SCIENCE CHINA Information Sciences, Volume 62 , Issue 9 : 192104(2019) https://doi.org/10.1007/s11432-018-9720-6

## Solving multi-scenario cardinality constrained optimization problems via multi-objective evolutionary algorithms

• AcceptedNov 24, 2018
• PublishedJul 30, 2019
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### Abstract

Cardinality constrained optimization problems (CCOPs) are fixed-size subset selection problems with applications in several fields. CCOPs comprising multiple scenarios, such as cardinality values that form an interval, can be defined as multi-scenario CCOPs (MSCCOPs). An MSCCOP is expected to optimize the objective value of each cardinality to support decision-making processes. When the computation is conducted using traditional optimization algorithms, an MSCCOP often requires several passes (algorithmic runs) to obtain all the (near-)optima, where each pass handles a specific cardinality. Such separate passes abandon most of the knowledge (including the potential superior solution structures) learned in one pass that can also be used to improve the results of other passes. By considering this situation, we propose a generic transformation strategy that can be referred to as the Mucard strategy, which converts an MSCCOP into a low-dimensional multi-objective optimization problem (MOP) to simultaneously obtain all the (near-)optima of the MSCCOP in a single algorithmic run. In essence, the Mucard strategy combines separate passes that deal with distinct variable spaces into a single pass, enabling knowledge reuse and knowledge interchange of each cardinality among genetic individuals. The performance of the Mucard strategy was demonstrated using two typical MSCCOPs. For a given number of evolved individuals, the Mucard strategy improved the accuracy of the obtained solutions because of the in-process knowledge than that obtained by untransformed evolutionary algorithms, while reducing the average runtime. Furthermore, the equivalence between the optimal solutions of the transformed MOP and the untransformed MSCCOP can be theoretically proved.

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• 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.

• Table 1   Configurations of the two algorithms compared in this study
 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
• Table A1   Multiplication factors of the average results with different running settings and different numbers of selected sites on each dataset. The values highlighted in bold font are plotted in Figure 3
 $p$ Köerkel Galv ao100 Galv ao150 A B C A B C A B C 10 0.74 0.98 0.95 0.82 0.99 1.00 0.83 1.00 1.00 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).

• Table A2   Multiplication factors of the standard deviations for different running settings and different numbers of selected sites on each dataset. The values highlighted in bold are plotted in Figure 3
 $p$ Köerkel Galv ao100 Galv ao150 A B C A B C A B C 10 0.12 0.49 0.68 0.33 r 0.10 0.52 0.09 1.11 0.71 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).

• Table 2   Comparison of the mean percentage deviation errors between Mucard and the algoriTheorem in . The small values in the first three rows are favorable
 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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