Jehan MML (2004) Differential evolution for multi-objective optimization. In: Proceedings of the 2004 Congress on Evolutionary Computation, 2004. CEC’04, pp 2696–2703

  26. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

    Article  Google Scholar 

  27. Wang Y, Cai Z, Guo G, Zhou Y (2007) Multiobjective optimization and hybrid evolutionary algorithm to solve constrained optimization problems. IEEE Trans Syst Man Cybern Syst 37(3):560–575

    Article  Google Scholar 

  28. Mezuramontes E, Coello CA (2011) Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm Evol Comput 1(4):173–194

    Article  Google Scholar 

  29. Coello CA (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Method Appl M 191(1112):1245–1287

    Article  MathSciNet  MATH  Google Scholar 

  30. Vargas DEC, Lemonge ACC, Barbosa HJC, Bernardino HS (2013) Differential evolution with the adaptive penalty method for constrained multiobjective optimization. In: Proceedings of the 2013 Congress on Evolutionary Computation, 2013. CEC’13, pp 1342–1349

  31. Runarsson TP, Yao X (2000) Stochastic ranking for constrained evolutionary optimization. IEEE Trans Evol Comput 4(3):284–294

    Article  Google Scholar 

  32. Runarsson TP, Yao X (2005) Search biases in constrained evolutionary optimization. IEEE T Syst Man Cy C 35(2):233–243

    Article  Google Scholar 

  33. Zhang M, Luo W, Wang X (2008) Differential evolution with dynamic stochastic selection for constrained optimization. Inform Sciences 178(15):3043–3074

    Article  Google Scholar 

  34. Takahama T. Sakai S, Iwane N (2005) Constrained optimization by the ε constrained hybrid algorithm of particle swarm optimization and genetic algorithm. In: proceedings of AI 2005: advances in artificial intelligence, 2005, pp 389-400

  35. Lin H, Fan Z, Cai X, Li W, Wang S, Li J, Zhang C (2014) Hybridizing infeasibility driven and constrained-domination principle with MOEA/D for constrained multiobjective evolutionary optimization. In: In: proceedings of international conference on technologies and applications of artificial intelligence, 2014, pp 249–261

    Google Scholar 

  36. Ray T, Kang T, Chye SK (2000) An evolutionary algorithm for constrained optimization. Proceedings of Genetic and Evolutionary Computation Conference 2000:771–777

    Google Scholar 

  37. Fan Z, Li W, Cai XY, Huang H, Fang Y, You YG, Mo JJ (2019) An improved epsilon constraint-handling method in moea/d for cmops with large infeasible regions. Soft Comput 23:12491–12510

    Article  Google Scholar 

  38. Fan Z, Fang Y, Li WJ, Cai XY, Goodman E (2018) MOEA/D with angle-based constrained dominance principle for constrained multi-objective optimization problems. Appl Soft Comput 74:621–633

    Article  Google Scholar 

  39. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82

    Article  Google Scholar 

  40. Niknam T, Azizipanahabarghooee R, Narimani MR (2012) An efficient scenario-based stochastic programming framework for multi-objective optimal micro-grid operation. Appl Energ 99:455–470

    Article  Google Scholar 

  41. Colson CM, Nehrir MH, Pourmousavi SA (2010) Towards real-time microgrid power management using computational intelligence methods. Proceedings of Power and Energy Society General Meeting 2010:1–8

    Google Scholar 

  42. Ho PY, Shimizu K (2007) Evolutionary constrained optimization using an addition of ranking method and a percentage-based tolerance value adjustment scheme. Inform Sciences 177(14):2985–3004

    Article  Google Scholar 

  43. Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506

    Article  MATH  Google Scholar 

  44. Lucken C, Baran B, Brizuela C (2014) A survey on multi-objective evolutionary algorithms for many-objective problems. Comput Optim Appl 58(3):707–756

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 51909199 and 51709215), by projects from Key Lab. of Marine Power Engineering and Tech. authorized by MOT (KLMPET2019-02 and KLMPET2019-03), the Green Intelligent Inland Ship Innovation Programme, the Fundamental Research Funds for the General Universities (WUT: 2020IVB012), and the Opening Foundation of Key Laboratory of Information Security of Zhejiang Province (Grant No. KF201912).

Author information

Authors and Affiliations

  1. School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, 430063, China

    Xin Li, Ruoli Tang & Xiaobing Mao

  2. Key Lab. of Marine Power Engineering and Tech. authorized by MOT, Wuhan, 430063, China

    Xin Li

  3. Artificial Intelligence School, Wuchang University of Technology, Wuhan, 430223, China

    Qing An

  4. Zhejiang Electronic Information Products Inspection and Research Institute (Key Laboratory of Information Security of Zhejiang Province), No.50 Tian Mu Shan Road, Hangzhou, China

    Jun Zhang

  5. School of Data Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, 990777, China

    Fan Xu

  6. School of Electrical Engineering and Automation, Wuhan University, Wuhan, 430072, China

    Zhengcheng Dong

  7. State Grid Beijing Electric Maintenance Company, Beijing, 100080, China

    Xiaodi Zhang

  8. E.ON Energy Research Center, RWTH Aachen University, 52074, Aachen, Germany

    Jingang Lai

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  1. Xin Li
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Correspondence to Jun Zhang or Xiaobing Mao.

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Appendices

Appendix 1

1.1 Test objective functions, constraints and parameter settings in this work

Table 11 Benchmark functions and corresponding constraints
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Table 12 Parameter settings of benchmark functions
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Appendix 2

1.1 True Pareto frontier of test function OSY in this work

Fig. 6
figure 6

True Pareto frontier of OSY

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Li, X., An, Q., Zhang, J. et al. A novel two-stage constraints handling framework for real-world multi-constrained multi-objective optimization problem based on evolutionary algorithm. Appl Intell 51, 8212–8229 (2021). https://doi.org/10.1007/s10489-020-02174-5

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