[1] Stephan R. Cardinality constrained combinatorial optimization: Complexity and polyhedra. Discrete Optimization, 2010, 7: 99-113 CrossRef Google Scholar
[2] Karp R M. Reducibility among combinatorial problems. In: Proceedings of Complexity of Computer Computations, 1972. 85--103. Google Scholar
[3] Banfield R E, Hall L O, Bowyer K W. Ensemble diversity measures and their application to thinning. Inf Fusion, 2005, 6: 49-62 CrossRef Google Scholar
[4] Moghaddam B, Weiss Y, Avidan S. Spectral bounds for sparse pca: exact and greedy algorithms. In: Proceedings of Advances in Neural Information Processing Systems, 2005. 915--922. Google Scholar
[5] Bruckstein A M, Donoho D L, Elad M. From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images. SIAM Rev, 2009, 51: 34-81 CrossRef Google Scholar
[6] Zhou X, Huaimin W, Bo D. How many robots are enough: a multi-objective genetic algorithm for the single-objective time-limited complete coverage problem. In: Proceedings of IEEE International Conference on Robotics and Automation, 2018. 2380--2387. Google Scholar
[7] Chai R, Li H, Meng F. Energy consumption optimization-based joint route selection and flow allocation algorithm for software-defined networking. Sci China Inf Sci, 2017, 60: 040306 CrossRef Google Scholar
[8] Deb K, Pratap A, Agarwal S. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Computat, 2002, 6: 182-197 CrossRef Google Scholar
[9] Wang R, Purshouse R C, Fleming P J. Preference-Inspired Coevolutionary Algorithms for Many-Objective Optimization. IEEE Trans Evol Computat, 2013, 17: 474-494 CrossRef Google Scholar
[10] Wang R, Zhou Z, Ishibuchi H. Localized Weighted Sum Method for Many-Objective Optimization. IEEE Trans Evol Computat, 2018, 22: 3-18 CrossRef Google Scholar
[11] Yong Wang , Han-Xiong Li , Yen G G. MOMMOP: multiobjective optimization for locating multiple optimal solutions of multimodal optimization problems.. IEEE Trans Cybern, 2015, 45: 830-843 CrossRef PubMed Google Scholar
[12] Gupta A, Ong Y S, Feng L. Multifactorial Evolution: Toward Evolutionary Multitasking. IEEE Trans Evol Computat, 2016, 20: 343-357 CrossRef Google Scholar
[13] Gupta A, Ong Y S, Feng L. Multiobjective Multifactorial Optimization in Evolutionary Multitasking.. IEEE Trans Cybern, 2017, 47: 1652-1665 CrossRef PubMed Google Scholar
[14] Knowles J D, Watson R A, Corne D W. Reducing local optima in single-objective problems by multi-objectivization. In: Proceedings of International Conference on Evolutionary Multi-Criterion Optimization, 2001. 269--283. Google Scholar
[15] Wu Song , Yong Wang , Han-Xiong Li . Locating Multiple Optimal Solutions of Nonlinear Equation Systems Based on Multiobjective Optimization. IEEE Trans Evol Computat, 2015, 19: 414-431 CrossRef Google Scholar
[16] Bienstock D. Computational study of a family of mixed-integer quadratic programming problems. Math Program, 1996, 74: 121--140. Google Scholar
[17] Burdakov O, Kanzow C, Schwartz A. On a reformulation of mathematical programs with cardinality constraints. In: Proceedings of Advances in Global Optimization, 2015. 3--14. Google Scholar
[18] Sun X, Zheng X, Li D. Recent Advances in Mathematical Programming with Semi-continuous Variables and Cardinality Constraint. J Oper Res Soc China, 2013, 1: 55-77 CrossRef Google Scholar
[19] Rifki O, Ono H. A survey of computational approaches to portfolio optimization by genetic algorithms. In: Proceedings of the 18th International Conference Computing in Economics and Finance, 2012. Google Scholar
[20] Ruiz-Torrubiano R, Garc'ıa-Moratilla S, Suárez A. Optimization problems with cardinality constraints. In: Proceedings of Computational Intelligence in Optimization, 2010. 105--130. Google Scholar
[21] Chang T J, Meade N, Beasley J E. Heuristics for cardinality constrained portfolio optimisation. Comput Operations Res, 2000, 27: 1271-1302 CrossRef Google Scholar
[22] Volgenant A. Solving the k-cardinality assignment problem by transformation. Eur J Operational Res, 2004, 157: 322-331 CrossRef Google Scholar
[23] Radcliffe N J, George F A. A study in set recombination. In: Proceedings of the 5th International Conference on Genetic Algorithms, 1993. 23--30. Google Scholar
[24] Kariv O, Hakimi S L. An algorithmic approach to network location problems. SIAM J Appl Math, 1979, 37: 539-560 CrossRef Google Scholar
[25] Reese J. Solution methods for the p-median problem: An annotated bibliography. Networks, 2006, 48: 125-142 CrossRef Google Scholar
[26] Mladenovi? N, Brimberg J, Hansen P. The p-median problem: A survey of metaheuristic approaches. Eur J Operational Res, 2007, 179: 927-939 CrossRef Google Scholar
[27] ReVelle C S, Eiselt H A, Daskin M S. A bibliography for some fundamental problem categories in discrete location science. Eur J Operational Res, 2008, 184: 817-848 CrossRef Google Scholar
[28] Hosage C M, Goodchild M F. Discrete space location-allocation solutions from genetic algorithms. Ann Oper Res, 1986, 6: 35-46 CrossRef Google Scholar
[29] Alp O, Erkut E, Drezner Z. An efficient genetic algorithm for the p-median problem. Ann Operations Res, 2003, 122: 21-42 CrossRef Google Scholar
[30] Li X, Xiao N, Claramunt C. Initialization strategies to enhancing the performance of genetic algorithms for the p-median problem. Comput Industrial Eng, 2011, 61: 1024-1034 CrossRef Google Scholar
[31] Lim A, Xu Z. A fixed-length subset genetic algorithm for the p-median problem. In: Proceedings of Genetic and Evolutionary Computation Conference, 2003. 1596--1597. Google Scholar
[32] Correa E S, Steiner M T A, Freitas A A. A Genetic Algorithm for Solving a Capacitated p-Median Problem. Numer Algorithms, 2004, 35: 373-388 CrossRef ADS Google Scholar
[33] Alba E, Domínguez E. Comparative analysis of modern optimization tools for the p-median problem. Stat Comput, 2006, 16: 251-260 CrossRef Google Scholar
[34] Hansen P, Mladenovi? N. Complement to a comparative analysis of heuristics for the p-median problem. Stat Comput, 2008, 18: 41-46 CrossRef Google Scholar
[35] Daskin M S, Maass K L. The p-median problem. In: Location Science. Berlin: Springer, 2015. 21--45. Google Scholar
[36] Daskin M S. Network and Discrete Location: Models, Algorithms, and Applications. Hoboken: John Wiley & Sons, 2013. Google Scholar
[37] Galv?o R D, ReVelle C. A Lagrangean heuristic for the maximal covering location problem. Eur J Operational Res, 1996, 88: 114-123 CrossRef Google Scholar
[38] K?rkel M. On the exact solution of large-scale simple plant location problems. Eur J Operational Res, 1989, 39: 157-173 CrossRef Google Scholar
[39] Cesarone F, Scozzari A, Tardella F. Efficient algorithms for mean-variance portfolio optimization with hard real-world constraints. In: Proceedings of the 18th AFIR Colloquium: Financial Risk in a Changing World, 2008. Google Scholar
[40] Ponsich A, Jaimes A L, Coello C A C. A Survey on Multiobjective Evolutionary Algorithms for the Solution of the Portfolio Optimization Problem and Other Finance and Economics Applications. IEEE Trans Evol Computat, 2013, 17: 321-344 CrossRef Google Scholar
[41] Metaxiotis K, Liagkouras K. Multiobjective Evolutionary Algorithms for Portfolio Management: A comprehensive literature review. Expert Syst Appl, 2012, 39: 11685-11698 CrossRef Google Scholar
[42] Markowitz H. Portfolio selection. J Financ, 1952, 7: 77--91. Google Scholar
[43] Fieldsend J E, Matatko J, Peng M. Cardinality constrained portfolio optimisation. In: Proceedings of International Conference on Intelligent Data Engineering and Automated Learning, 2004. 788--793. Google Scholar
[44] Anagnostopoulos K P, Mamanis G. A portfolio optimization model with three objectives and discrete variables. Comput Operations Res, 2010, 37: 1285-1297 CrossRef Google Scholar
[45] Ruiz-Torrubiano R, Suarez A. Hybrid Approaches and Dimensionality Reduction for Portfolio Selection with Cardinality Constraints. IEEE Comput Intell Mag, 2010, 5: 92-107 CrossRef Google Scholar
[46] Cesarone F, Scozzari A, Tardella F. A new method for mean-variance portfolio optimization with cardinality constraints. Ann Oper Res, 2013, 205: 213-234 CrossRef Google Scholar
[47] Zitzler E, Thiele L, Laumanns M. Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Computat, 2003, 7: 117-132 CrossRef Google Scholar
Figure A1
(Color online) Solution procedures of conventional EAs and Mucard. The red circles are the real optima; the other circles are the objective values of the EA individuals. (a1) Initialization and evolution direction of conventional EAs; (a2) final result of conventional EAs; (b1) initialization and evolution direction of Mucard; (b2) evolution of Mucard;protect łinebreak (b3) final results of Mucard.
| Settings | SITATION GA | Mucard |
| Representation | Set | Set |
| Crossover | Uniform crossover | RRR operator |
| Mutation | Greedy interchange | Greedy interchange |
| Repair | Yes | No |
| Reproduction selection | Tournament | Tournament |
| Survival selection | Elitist | Crowding distance operator |
| $p$ | Köerkel | Galv ao100 | Galv ao150 | ||||||
| A | B | C | A | B | C | A | B | C | |
| 10 | |||||||||
| 11 | 0.80 | 0.98 | 0.94 | 0.82 | 1.00 | 1.00 | 0.83 | 1.00 | 1.00 |
| 12 | 0.76 | 0.98 | 0.94 | 0.82 | 1.00 | 1.00 | 0.81 | 1.00 | 1.00 |
| 13 | 0.74 | 0.98 | 0.93 | 0.82 | 1.00 | 1.00 | 0.81 | 1.00 | 1.00 |
| 14 | 0.76 | 0.98 | 0.93 | 0.83 | 1.00 | 1.00 | 0.81 | 1.00 | 1.00 |
| 15 | 0.72 | 1.00 | 0.93 | 0.83 | 1.00 | 1.00 | 0.83 | 1.00 | 1.00 |
| 16 | 0.78 | 0.98 | 0.92 | 0.83 | 1.00 | 1.00 | 0.82 | 1.00 | 1.00 |
| 17 | 0.76 | 0.99 | 0.93 | 0.87 | 1.00 | 1.00 | 0.81 | 1.00 | 0.99 |
| 18 | 0.77 | 0.99 | 0.94 | 0.85 | 1.00 | 1.00 | 0.80 | 1.00 | 0.99 |
| 19 | 0.78 | 0.99 | 0.93 | 0.87 | 1.00 | 1.00 | 0.81 | 0.99 | 0.99 |
| 20 | 0.78 | 0.98 | 0.92 | 0.87 | 1.00 | 1.00 | 0.81 | 0.99 | 0.99 |
A: setting (20, 1) to setting (20, 100); B: setting (20, 100) to setting (220, 100); C: setting (20, 100) to setting (20, 1100).
| $p$ | Köerkel | Galv ao100 | Galv ao150 | ||||||
| A | B | C | A | B | C | A | B | C | |
| 10 | r | ||||||||
| 11 | 0.08 | 1.31 | 0.41 | 0.13 | 0.10 | 0.41 | 0.07 | 1.12 | 1.13 |
| 12 | 0.11 | 0.72 | 0.62 | 0.37 | 0.38 | 0.36 | 0.07 | 0.85 | 0.81 |
| 13 | 0.07 | 0.87 | 0.53 | 0.18 | 0.14 | 1.02 | 0.13 | 0.77 | 0.64 |
| 14 | 0.07 | 0.66 | 0.34 | 0.08 | 0.04 | 0.16 | 0.20 | 0.41 | 0.24 |
| 15 | 0.12 | 0.87 | 0.40 | 0.06 | 0.22 | 0.06 | 0.07 | 0.93 | 0.69 |
| 16 | 0.18 | 0.46 | 0.26 | 0.04 | 0.34 | 0.42 | 0.08 | 0.71 | 1.16 |
| 17 | 0.16 | 0.96 | 0.58 | 0.03 | 1.52 | 2.40 | 0.24 | 0.51 | 0.56 |
| 18 | 0.16 | 0.81 | 0.90 | 0.02 | 2.09 | 1.67 | 0.18 | 0.70 | 0.70 |
| 19 | 0.22 | 0.82 | 0.61 | 0.03 | 1.10 | 0.86 | 0.15 | 1.21 | 1.01 |
| 20 | 0.26 | 0.40 | 0.60 | 0.09 | 0.75 | 0.36 | 0.31 | 0.91 | 0.73 |
A: setting (20, 1) to setting (20, 100); B: setting (20, 100) to setting (220, 100); C: setting (20, 100) to setting (20, 1100).
| Port1 | Port2 | Port3 | Port4 | Port5 | |
| Row 1 | 0.01097 | 0.02524 | 0.01108 | 0.01933 | 0.00796 |
| Row 2 | 0.01560 | 0.03616 | 0.01680 | 0.03365 | 0.01066 |
| Row 3 | 0.01095 | 0.02464 | 0.00738 | 0.01672 | 0.00504 |
| Row 4 | 0.00463 | 0.01092 | 0.00573 | 0.01432 | 0.00270 |
| Row 5 | $-$0.00002 | $-$0.00060 | $-$0.00369 | $-$0.00260 | $-$0.00292 |
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