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摘要: 代理模型能够辅助进化算法在计算资源有限的情况下加快找到问题的最优解集, 因此建立高效的代理模型辅助多目标进化搜索逐渐受到了重视. 然而随着目标数量的增加, 对每个目标分别建立高斯过程模型时个体整体估值的不确定度会随之增加. 因此通过对模型最优解集的搜索探索原问题潜在的非支配解集, 并基于个体的收敛性, 种群的多样性和估值的不确定度, 提出了一种新的期望提高计算方法, 用于辅助从潜在的非支配解集中选择使用真实目标函数计算的个体, 从而更新代理模型, 能够在有限的计算资源下更有效地辅助优化算法找到好的非支配解集. 在7个DTLZ 基准测试问题上的实验对比结果表明, 该算法在求解计算费时高维多目标优化问题上是有效的, 且具有较强的竞争力.Abstract: Surrogate models have attracted increasing attention in assisting evolutionary many-objective optimization when the computational budget is limited since a surrogate model can assist to accelerate the search for a set of Pareto solutions. However, the approximation uncertainty of an individual on the objective approximated values will be increased when the number of objectives increases in the case that a surrogate model is trained for each objective. Therefore, we propose to explore the potential non-dominated solutions of the original optimization problem by searching for the optimal solutions of the surrogate models, and a new expected improvement in this paper, which takes into account on the convergence of the solution, the diversity of the population, and the uncertainty of the approximation, for assisting selecting solutions from the potential non-dominated solutions to be evaluated using the exact expensive objective function. The surrogate model will be updated using the new evaluated solutions and is expected to assist the optimization algorithms to efficiently find a good set of non-dominated solutions within a limited computational budget. The experimental results on seven DTLZ test problems show that our proposed method is efficient and competitive to solve expensive many-objective problems.
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图 1 不同模型评价次数下算法的性能结果对比图
Fig. 1 Performance comparison of the proposed method with different number of evaluations on surrogate model
图 2 不同算法在DTLZ1上的性能结果对比
Fig. 2 Performance comparison of different methods on three-objective DTLZ1 problem
表 1 SAExp-EMO和ParEGO在3个和4个目标函数的DTLZ测试问题上获得的平均IGD统计结果
Table 1 Average IGD statistical results of SAExp-EMO and ParEGO on DTLZ test problems of 3 and 4 objective functions
测试问题 目标数 ParEGO SAExp-EMO DTLZ1 3 $4.84\times 10^{1}(8.51\times 10^{0})$− $\bf{1.09\times10^{1}(4.16\times 10^{0})}$ 4 $5.49\times10^{1}(1.09\times10^{1})$− $\bf{1.25\times10^{1}(5.06\times10^{1})}$ DTLZ2 3 $4.76\times10^{-1}(3.63\times10^{-2})$− $\bf{1.72\times10^{-1}(4.60\times10^{-2})} $ 4 $5.77\times10^{-1}(2.98\times10^{-2})$− $\bf{3.65\times10^{-1}(4.40\times10^{-1})}$ DTLZ3 3 $4.61\times10^{0}(5.38\times10^{1})$− $\bf{1.65\times10^{2}(6.07\times10^{1})} $ 4 $4.45\times10^{2}(7.57\times10^{1})$− $\bf{2.40\times10^{2}(1.19\times10^{2})}$ DTLZ4 3 $7.80\times10^{-1}(7.38\times10^{-2})$$\approx$ $\bf{5.59\times10^{-1}(4.80\times10^{-2})}$ 4 $8.92\times10^{-1}(9.00\times10^{-1})$− $\bf{7.12\times10^{-1}(1.48\times10^{\rm{-}1})} $ DTLZ5 3 $3.74\times10^{-1}(7.78\times10^{-2})$− $\bf{4.01\times10^{-2}(7.00\times10^{-2})}$ 4 $4.09\times10^{-1}(4.52\times10^{-2})$− $\bf{7.80\times10^{-2}(0.00\times10^{0})}$ DTLZ6 3 $8.04\times10^{0}(2.44\times10^{-1})$− $\bf{3.63\times10^{0}(2.61\times10^{0})}$ 4 $8.16\times10^{0}(2.52\times10^{-1})$− $\bf{3.84\times10^{0}(4.61\times10^{-1})}$ DTLZ7 3 $7.28\times10^{0}(2.16\times10^{0})$− $\bf{7.70\times10^{-1}(1.35\times10^{-1})}$ 4 $ 1.11\times10^{1}(7.97\times10^{-1})$− $\bf{1.09\times10^{0}(3.18\times10^{-1})}$ +/−/≈ 0/13/1 
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表 2 SAExp-EMO、RVEA、K-RVEA和CSEA得到的平均IGD值
Table 2 Average IGD values obtained by SAExp-EMO, RVEA, K-RVEA and CSEA
测试问题 目标数 RVEA K-RVEA CSEA SAExp-EMO DTLZ1 3 $3.65\times10^{1}(1.10\times10^{1})$− $2.48\times10^{1}(8.56\times10^{0})$− $1.97\times10^{1}(5.82\times10^{0})$− $\bf{1.33\times10^{1}(4.53\times10^{0})} $ 4 $3.18\times10^{1}(1.03\times10^{1})$− $3.01\times10^{1}(1.18\times10^{1})$− $1.71\times10^{1}(5.31\times10^{0})$− $\bf{1.35\times10^{1}(5.03\times10^{0})} $ 6 $2.96\times10^{1}(8.16\times10^{0})$− $3.18\times10^{1}(6.94\times10^{0})$− $1.43\times10^{1}(6.68\times10^{0})$− $\bf{1.15\times10^{1}(6.29\times10^{0})}$ 8 $2.00\times10^{1}(9.31\times10^{0})$− $3.22\times10^{1}(1.12\times10^{1})$$\approx$ $1.44\times10^{1}(6.01\times10^{0})$− $\bf{1.17\times10^{1}(4.46\times10^{0})} $ 10 $2.15\times10^{1}(8.45\times10^{0})$− $2.48\times10^{1}(9.28\times10^{0})$− $1.45\times10^{1}(5.70\times10^{0})$− $\bf{1.28\times10^{1}(5.45\times10^{0})}$ DTLZ2 3 $4.09\times10^{-1}(3.22\times10^{-2})$− $2.66\times10^{-1}(4.88\times10^{-2})$− $2.69\times10^{-1}(1.13\times10^{-1})$− $\bf{1.38\times10^{-1}(6.13\times10^{-2} )}$ 4 $5.16\times10^{-1}(3.61\times10^{-2})$− $3.95\times10^{-1}(4.94\times10^{-2})$− $4.76\times10^{-1}(1.04\times10^{-1})$− $\bf{3.27\times10^{-1}(5.53\times10^{-2})}$ 6 $6.97\times10^{-1}(6.73\times10^{-2})$− $5.93\times10^{-1}(4.96\times10^{-2})$− $\bf{5.76\times10^{-1}(4.01\times10^{-2})}$$\approx$ ${6.15\times10^{-1}(4.93\times10^{-2})}$ 8 $7.93\times10^{-1}(3.69\times10^{-2})$− $6.54\times10^{-1}(4.95\times10^{-2})$− $7.57\times10^{-1}(3.52\times10^{-2})$− $\bf{5.45\times10^{-1}(2.42\times10^{-1})}$ 10 $9.54\times10^{-1}(5.16\times10^{-2})$− $7.36\times10^{-1}(4.59\times10^{-2})$− $8.44\times10^{-1}(5.65\times10^{-2})$− $\bf{6.08\times10^{-1}(3.09\times10^{-1})}$ DTLZ3 3 $4.18\times10^{2}(6.66\times10^{1})$− $3.38\times10^{2}(7.51\times10^{1})$− $2.12\times10^{2}(4.37\times10^{1})$− $\bf{1.13\times10^{2}(2.96\times10^{1})}$ 4 $4.17\times10^{2}(7.54\times10^{1})$− $3.56\times10^{2}(7.56\times10^{1})$− $2.17\times10^{2}(4.94\times10^{1})$− $\bf{1.26\times10^{2}(6.16\times10^{1})}$ 6 $3.85\times10^{2}(7.05\times10^{1})$− $3.45\times10^{2}(7.90\times10^{1})$− $2.09\times10^{2}(5.44\times10^{1})$− $\bf{1.46\times10^{2}(7.89\times10^{1})}$ 8 $3.57\times10^{2}(7.05\times10^{1})$− $3.38\times10^{2}(5.74\times10^{1})$− $2.08\times10^{2}(5.09\times10^{1})$− $\bf{1.49\times10^{2}(7.88\times10^{1} )}$ 10 $3.77\times10^{2}(1.02\times10^{2})$− $3.24\times10^{2}(7.92\times10^{1})$− $2.18\times10^{2}(5.85\times10^{1})$ $\approx$ $\bf{1.10\times10^{2}(2.87\times10^{1})}$ DTLZ4 3 $5.58\times10^{-1}(6.90\times10^{-2})$− $\bf{4.17\times10^{-1}(1.12\times10^{-1})}$ $\approx$ $7.22\times10^{-1}(1.53\times10^{-1})$− $4.81\times10^{-1}(1.40\times10^{-1})$ 4 $6.96\times10^{-1}(8.80\times10^{-2})$ $\approx$ $5.46\times10^{-1}(1.13\times10^{-1})$+ $\bf{5.43\times10^{-1}(1.02\times10^{-1})}$+ $6.64\times10^{-1}(1.45\times10^{-1})$ 6 $8.53\times10^{-1}(8.13\times10^{-2})$− $6.84\times10^{-1}(8.64\times10^{-2})$+ $\bf{5.74\times10^{-1}(1.01\times10^{-1})}$ $\approx$ $8.52\times10^{-1}(8.99\times10^{-2})$ 8 $9.32\times10^{-1}(7.75\times10^{-2})$− $8.34\times10^{-1}(9.14\times10^{-2})$+ $\bf{7.39\times10^{-1}(3.42\times10^{-2})}$+ $8.36\times10^{-1}(1.76\times10^{-1})$ 10 $1.03\times10^{0}(7.03\times10^{-2})$− $8.89\times10^{-1}(6.96\times10^{-2})$ $\approx$ $\bf{8.12\times10^{-1}(4.54\times10^{-2})}$+ $8.17\times10^{-1}(2.43\times10^{-1})$ DTLZ5 3 $3.45\times10^{-1}(4.41\times10^{-2})$− $1.81\times10^{-1}(4.44\times10^{-2})$− $1.46\times10^{-1}(4.29\times10^{-2})$− $\bf{3.84\times10^{-2}(8.64\times10^{-3})}$ 4 $3.79\times10^{-1}(7.42\times10^{-2})$− $1.90\times10^{-1}(3.12\times10^{-2})$− $2.00\times10^{-1}(4.29\times10^{-2})$− $\bf{6.98\times10^{-2}(1.43\times10^{-2})}$ 6 $4.28\times10^{-1}(6.52\times10^{-2})$− $2.29\times10^{-1}(3.40\times10^{-1})$ $\approx$ $2.17\times10^{-1}(7.87\times10^{-1})$− $\bf{1.30\times10^{-1}(4.04\times10^{-2})}$ 8 $4.26\times10^{-1}(5.83\times10^{-2})$− $2.19\times10^{-1}(4.87\times10^{-2})$− $2.43\times10^{-1}(6.08\times10^{-2})$− $\bf{8.31\times10^{-2}(2.32\times10^{-2})}$ 10 $4.06\times10^{-1}(1.02\times10^{-1})$− $2.23\times10^{-1}(5.87\times10^{-2})$ $\approx$ $2.54\times10^{-1}(5.35\times10^{-2})$− $\bf{9.97\times10^{-2}(4.07\times10^{-2})} $ DTLZ6 3 $7.94\times10^{0}(2.75\times10^{-1})$− $4.42\times10^{0}(5.40\times10^{-1})$− $4.54\times10^{0}(5.84\times10^{-1})$− $\bf{3.07\times10^{0}(7.11\times10^{-1})}$ 4 $8.02\times10^{0}(2.61\times10^{-1})$− $4.35\times10^{0}(4.66\times10^{-1})$− $6.99\times10^{0}(7.87\times10^{-1})$− $\bf{3.46\times10^{0}(4.82\times10^{-1})}$ 6 $8.19\times10^{0}(3.42\times10^{-1})$− $4.58\times10^{0}(7.79\times10^{-1})$− $7.11\times10^{0}(1.76\times10^{-1})$− $\bf{4.19\times10^{0}(6.93\times10^{-1})} $ 8 $8.18\times10^{0}(2.75\times10^{-1})$− $5.78\times10^{0}(4.49\times10^{-1})$− ${7.21\times10^{0}(5.28\times10^{-1})}$$\approx$ $\bf{3.60\times10^{0}(5.11\times10^{-1})}$ 10 $8.22\times10^{0}(4.07\times10^{-1})$− $6.32e\times10^{0}(6.35\times10^{-1})$ $\approx$ $7.44\times10^{0}(4.18\times10^{-1})$ $\approx$ $\bf{3.95\times10^{0}(1.16\times10^{0})}$ DTLZ7 3 $6.85\times10^{0}(7.29\times10^{-1})$− $1.15\times10^{0}(1.69\times10^{0})$− $4.02\times10^{0}(4.82\times10^{0})$− $\bf{5.36\times10^{-1}(2.26\times10^{-1})}$ 4 $8.81\times10^{0}(1.31\times10^{0})$− $2.14\times10^{0}(3.24\times10^{0})$ $\approx$ $7.54\times10^{0}(9.33\times10^{-1})$ $\approx$ $\bf{6.92\times10^{-1}(1.37\times10^{-1})}$ 6 $1.29\times10^{1}(1.44\times10^{0})$− $3.49\times10^{0}(2.76\times10^{0})$− $1.36\times10^{1}(1.65\times10^{0})$− $ \bf{1.13\times10^{0}(2.42\times10^{-1})}$ 8 $1.72\times10^{1}(2.18\times10^{0})$− $4.18\times10^{0}(2.18\times10^{0})$− $2.26\times10^{1}(2.27\times10^{0})$− $\bf{6.82\times10^{-1}(1.38\times10^{-1})}$ 10 $2.18\times10^{1}(3.56\times10^{0})$− $7.83\times10^{0}(3.32\times10^{0})$− $2.86\times10^{1}(2.30\times10^{0})$− $\bf{1.19\times10^{0}(5.68\times10^{-1})} $ +/−/≈ 0/34/1 3/25/7 3/26/6
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