基于剪枝堆栈泛化的离线数据驱动进化优化

梁正平; 黄锡均; 李燊钿; 王喜瑜; 朱泽轩

doi:10.16383/j.aas.c220387

基于剪枝堆栈泛化的离线数据驱动进化优化

doi: 10.16383/j.aas.c220387

1.
深圳大学计算机与软件学院深圳 518060
2.
中兴通讯深圳 518057

基金项目: 国家重点研发计划(2021YFB2900800), 广东省自然科学基金(2021A1515011911), 深圳市科技计划项目(20200811181752003, JCYJ20220531102617039)资助

详细信息

作者简介:
梁正平：深圳大学计算机与软件学院副教授. 2006年获武汉大学博士学位. 主要研究方向为计算智能, 大数据分析与应用. E-mail: liangzp@szu.edu.cn

黄锡均：深圳大学计算机与软件学院硕士研究生. 主要研究方向为计算智能与应用. E-mail: 1910273026@email.szu.edu.cn

李燊钿：深圳大学计算机与软件学院硕士研究生. 主要研究方向为计算智能与应用. E-mail: 2070276194@email.szu.edu.cn

王喜瑜：中兴通讯首席技术官, 高级工程师. 1998年获北京交通大学硕士学位. 主要研究方向为网络通讯和人工智能. 本文通信作者. E-mail: 13312902080@189.cn

朱泽轩：深圳大学计算机与软件学院教授. 2008年获新加坡南洋理工大学博士学位. 主要研究方向为计算智能, 机器学习和生物信息学. E-mail: zhuzx@szu.edu.cn

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出版历程
- 收稿日期: 2022-05-12
- 录用日期: 2022-09-21
- 网络出版日期: 2023-02-20
- 刊出日期: 2023-06-20

Offline Data Driven Evolutionary Optimization Based on Pruning Stacked Generalization

1.
College of Computer Science and Software Engineering, Shenzhen University, Shenzhen 518060
2.
ZTE Corporation, Shenzhen 518057

Funds: Supported by National Key Research and Development Program of China (2021YFB2900800), Natural Science Foundation of Guangdong Province (2021A1515011911), and Shenzhen Fundamental Research Program (20200811181752003, JCYJ20220531102617039)

More Information

Author Bio:
LIANG Zheng-Ping　Associate professor at the College of Computer Science and Software Engineering, Shenzhen University. He received his Ph.D. degree from Wuhan University in 2006. His research interest covers computational intelligence, big data analysis and application

HUANG Xi-Jun　Master student at the College of Computer Science and Software Engineering, Shenzhen University. His research interest covers computational intelligence and applications

LI Shen-Tian　Master student at the College of Computer Science and Software Engineering, Shenzhen University. His research interest covers computational intelligence and applications

Wang Xi-Yu　Senior engineer of ZTE Corporation, China. He received his master degree from Beijing Jiaotong University in 1998. His research interest covers network communication and artificial intelligence. Corresponding author of this paper

ZHU Ze-Xuan　Professor at the College of Computer Science and Software Engineering, Shenzhen University. He received his Ph.D. degree from Nanyang Technological University in 2008. His research interest covers computational intelligence, machine learning, and bioinformatics

摘要

摘要: 现实世界中存在很多目标函数的计算非常昂贵, 甚至目标函数难以建模的复杂优化问题. 常规优化方法在解决此类问题时要么无从入手, 要么效率低下. 离线数据驱动的进化优化方法不需对真实目标函数进行评估, 跳出了传统优化方法的固铚, 极大推动了昂贵优化问题和不可建模优化问题的求解. 但离线数据驱动进化优化的效果严重依赖于所采用代理模型的质量. 为提升离线数据驱动进化优化的性能, 提出了一个基于剪枝堆栈泛化(Stacked generalization, SG)代理模型构建方法. 具体而言, 一方面基于异构的基学习器建立初级模型池, 再采用学习方式对各初级模型进行组合, 以提升代理模型的通用性和准确率. 另一方面基于等级保护指标对初级模型进行剪枝, 在提高初级模型集成效率的同时进一步提升最终代理模型的准确率, 并更好地指导种群的搜索. 为验证所提方法的有效性, 与7个最新的离线数据驱动的进化优化算法在12个基准测试问题上进行对比, 实验结果表明所提出的方法具有明显的优势.
- 堆栈泛化 /
- 代理模型 /
- 离线数据驱动优化 /
- 进化计算
Abstract: In the real world, there are many complex optimization problems in which the objective function is time-consuming or even unavailable. Traditional optimization methods are either unable to start or inefficient in solving such problems. Offline data driven evolutionary optimization method is no need to evaluate the real objective function in the process of evolutionary, which greatly promotes the solution of expensive optimization problems and unmodeled optimization problems. However, the effectiveness of offline data driven evolutionary optimization depends heavily on surrogate model. In order to improve the quality of surrogate model, this paper proposes a surrogate model construction method based on pruning stack generalization (SG). Specifically, on the one hand, the primary model pool is established based on heterogeneous base learners, and then the primary models are conducted by learning methods to improve the generality and accuracy of surrogate model. On the other hand, the primary models are pruned based on the ranking preservation indicator, which can not only improve the ensemble efficiency of the primary models, but also further improve the accuracy of the final surrogate model, and better guide the evolutionary search. In order to verify the effectiveness of the proposed method, it is compared with 7 state-of-the-arts offline data driven evolutionary optimization algorithms on 12 benchmark problems. The experimental results demonstrate the superior performance of proposed method over compared algorithms.
- Stacked generalization (SG) /
- surrogate model /
- offline data driven optimization /
- evolutionary computation

HTML全文

图 1 两层结构的堆栈泛化框架

Fig. 1 Framework of 2-level stacked generalization

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图 2 DDEA-PSG的整体框架

Fig. 2 Framework of DDEA-PSG

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图 3 DDEA-SE, BDDEA-LDG, DDEA-PES, SRK-DDEA, TT-DDEA和DDEA-PSG在10, 30, 50, 100维Ellipsoid和Rastrigin上的收敛图

Fig. 3 Convergence profiles of DDEA-SE, BDDEA-LDG, DDEA-PES, SRK-DDEA, TT-DDEA and DDEA-PSG over 30 independent runs on Ellipsoid and Rastrigin of 10, 30, 50 and 100 dimensions

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图 4 DDEA-PSG和NSGAII_GP, MS-RV, BDDEA-LDG, DDEA-PES, TT-DDEA在决策变量维度为10, 30, 50和100, 2目标和3目标MaF1问题实例上, 30次独立运行获得的非支配解集

Fig. 4 Non-dominated solutions of NSGAII_GP, MS-RV, BDDEA-LDG, DDEA-PES, TT-DDEA and DDEA-PSG over 30 independent runs on 2-objective MaF1 and 3-objective MaF1 of 10, 30, 50 and 100 dimensions

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图 5 DDEA-PSG和各对比算法在100维Ellipsoid和Rastrigin测试实例, 以及100维2目标MaF1测试实例的2个目标函数上, 30次独立运行, 每次迭代过程中各算法代理模型的平均预测误差

Fig. 5 The average prediction error of surrogate model of DDEA-PSG and all comparison algorithms over 30 independent runs on 100 dimensions Ellipsoid, Rastrigin and two objective function of 2-objctive MaF1

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图 6 DDEA-PR, DDEA-SVM, DDEA-RBFN,DDEA-PSR, DDEA-SG和DDEA-PSG在所有测试实例上30次独立运行的平均性能排名

Fig. 6 Average performance ranking of DDEA-PR, DDEA-SVM, DDEA-RBFN, DDEA-PSR, DDEA-SG and DDEA-PSG over 30 independent runs on all test cases

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图 7 参数$ k $取值不同时, DDEA-PSG在10, 50和100维Ellipsoid, Rastrigin和3目标MaF1, MaF7测试实例上, 30次独立运行的平均性能变化曲线图

Fig. 7 Average performance of DDEA-PSG over 30 independent runs on 10, 50 and 100 dimensions Ellipsoid, Rastrigin and 3-objective MaF1 and MaF7 with different values of parameter $ k $

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图 8 参数$ l $取值不同时, DDEA-PSG在10, 50和100维Ellipsoid, Rastrigin和3目标MaF1, MaF7测试实例上, 30次独立运行的平均性能变化曲线图

Fig. 8 Average performance of DDEA-PSG over 30 independent runs on 10, 50 and 100 dimensions Ellipsoid, Rastrigin and 3-objective MaF1 and MaF7 with different values of parameter $ l $

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表 1 DDEA-SE, BDDEA-LDG, DDEA-PES, SRK-DDEA, TT-DDEA和DDEA-PSG在5个单目标测试问题上, 决策变量维度分别为10, 30, 50和100, 30次独立运行的平均值和标准差

Table 1 The mean and standard deviation of optimum solution of DDEA-SE, BDDEA-LDG, DDEA-PES, SRK-DDEA, TT-DDEA and DDEA-PSG over 30 independent runs on 5 single objective test problems of 10, 30, 50 and 100 dimensions

问题	维度	DDEA-SE	BDDEA-LDG	DDEA-PES	SRK-DDEA	TT-DDEA	DDEA-PSG
Ellipsoid	10	9.89×10⁻¹−	1.22×10⁰−	2.22×10⁰−	6.60×10⁰−	1.38×10⁰−	4.67×10⁻⁵
	10	(3.52×10⁻¹)	(3.79×10⁻¹)	(6.38×10⁻¹)	(5.22×10⁻¹)	(4.91×10⁻¹)	(5.41×10⁻⁵)
	30	4.73×10⁰−	6.60×10⁰−	8.72×10⁰−	9.29×10⁰−	3.43×10⁰−	1.26×10⁰
	30	(1.96×10⁰)	(1.30×10⁰)	(2.44×10⁰)	(1.65×10⁰)	(1.33×10⁰)	(1.19×10⁰)
	50	1.47×10¹−	1.90×10¹−	2.27×10¹−	1.81×10¹−	9.72×10⁰−	2.38×10⁰
	50	(4.13×10⁰)	(3.37×10⁰)	(4.95×10⁰)	(3.40×10⁰)	(2.75×10⁰)	(2.20×10⁰)
	100	3.21×10²−	3.00×10²−	3.16×10²−	7.91×10¹−	6.79×10¹−	3.71×10¹
	100	(7.56×10¹)	(6.13×10¹)	(7.51×10¹)	(1.72×10¹)	(1.24×10¹)	(9.81×10⁰)
Rosenbrock	10	2.69×10¹+	3.44×10¹−	4.11×10¹−	6.05×10¹−	3.57×10¹−	3.18×10¹
	10	(8.63×10⁰)	(6.04×10⁰)	(6.37×10⁰)	(1.61×10¹)	(9.41×10⁰)	(1.32×10¹)
	30	5.77×10¹+	7.40×10¹−	8.01×10¹−	1.20×10²−	5.85×10¹+	6.94×10¹
	30	(9.95×10⁰)	(6.57×10⁰)	(1.25×10¹)	(1.04×10¹)	(9.39×10⁰)	(1.31×10¹)
	50	8.27×10¹−	9.82×10¹−	1.04×10²−	1.45×10²−	7.56×10¹−	6.11×10¹
	50	(6.34×10⁰)	(4.83×10⁰)	(8.96×10⁰)	(6.96×10⁰)	(5.96×10⁰)	(4.59×10⁰)
	100	2.43×10²−	2.54×10²−	2.42×10²−	2.36×10²−	1.47×10²−	1.13×10²
	100	(2.97×10¹)	(3.76×10¹)	(5.61×10¹)	(1.71×10¹)	(9.71×10⁰)	(3.13×10⁰)
Ackley	10	5.91×10⁰−	7.24×10⁰−	8.43×10⁰−	8.30×10⁰−	6.12×10⁰−	5.61×10⁰
	10	(1.19×10⁰)	(1.09×10⁰)	(1.52×10⁰)	(1.24×10⁰)	(1.10×10⁰)	(1.43×10⁰)
	30	4.77×10⁰−	5.71×10⁰−	6.13×10⁰−	6.54×10⁰−	4.71×10⁰−	2.91×10⁰
	30	(6.52×10⁻¹)	(4.13×10⁻¹)	(4.56×10⁻¹)	(6.14×10⁻¹)	(4.35×10⁻¹)	(5.78×10⁻¹)
	50	4.61×10⁰−	5.41×10⁰−	5.87×10⁰−	5.61×10⁰−	4.34×10⁰−	2.49×10⁰
	50	(4.14×10⁻¹)	(3.09×10⁻¹)	(5.28×10⁻¹)	(3.97×10⁻¹)	(2.49×10⁻¹)	(4.22×10⁻¹)
	100	6.89×10⁰−	7.19×10⁰−	7.28×10⁰−	5.34×10⁰−	4.72×10⁰−	3.85×10⁰
	100	(6.01×10⁻¹)	(4.01×10⁻¹)	(4.56×10⁻¹)	(5.46×10⁻¹)	(2.86×10⁻¹)	(2.52×10⁻¹)
Griewank	10	1.25×10⁰−	1.36×10⁰−	2.34×10⁰−	7.07×10⁻¹−	1.13×10⁰−	2.07×10⁻²
	10	(1.72×10⁻¹)	(1.55×10⁻¹)	(1.98×10⁻¹)	(2.28×10⁻¹)	(1.58×10⁻¹)	(2.53×10⁻²)
	30	1.23×10⁰≈	1.36×10⁰−	2.67×10⁰−	1.05×10⁰+	1.14×10⁰≈	1.20×10⁰
	30	(1.30×10⁻¹)	(6.37×10⁻²)	(1.18×10⁻¹)	(6.79×10⁻²)	(6.41×10⁻²)	(1.70×10⁻¹)
	50	1.77×10⁰−	2.05×10⁰−	4.16×10⁰−	1.38×10⁰+	1.55×10⁰−	1.44×10⁰
	50	(2.43×10⁻¹)	(1.96×10⁻¹)	(2.01×10⁻¹)	(1.73×10⁻¹)	(1.14×10⁻¹)	(2.05×10⁻¹)
	100	1.80×10¹−	1.83×10¹−	2.20×10¹−	4.47×10⁰−	4.40×10⁰−	3.54×10⁰
	100	(4.12×10⁰)	(3.91×10⁰)	(4.96×10⁰)	(1.14×10⁰)	(7.23×10⁻¹)	(4.09×10⁻¹)
Rastrigin	10	6.51×10¹−	8.13×10¹−	8.65×10¹−	7.89×10¹−	6.39×10¹−	4.12×10¹
	10	(2.87×10¹)	(2.62×10¹)	(4.19×10¹)	(2.50×10¹)	(2.21×10¹)	(2.64×10¹)
	30	1.15×10²−	1.41×10²−	1.59×10²−	1.94×10²−	8.06×10¹−	2.36×10¹
	30	(2.94×10¹)	(2.57×10¹)	(2.98×10¹)	(3.51×10¹)	(2.11×10¹)	(2.30×10¹)
	50	2.04×10²−	2.08×10²−	2.36×10²−	2.57×10²−	1.20×10²−	3.85×10¹
	50	(3.14×10¹)	(3.10×10¹)	(5.36×10¹)	(2.78×10¹)	(2.40×10¹)	(1.64×10¹)
	100	8.57×10²−	7.61×10²−	7.64×10²−	4.16×10²−	2.98×10²−	1.39×10²
	100	(9.44×10¹)	(1.12×10²)	(1.31×10²)	(5.81×10¹)	(4.54×10¹)	(3.06×10¹)
+/≈/−		2/1/17	0/0/20	0/0/20	2/0/18	1/1/18

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表 2 NSGAII_GP, MS-RV, BDDEA-LDG, DDEA-PES, TT-DDEA和DDEA-PSG在决策变量维度为10, 30, 50和100, 2目标MaF测试实例上, 30次独立运行的IGD平均值和标准差

Table 2 The mean and standard deviation of IGD of NSGAII_GP, MS-RV, BDDEA-LDG, DDEA-PES, TT-DDEA and DDEA-PSG over 30 independent runs on 2-objective MaF of 10, 30, 50 and 100 dimensions

问题	目标数	维度	NSGAII_GP	MS-RV	BDDEA-LDG	DDEA-PES	TT-DDEA	DDEA-PSG
MaF1	2	10	3.19×10¹−	2.79×10¹−	7.94×10²−	2.42×10¹−	1.17×10¹−	4.11×10²
		10	(3.81×10²)	(3.91×10²)	(2.03×10²)	(7.91×10²)	(2.35×10²)	(4.09×10²)
		30	1.23×10⁰−	1.12×10⁰−	2.59×10¹−	2.87×10¹−	2.52×10¹−	1.44×10¹
		30	(1.19×10¹)	(8.53×10²)	(8.89×10²)	(8.76×10²)	(6.78×10²)	(6.35×10²)
		50	2.32×10⁰−	2.25×10⁰−	8.01×10¹−	8.12×10¹−	2.88×10¹−	1.71×10¹
		50	(1.67×10¹)	(1.43×10¹)	(1.12×10¹)	(2.16×10¹)	(8.34×10²)	(3.97×10²)
		100	5.10×10⁰−	8.44×10¹−	2.74×10⁰−	8.24×10¹−	6.98×10¹−	4.43×10¹
		100	(2.48×10¹)	(8.50×10²)	(5.00×10²)	(3.55×10¹)	(4.30×10²)	(1.04×10¹)
MaF2	2	10	6.15×10²−	5.78×10⁻³≈	1.92×10²−	4.50×10²−	6.96×10⁻³−	5.33×10⁻³
		10	(4.79×10⁻³)	(1.37×10⁻³)	(6.10×10⁻³)	(1.90×10²)	(1.20×10⁻³)	(8.71×10⁻⁴)
		30	2.19×10¹−	7.81×10²−	4.01×10²−	4.85×10²−	1.03×10²+	1.67×10²
		30	(1.49×10²)	(8.35×10⁻³)	(1.06×10²)	(1.67×10²)	(5.38×10⁻³)	(3.52×10⁻³)
		50	3.86×10¹−	2.01×10¹−	8.89×10²−	1.14×10¹−	2.96×10²+	3.27×10²
		50	(2.12×10²)	(1.80×10²)	(5.65×10⁻³)	(4.92×10²)	(1.25×10²)	(7.95×10⁻³)
		100	8.58×10¹−	1.86×10¹−	3.51×10¹−	1.07×10¹−	5.56×10²≈	4.90×10²
		100	(2.16×10²)	(1.80×10²)	(4.90×10²)	(6.98×10⁻³)	(8.13×10⁻³)	(9.72×10⁻³)
MaF3	2	10	6.77×10⁵−	2.38×10⁵+	6.39×10⁵−	3.03×10⁵−	3.86×10⁵−	3.53×10⁵
		10	(2.71×10⁵)	(1.03×10⁵)	(9.00×10⁴)	(1.82×10⁵)	(6.24×10⁴)	(2.34×10⁵)
		30	1.74×10⁷−	5.64×10⁶≈	9.17×10⁶−	7.12×10⁶−	6.20×10⁶−	5.37×10⁶
		30	(9.77×10⁶)	(1.16×10⁶)	(9.94×10⁵)	(2.42×10⁵)	(1.75×10⁶)	(2.17×10⁶)
		50	1.57×10⁸−	1.89×10⁷+	2.25×10⁷−	2.07×10⁷−	2.96×10⁷−	1.97×10⁷
		50	(1.94×10⁸)	(2.77×10⁶)	(6.26×10⁶)	(9.46×10⁶)	(3.29×10⁶)	(3.40×10⁶)
		100	2.15×10⁹−	1.64×10⁷+	3.32×10⁸−	9.80×10⁷−	7.17×10⁷−	1.80×10⁷
		100	(2.75×10⁹)	(4.27×10⁶)	(3.23×10⁷)	(3.46×10⁶)	(3.80×10⁷)	(3.97×10⁶)
MaF4	2	10	6.65×10²−	4.91×10²≈	7.04×10²−	5.24×10²−	5.53×10²−	4.36×10²
		10	(1.62×10²)	(8.03×10¹)	(8.53×10¹)	(2.62×10²)	(1.18×10²)	(1.25×10²)
		30	2.87×10³−	2.49×10³≈	3.21×10³−	3.33×10³−	3.01×10³−	2.36×10³
		30	(2.70×10²)	(2.36×10²)	(7.60×10¹)	(3.87×10²)	(3.81×10²)	(2.52×10²)
		50	5.29×10³≈	4.62×10³+	5.20×10³≈	5.00×10³≈	5.05×10³≈	5.14×10³
		50	(2.50×10²)	(2.36×10²)	(1.24×10²)	(1.97×10²)	(4.13×10²)	(3.33×10²)
		100	1.12×10⁴≈	4.61×10³+	1.10×10⁴≈	1.20×10⁴−	1.07×10⁴≈	1.08×10⁴
		100	(4.37×10²)	(3.48×10²)	(3.19×10²)	(5.11×10²)	(7.62×10²)	(4.45×10²)
MaF5	2	10	2.69×10⁰−	1.90×10⁰+	2.01×10⁰≈	2.02×10⁰≈	2.01×10⁰≈	2.04×10⁰
		10	(4.02×10¹)	(3.76×10¹)	(4.51×10⁻⁴)	(4.04×10⁻³)	(1.42×10⁻³)	(2.66×10²)
		30	6.83×10⁰−	1.81×10⁰≈	2.03×10⁰−	2.06×10⁰−	2.02×10⁰−	1.86×10⁰
		30	(1.61×10⁰)	(3.66×10¹)	(6.91×10⁻³)	(3.22×10²)	(9.57×10⁻³)	(7.55×10¹)
		50	1.19×10¹−	4.33×10⁰−	2.09×10⁰−	2.51×10⁰−	2.04×10⁰−	1.75×10⁰
		50	(1.57×10⁰)	(5.12×10¹)	(7.47×10⁻³)	(3.01×10¹)	(6.73×10⁻³)	(9.65×10¹)
		100	2.40×10¹−	5.46×10⁰−	2.44×10⁰−	3.19×10⁰−	2.18×10⁰−	5.45×10¹
		100	(5.19×10⁰)	(3.09×10¹)	(2.16×10¹)	(4.24×10¹)	(7.56×10²)	(1.37×10¹)
MaF6	2	10	3.08×10¹−	2.85×10¹−	1.13×10⁰−	2.81×10⁰−	2.65×10⁰−	1.65×10⁰
		10	(7.17×10⁰)	(7.47×10⁰)	(8.33×10²)	(1.55×10⁰)	(1.54×10⁰)	(1.08×10⁰)
		30	1.61×10²−	1.55×10²−	2.22×10¹−	1.80×10¹−	8.69×10⁰−	8.08×10⁰
		30	(1.05×10¹)	(1.30×10¹)	(5.34×10⁰)	(4.15×10⁰)	(2.79×10⁰)	(2.00×10⁰)
		50	2.98×10²−	2.87×10²−	6.77×10¹−	7.18×10¹−	1.81×10¹−	1.73×10¹
		50	(1.88×10¹)	(2.50×10¹)	(5.31×10⁰)	(1.66×10¹)	(4.30×10⁰)	(3.52×10⁰)
		100	6.74×10²−	1.46×10²−	3.34×10²−	1.63×10²−	7.61×10¹−	4.99×10¹
		100	(3.10×10¹)	(1.43×10¹)	(9.68×10⁰)	(6.79×10¹)	(9.76×10⁰)	(5.94×10⁰)
MaF7	2	10	5.40×10⁰−	3.84×10¹−	5.82×10⁰−	4.46×10⁰−	1.14×10⁰−	1.56×10²
		10	(5.33×10¹)	(6.70×10²)	(8.45×10²)	(1.11×10¹)	(4.24×10¹)	(1.23×10⁻³)
		30	6.78×10⁰−	2.38×10⁰−	5.55×10⁰−	5.12×10⁰−	2.93×10⁰−	2.26×10¹
		30	(4.36×10¹)	(3.39×10¹)	(1.76×10¹)	(6.22×10²)	(9.83×10²)	(1.45×10¹)
		50	6.98×10⁰−	3.61×10⁰−	5.80×10⁰−	5.32×10⁰−	3.33×10⁰−	8.91×10¹
		50	(3.69×10¹)	(4.17×10¹)	(6.82×10¹)	(1.75×10¹)	(3.95×10¹)	(3.14×10¹)
		100	7.48×10⁰−	3.63×10⁰−	5.66×10⁰−	6.19×10⁰−	3.57×10⁰−	1.51×10⁰
		100	(2.24×10¹)	(3.33×10¹)	(2.89×10¹)	(6.13×10¹)	(4.26×10¹)	(2.07×10¹)
+/≈/−			0/2/26	6/5/17	0/3/25	0/2/26	2/4/22

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表 3 NSGAII_GP, MS-RV, BDDEA-LDG, DDEA-PES, TT-DDEA和DDEA-PSG在决策变量维度为10, 30, 50和100, 3目标MaF测试实例上, 30次独立运行的IGD平均值和标准差

Table 3 The mean and standard deviation of IGD of NSGAII_GP, MS-RV, BDDEA-LDG, DDEA-PES, TT-DDEA and DDEA-PSG over 30 independent runs on 3-objective MaF of 10, 30, 50 and 100 dimensions

问题	目标数	维度	NSGAII_GP	MS-RV	BDDEA-LDG	DDEA-PES	TT-DDEA	DDEA-PSG
MaF1	3	10	8.91×10¹−	4.44×10¹−	9.62×10¹−	1.18×10⁰−	5.46×10¹−	7.86×10⁻²
		10	(3.14×10¹)	(5.14×10⁻²)	(6.62×10⁻²)	(7.41×10⁻²)	(2.30×10¹)	(1.01×10⁻²)
		30	1.99×10⁰−	1.48×10⁰−	1.56×10¹≈	1.81×10¹−	1.68×10¹−	1.49×10¹
		30	(2.23×10¹)	(1.76×10¹)	(3.09×10⁻²)	(2.38×10⁻²)	(2.39×10⁻²)	(1.84×10⁻²)
		50	3.72×10⁰−	2.57×10⁰−	4.42×10¹−	4.86×10¹−	2.53×10¹−	2.34×10¹
		50	(2.26×10¹)	(2.71×10¹)	(1.94×10⁻²)	(7.53×10⁻²)	(2.35×10⁻²)	(2.11×10⁻²)
		100	8.26×10⁰−	5.63×10⁰−	2.00×10⁰−	1.33×10⁰−	1.44×10⁰−	6.25×10¹
		100	(3.65×10¹)	(1.78×10¹)	(1.91×10¹)	(3.41×10¹)	(7.85×10⁻²)	(5.62×10⁻²)
MaF2	3	10	7.52×10⁻²−	4.03×10⁻²+	6.31×10⁻²−	1.22×10¹−	4.53×10⁻²−	4.89×10⁻²
		10	(1.64×10³)	(2.79×10³)	(8.54×10³)	(1.32×10⁻²)	(5.58×10³)	(3.12×10³)
		30	1.78×10¹−	1.04×10¹−	8.56×10⁻²−	9.73×10⁻²−	8.40×10⁻²−	6.78×10⁻²
		30	(2.93×10³)	(8.61×10³)	(2.49×10⁻²)	(1.13×10⁻²)	(6.39×10³)	(2.54×10³)
		50	2.91×10¹−	1.99×10¹−	1.33×10¹−	1.16×10¹−	1.15×10⁰−	8.55×10⁻²
		50	(6.63×10³)	(1.23×10⁻²)	(3.42×10⁻²)	(9.67×10³)	(1.68×10⁻²)	(6.87×10³)
		100	5.96×10¹−	2.21×10¹−	2.59×10¹−	2.12×10¹−	7.49×10¹−	1.63×10¹
		100	(2.14×10⁻²)	(1.56×10⁻²)	(2.52×10⁻²)	(2.58×10⁻²)	(2.83×10⁻²)	(1.20×10⁻²)
MaF3	3	10	8.27×10⁵−	8.92×10⁴+	1.11×10⁵−	2.63×10⁵−	5.12×10⁵−	2.66×10⁵
		10	(3.35×10⁴)	(4.67×10⁴)	(4.56×10⁴)	(4.30×10⁴)	(8.40×10⁴)	(1.64×10⁴)
		30	1.36×10⁷−	4.58×10⁶+	2.55×10⁷−	4.91×10⁶−	3.85×10⁷−	8.78×10⁶
		30	(1.22×10⁷)	(7.28×10⁵)	(3.52×10⁶)	(1.10×10⁵)	(3.75×10⁶)	(2.35×10⁵)
		50	1.06×10⁸−	1.54×10⁷+	2.61×10⁷−	5.21×10⁷−	8.29×10⁷−	2.86×10⁷
		50	(1.38×10⁸)	(1.44×10⁶)	(7.54×10⁶)	(5.00×10⁸)	(1.11×10⁹)	(3.53×10⁶)
		100	2.42×10⁸−	1.61×10⁷+	1.30×10⁸−	6.65×10⁸−	2.65×10⁸−	7.86×10⁷
		100	(3.21×10⁶)	(2.79×10⁶)	(7.09×10⁶)	(7.93×10⁶)	(5.15×10⁶)	(1.05×10⁶)
MaF4	3	10	1.61×10³−	9.52×10²+	1.62×10³−	1.60×10³−	1.94×10³−	1.37×10³
		10	(3.16×10²)	(2.83×10²)	(6.54×10²)	(4.54×10²)	(2.54×10²)	(3.05×10²)
		30	7.18×10³≈	6.32×10³+	7.43×10³−	8.23×10³−	7.02×10³≈	7.04×10³
		30	(5.30×10²)	(6.04×10²)	(9.58×10²)	(2.62×10²)	(1.33×10³)	(8.79×10²)
		50	1.32×10⁴≈	1.21×10⁴+	1.15×10⁴−	1.26×10⁴≈	1.23×10⁴−	1.32×10⁴
		50	(9.63×10²)	(8.51×10²)	(9.32×10²)	(9.87×10²)	(1.52×10³)	(1.09×10³)
		100	2.81×10⁴−	1.17×10⁴+	4.07×10⁴−	3.05×10⁴−	2.81×10⁴−	1.85×10⁴
		100	(1.72×10³)	(1.02×10³)	(2.83×10²)	(2.19×10³)	(1.78×10³)	(1.08×10³)
MaF5	3	10	5.08×10⁰−	8.89×10¹−	4.90×10⁰−	4.92×10⁰−	4.92×10⁰−	7.86×10⁻²
		10	(1.54×10⁰)	(2.72×10¹)	(5.72×10³)	(1.23×10⁻²)	(2.63×10⁻²)	(1.01×10⁻²)
		30	1.45×10¹−	3.19×10⁰−	5.00×10⁰−	5.30×10⁰−	4.96×10⁰−	1.57×10⁰
		30	(3.60×10⁰)	(2.42×10¹)	(3.00×10⁻²)	(1.76×10¹)	(5.25×10⁻²)	(6.27×10¹)
		50	2.39×10¹−	6.00×10⁰−	5.27×10⁰−	5.59×10⁰−	5.00×10⁰−	2.28×10⁰
		50	(5.24×10⁰)	(4.15×10¹)	(5.27×10⁻²)	(2.85×10¹)	(6.13×10⁻²)	(1.40×10⁰)
		100	4.09×10¹−	6.90×10⁰−	7.36×10⁰−	9.14×10⁰−	5.56×10⁰−	1.96×10⁰
		100	(1.21×10¹)	(1.44×10⁰)	(1.84×10⁰)	(9.93×10¹)	(5.66×10¹)	(6.27×10¹)
MaF6	3	10	2.34×10¹−	2.29×10¹−	6.24×10¹−	1.09×10⁰−	3.39×10¹−	4.89×10⁻²
		10	(3.82×10⁰)	(5.85×10⁰)	(2.44×10¹)	(8.98×10¹)	(9.68×10⁻²)	(3.12×10³)
		30	1.54×10²−	1.43×10²−	1.37×10¹−	1.86×10¹−	1.28×10¹−	9.05×10⁰
		30	(1.87×10¹)	(1.62×10¹)	(1.69×10⁰)	(2.30×10⁰)	(4.69×10¹⁴)	(1.60×10⁰)
		50	2.91×10²−	2.91×10²−	4.36×10¹−	3.76×10¹−	3.96×10¹−	3.50×10¹
		50	(1.90×10¹)	(1.86×10¹)	(1.02×10¹)	(9.97×10⁰)	(7.44×10⁰)	(1.09×10¹)
		100	6.72×10²−	1.57×10²−	1.93×10²−	2.97×10²−	1.91×10²−	1.20×10²
		100	(1.92×10¹)	(1.86×10¹)	(4.08×10¹)	(6.88×10¹)	(2.27×10¹)	(2.05×10¹)
MaF7	3	10	7.23×10⁰−	5.53×10¹−	7.78×10⁰−	6.04×10⁰−	1.85×10⁰−	2.69×10¹
		10	(1.36×10⁰)	(1.67×10¹)	(2.49×10⁰)	(1.82×10⁰)	(2.10×10¹)	(2.60×10¹)
		30	9.75×10⁰−	3.64×10⁰−	8.46×10⁰−	7.86×10⁰−	4.80×10⁰−	7.51×10¹
		30	(5.76×10¹)	(8.02×10¹)	(9.23×10¹)	(7.07×10⁻²)	(4.21×10¹)	(7.58×10⁻²)
		50	1.05×10¹−	5.50×10⁰−	8.40×10⁰−	7.43×10⁰−	5.68×10⁰−	1.38×10⁰
		50	(6.66×10¹)	(6.48×10¹)	(3.43×10¹)	(2.57×10¹)	(7.08×10¹)	(1.82×10¹)
		100	1.13×10¹−	5.73×10⁰−	9.57×10⁰−	8.77×10⁰−	5.88×10⁰−	2.79×10⁰
		100	(2.68×10¹)	(5.92×10¹)	(7.87×10¹)	(5.57×10¹)	(1.16×10⁰)	(3.62×10¹)
+/≈/−			0/2/26	9/0/19	0/1/27	0/1/27	0/1/27

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表 4 DDEA-PR, DDEA-SVM, DDEA-RBFN, DDEA-PSR, DDEA-SG和DDEA-PSG在Ellipsoid和Rastrigin上, 决策变量维度分别为10, 50和100, 30次独立运行的最优解的平均值和标准差

Table 4 The mean and standard deviation of optimum solution of DDEA-PR, DDEA-SVM, DDEA-RBFN, DDEA-PSR, DDEA-SG and DDEA-PSG over 30 independent runs on Ellipsoid and Rastirgin of 10, 50 and 100 dimensions

问题	维度	DDEA-PR	DDEA-SVM	DDEA-RBFN	DDEA-PSR	DDEA-SG	DDEA-PSG
Ellipsoid	10	5.03×10⁻⁵−	1.70×10⁰−	1.39×10⁰−	1.35×10⁰−	6.32×10⁻⁵−	4.67×10⁻⁵
	10	(4.75×10⁻⁵)	(6.57×10⁻¹)	(4.61×10⁻¹)	(4.09×10⁻¹)	(1.44×10⁻⁵)	(5.41×10⁻⁵)
	50	1.91×10⁴−	1.26×10²−	4.84×10⁰−	1.69×10¹−	2.92×10⁰−	2.38×10⁰
	50	(2.09×10³)	(3.01×10¹)	(1.85×10⁰)	(1.05×10¹)	(6.24×10⁻¹)	(2.20×10⁰)
	100	9.18×10⁴−	6.43×10²−	8.94×10¹−	1.47×10²−	4.05×10¹−	3.71×10¹
	100	(4.36×10³)	(8.56×10¹)	(1.13×10¹)	(3.59×10¹)	(7.26×10⁰)	(9.81×10⁰)
Rastrigin	10	2.25×10²−	1.06×10²−	5.51×10¹−	1.01×10²−	5.22×10¹−	4.12×10¹
	10	(3.29×10¹)	(2.44×10¹)	(2.18×10¹)	(2.45×10¹)	(1.47×10¹)	(2.64×10¹)
	50	1.30×10³−	5.11×10²−	8.23×10¹−	2.51×10²−	3.00×10¹+	3.85×10¹
	50	(5.27×10¹)	(5.60×10¹)	(2.88×10¹)	(7.96×10¹)	(1.39×10¹)	(1.64×10¹)
	100	2.53×10³−	9.97×10²−	2.64×10²−	4.72×10²−	1.60×10²−	1.39×10²
	100	(7.35×10¹)	(7.29×10¹)	(6.05×10¹)	(1.22×10²)	(3.19×10¹)	(3.06×10¹)
+/≈/−		0/0/6	0/0/6	0/0/6	0/0/6	1/0/5

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表 5 DDEA-PR, DDEA-SVM, DDEA-RBFN, DDEA-PSR, DDEA-SG和DDEA-PSG在决策变量维度分别为10, 50和100, 2目标和3目标MaF1和MaF7上, 30次独立运行的IGD平均值和标准差

Table 5 The mean and standard deviation of IGD of DDEA-PR, DDEA-SVM, DDEA-RBFN, DDEA-PSR, DDEA-SG and DDEA-PSG over 30 independent runs on 2-objective 3-objective MaF1 and MaF7 of 10, 50 and 100 dimensions

问题	目标数	维度	DDEA-PR	DDEA-SVM	DDEA-RBFN	DDEA-PSR	DDEA-SG	DDEA-PSG
MaF1	2	10	4.74×10⁻²−	7.95×10⁻²−	1.57×10⁻¹−	5.52×10⁻²−	3.20×10⁻²+	4.11×10⁻²
		10	(1.26×10⁻³)	(1.16×10⁻³)	(5.05×10⁻²)	(3.82×10⁻³)	(1.62×10⁻³)	(4.09×10⁻²)
		50	4.40×10⁰−	4.43×10⁻¹−	2.79×10⁻¹−	2.78×10⁻¹−	2.19×10⁻¹−	1.71×10⁻¹
		50	(4.00×10⁻¹)	(3.94×10⁻³)	(2.94×10⁻²)	(1.10×10⁻²)	(3.59×10⁻²)	(3.97×10⁻²)
		100	1.05×10¹−	6.53×10⁻¹−	5.19×10⁻¹−	5.62×10⁻¹−	5.02×10⁰−	4.43×10⁻¹
		100	(6.03×10⁻¹)	(7.76×10⁻³)	(2.95×10⁻¹)	(3.50×10⁻²)	(4.23×10⁻¹)	(1.04×10⁻¹)
	3	10	9.42×10⁻²−	9.80×10⁻²−	3.06×10⁻¹−	9.70×10⁻²−	6.26×10⁻²+	7.86×10⁻²
		10	(1.26×10⁻²)	(1.46×10⁻²)	(6.44×10⁻²)	(1.52×10⁻²)	(2.82×10⁻³)	(1.01×10⁻²)
		50	5.93×10⁰−	3.16×10⁻¹−	3.29×10⁻¹−	3.19×10⁻¹−	2.86×10⁻¹−	2.34×10⁻¹
		50	(6.78×10⁻¹)	(1.42×10⁻²)	(1.98×10⁻²)	(1.30×10⁻²)	(1.89×10⁻²)	(2.11×10⁻²)
		100	1.37×10¹−	8.47×10⁻¹−	7.28×10⁻¹−	7.60×10⁻¹−	6.83×10⁻¹−	6.25×10⁻¹
		100	(5.92×10⁻¹)	(4.10×10⁻²)	(1.90×10⁻¹)	(2.65×10⁻²)	(7.66×10⁻²)	(5.62×10⁻²)
MaF7	2	10	3.34×10⁻¹−	4.11×10⁰−	7.40×10⁰−	1.69×10⁰−	3.90×10⁻¹−	1.56×10⁻²
		10	(4.65×10⁻¹)	(4.74×10⁻¹)	(6.25×10⁻¹)	(1.28×10⁰)	(5.01×10⁻¹)	(1.23×10⁻³)
		50	3.23×10⁰−	4.26×10⁰−	8.77×10⁰−	6.11×10⁰−	1.25×10⁰−	8.91×10⁻¹
		50	(1.06×10⁰)	(4.56×10⁻²)	(2.86×10⁻¹)	(1.83×10⁻¹)	(7.72×10⁻¹)	(3.14×10⁻¹)
		100	4.03×10⁰−	3.29×10⁰−	2.51×10⁰−	7.10×10⁰−	2.21×10⁰−	1.51×10⁰
		100	(6.11×10⁻¹)	(7.66×10⁻¹)	(1.32×10⁰)	(2.55×10⁰)	(4.34×10⁻¹)	(2.07×10⁻¹)
	3	10	5.46×10⁻¹−	6.23×10⁰−	9.26×10⁰−	1.40×10⁰−	5.63×10⁻¹−	2.69×10⁻¹
		10	(6.23×10⁻¹)	(5.87×10⁻¹)	(1.01×10⁰)	(6.98×10⁻¹)	(1.21×10⁻¹)	(2.60×10⁻¹)
		50	3.82×10⁰−	6.31×10⁰−	1.26×10¹−	8.84×10⁰−	2.11×10⁰−	1.38×10⁰
		50	(8.01×10⁻¹)	(3.78×10⁻¹)	(9.94×10⁻¹)	(1.44×10⁰)	(3.00×10⁻¹)	(1.82×10⁻¹)
		100	4.55×10⁰−	1.27×10¹−	3.23×10¹−	1.18×10¹−	3.21×10⁰−	2.79×10⁰
		100	(3.97×10⁻¹)	(3.88×10⁻¹)	(4.42×10⁻¹)	(1.02×10⁰)	(5.23×10⁻¹)	(3.62×10⁻¹)
+/≈/−			0/0/12	0/0/12	0/0/12	0/0/12	2/0/10

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表 6 DDEA-SG和DDEA-PSG在决策变量维度为10, 50和100的Ellipsoid, Rastrigin, 以及2目标和3目标MaF1和MaF7上, 30次独立运行的运行时间平均值和标准差

Table 6 The mean and standard deviation of run time of DDEA-SG and DDEA-PSG over 30 independent runs on Ellipsoid, Rastrigin, 2-objctive/3-objective MaF1 and MaF7 of 10, 50 and 100 dimensions

问题	目标数	维度	DDEA-SG	DDEA-PSG
Ellipsoid	1	10	7.65×10¹−	4.55×10¹
		10	(2.50×10⁰)	(1.30×10⁰)
		50	1.20×10²−	7.44×10¹
		50	(7.73×10⁻¹)	(6.20×10⁻¹)
		100	2.22×10²−	1.39×10²
		100	(2.83×10⁰)	(8.23×10⁻¹)
Rastrigin	1	10	7.70×10¹−	4.55×10¹
		10	(2.05×10⁰)	(1.21×10⁰)
		50	1.22×10²−	7.39×10¹
		50	(2.78×10⁰)	(9.38×10⁻¹)
		100	2.24×10²−	1.41×10²
		100	(2.65×10⁰)	(3.72×10⁻¹)
MaF1	2	10	8.77×10¹−	5.34×10¹
		10	(2.37×10⁻¹)	(1.45×10⁻¹)
		50	1.51×10²−	9.22×10¹
		50	(1.11×10⁰)	(6.76×10⁻¹)
		100	7.39×10²−	4.51×10²
		100	(1.35×10¹)	(8.24×10⁰)
	3	10	1.05×10²−	6.43×10¹
		10	(2.68×10⁻¹)	(1.63×10⁻¹)
		50	2.05×10²−	1.25×10²
		50	(1.19×10¹)	(7.25×10⁰)
		100	1.14×10³−	6.94×10²
		100	(5.85×10¹)	(3.57×10¹)
MaF7	2	10	8.95×10¹−	5.46×10¹
		10	(1.50×10⁻¹)	(9.15×10⁻²)
		50	1.55×10²−	9.46×10¹
		50	(7.64×10⁰)	(4.66×10⁰)
		100	7.75×10²−	4.73×10²
		100	(2.33×10¹)	(1.42×10¹)
	3	10	1.03×10²−	6.31×10¹
		10	(4.83×10⁻¹)	(2.94×10⁻¹)
		50	2.10×10²−	1.28×10²
		50	(1.24×10¹)	(7.58×10⁰)
		100	1.13×10³−	6.89×10²
		100	(5.25×10¹)	(3.20×10¹)
+/≈/−			0/0/18

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表 7 不同训练数据场景下, DDEA-PSG在Ellipsoid和Rastrigin上, 决策变量维度分别为10, 50和100, 30次独立运行的最优解的平均值和标准差

Table 7 The mean and standard deviation of optimum solution of DDEA-PSG over 30 independent runs on different offline data of Ellipsoid and Rastrigin of 10, 50 and 100 dimensions

问题	维度	5$ d $ rand	11$ d $ rand	5$ d $ LHS	11$ d $ LHS
Ellipsoid	10	4.36×10⁰−	6.86×10⁻⁵−	1.20×10⁰−	4.67×10⁻⁵
	10	(2.56×10⁰)	(5.32×10⁻⁵)	(1.33×10⁰)	(5.41×10⁻⁵)
	50	4.19×10¹−	1.78×10¹−	5.73×10⁰−	2.38×10⁰
	50	(7.79×10⁰)	(3.65×10⁰)	(1.93×10⁰)	(2.20×10⁰)
	100	1.35×10²−	7.82×10¹−	4.97×10¹−	3.71×10¹
	100	(2.60×10¹)	(1.17×10¹)	(1.32×10¹)	(9.81×10⁰)
Rastrigin	10	9.93×10¹−	8.00×10¹−	7.01×10¹−	4.12×10¹
	10	(2.26×10¹)	(2.07×10¹)	(2.99×10¹)	(2.64×10¹)
	50	2.50×10²−	1.45×10²−	6.14×10¹−	3.85×10¹
	50	(3.56×10¹)	(2.45×10¹)	(1.96×10¹)	(1.64×10¹)
	100	4.26×10²−	2.56×10²−	1.75×10²−	1.39×10²
	100	(5.55×10¹)	(3.59×10¹)	(3.43×10¹)	(3.06×10¹)
+/≈/−		0/0/6	0/0/6	0/0/6

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表 8 不同训练数据场景下, DDEA-PSG在决策变量维度分别为10, 50和100, 2目标和3目标MaF1和MaF7上, 30次独立运行的IGD平均值和标准差

Table 8 The mean and standard deviation of IGD of DDEA-PSG over 30 independent runs on different offline data of 2-objective/3-objective MaF1 and MaF7 of 10, 50 and 100 dimensions

问题	目标数	维度	5$ d $ rand	11$ d $ rand	5$ d $ LHS	11$ d $ LHS
MaF1	2	10	5.95×10⁻²−	3.12×10⁻²−	4.37×10⁻²−	4.11×10⁻²
		10	(3.18×10⁻²)	(1.43×10⁻²)	(2.32×10⁻²)	(4.09×10⁻²)
		50	2.43×10⁻¹−	1.77×10⁻¹≈	1.81×10⁻¹−	1.71×10⁻¹
		50	(2.11×10⁻²)	(2.98×10⁻²)	(5.34×10⁻²)	(3.97×10⁻²)
		100	5.28×10⁻¹−	4.53×10⁻¹≈	5.74×10⁻¹−	4.43×10⁻¹
		100	(6.93×10⁻²)	(5.16×10⁻²)	(1.76×10⁻¹)	(1.04×10⁻¹)
	3	10	1.09×10⁻¹−	9.66×10⁻²−	1.23×10⁻¹−	7.86×10⁻²
		10	(1.53×10⁻²)	(9.51×10⁻³)	(3.45×10⁻²)	(1.01×10⁻²)
		50	3.38×10⁻¹−	2.82×10⁻¹−	3.23×10⁻¹−	2.34×10⁻¹
		50	(3.36×10⁻²)	(2.32×10⁻²)	(3.23×10⁻²)	(2.11×10⁻²)
		100	8.22×10⁻¹−	6.87×10⁻¹−	7.34×10⁻¹−	6.25×10⁻¹
		100	(1.02×10⁻¹)	(4.39×10⁻²)	(8.14×10⁻²)	(5.62×10⁻²)
MaF7	2	10	2.26×10⁻²−	3.63×10⁻¹−	1.92×10⁻¹−	1.56×10⁻²
		10	(2.24×10⁻³)	(7.56×10⁻¹)	(3.71×10⁻¹)	(1.23×10⁻³)
		50	1.05×10⁰−	9.65×10⁻¹−	1.56×10⁰−	8.91×10⁻¹
		50	(1.81×10⁻¹)	(4.42×10⁻¹)	(1.65×10⁰)	(3.14×10⁻¹)
		100	1.72×10⁰−	1.66×10⁰−	1.92×10⁻¹−	1.51×10⁰
		100	(1.74×10⁻¹)	(1.78×10⁻¹)	(3.71×10⁻¹)	(2.07×10⁻¹)
	3	10	4.49×10⁻¹−	2.89×10⁻¹−	3.85×10⁻¹−	2.69×10⁻¹
		10	(6.60×10⁻²)	(3.28×10⁻²)	(7.30×10⁻²)	(2.60×10⁻¹)
		50	3.05×10⁰−	2.40×10⁰−	2.73×10⁰−	1.38×10⁰
		50	(7.24×10⁻¹)	(1.96×10⁻¹)	(1.54×10⁻¹)	(1.82×10⁻¹)
		100	2.92×10⁰−	2.85×10⁰≈	2.90×10⁰−	2.79×10⁰
		100	(2.06×10⁻¹)	(2.93×10⁻¹)	(4.27×10⁻¹)	(3.62×10⁻¹)
+/≈/−			0/0/12	0/3/9	0/0/12

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