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An artificial bee colony algorithm with the feasibility rule for portfolio investment optimizations
LIU Yongbo
Department of Information Engineering,Luzhou Vocational and Technical College,Luzhou 646005,China
Abstract: This current work was carried out to approach the portfolio investment optimization problem by using an artificial bee colony(ABC)algorithm,in order to provide references for related researches. A constrained optimization model was constructed to formulate the portfolio investment optimization problem concerning securities subject to transaction fees and risk preferences of investors. This study employs feasibility rules to handle the constrained conditions of the optimization problem and forms an ABC algorithm with the feasibility rule(FRABC). It has been concluded by means of the Markov chain theory that the developed FRABC algorithm is globally convergent. A realistic case of the portfolio investment optimization is given to show that this method is valid and feasible,and the results are better than the ones obtained by the adaptive genetic algorithm(AGA). The proposed FRABC algorithm performs better,in terms of the final results,than the compared algorithms such as the genetic algorithm,particle swarm optimization algorithm and the basic ABC algorithm with the feasibility rule,under the assumed condition that the computational costs for the two algorithms are the same.
Key words: portfolio investment     constrained optimization     artificial bee colony algorithm     feasibility rule     Markov chain

1 投资组合优化数学模型

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2 约束优化的人工蜂群算法 2.1 二级标题约束优化的可行性规则

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2.2 可行性规则人工蜂群算法

1)初始化种群。按式(7)随机生成NL个候选解，并置采蜜时刻t=0。

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2)引领蜂采蜜。采蜜至第t时刻，每只引领蜂均在其当前蜜源的邻域内搜索，按式(8)生成新蜜源：

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NL只引领蜂(k=1,2,…,NL)，执行NL次上述邻域搜索。

3)跟随蜂采蜜。跟随蜂的邻域搜索与蜜源更新方式与引领蜂一致。采蜜至第t时刻，每只跟随蜂均应用轮赌法选择被跟随的引领蜂。第k只引领蜂x(t+1)k被选中的概率为

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NF只跟随蜂(k=1,2,…,NF)，执行NF次上述邻域搜索。

4)侦察蜂采蜜。当δtk达到预设阈值Δt时，该蜜源对应的引领蜂转变为侦察蜂，应用式(7)重新随机产生新蜜源，并置δtk=0。

5)更新最优蜜源。应用可行性规则确定新一代蜂群中的最优蜜源x(t+1)best

6)终止判断。若满足终止条件，则输出最优蜜源x(t+1)best及其相应目标函数值F(x(t+1)best)；否则，置t=t+1，返回2)。

2.3 计算复杂度分析

2)的计算复杂度为：引领蜂执行邻域搜索生成新蜜源：O(N×n/2)，评价新蜜源：O(N/2)，更新蜜源：O(N/2)，更新计数器δtkO(N/2)。3)的计算复杂度为：计算引领蜂被选择的概率：O(N/2)，选择被跟随的引领蜂：O(N/2)，跟随蜂执行邻域搜索生成新蜜源：O(N×n/2)，评价新蜜源：O(N/2)，更新蜜源：O(N/2)，更新计数器δtkO(N/2)。5)的计算复杂度为：O(N/2)，因此步存在不确定性，可忽略其复杂度5)的计算复杂度为：确定每代的最优蜜源：O(N/2)，更新最优蜜源：O(1)。略去上述各步中的低阶项，则FRABC算法的计算复杂度为O(tmax×N×n)，为立方阶复杂度[17]

3 算法收敛性分析

1)引领蜂执行邻域搜索的一步转移概率。

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x(t)ky执行交叉操作，生成的概率为

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x(t)k执行贪婪选择，得到x’的概率为

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2)跟随蜂执行领域搜索的一步转移概率。

3)侦察蜂执行随机搜索的一步转移概率。

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1)当X1BX2B时，Pr{TX1=X2}>0，Pr{TX2=X1}>0，即X1X2可互通；

2)当X1BX2B时，Pr{TX1=X2}=0，Pr{TX2=X1}>0，即X1不能通向X2

4 算例与讨论

R=[0.01675  0.00859  0.05146  0.04227  0.09462]T

5支股票的风险方差方阵为

 风险偏好因子w 算法 股票交易量x/手 总投资金额C/元 总收益率f/% 风险率g/% 加权值F/% x1 x2 x3 x4 x5 0.0 AGA 119 81 2 67 25 99 912 2.260 8 0.609 1 -0.609 1 FRABC 36 162 0 92 0 99 816 1.766 1 0.254 6 -0.254 6 0.1 AGA 22 114 63 51 68 99 987 3.441 6 1.068 7 -0.617 6 FRABC 30 162 0 100 0 99 804 1.822 9 0.259 1 -0.050 9 0.2 AGA 16 99 100 35 69 99 936 3.807 5 1.307 6 -0.284 6 FRABC 0 162 39 95 0 99 807 2.235 8 0.341 5 0.174 0 0.3 AGA 17 76 125 32 73 99 927 4.205 2 1.671 8 0.091 3 FRABC 0 136 82 80 0 99 966 2.694 0 0.506 4 0.453 7 0.4 AGA 4 81 131 30 81 99 951 4.371 8 1.791 7 0.673 7 FRABC 0 91 153 57 0 99 957 3.471 4 0.933 0 0.828 7 0.5 AGA 15 70 131 28 83 99 873 4.427 2 1.904 2 1.261 5 FRABC 0 67 184 21 42 99 834 4.323 4 1.628 6 1.347 4 0.6 AGA 14 39 194 15 59 99 858 4.745 3 2.160 5 1.983 0 FRABC 0 46 184 0 108 99 960 5.313 2 2.843 4 2.050 6 0.7 AGA 9 42 185 8 86 99 837 5.025 3 2.508 7 2.765 1 FRABC 0 0 184 0 189 99 858 6.786 5 5.630 7 3.061 3 0.8 AGA 4 33 117 9 216 99 945 6.408 9 5.308 3 4.065 5 FRABC 0 0 122 0 286 99 954 7.664 4 8.388 6 4.453 8 0.9 AGA 5 10 109 35 232 99 843 6.841 9 6.583 9 5.499 4 FRABC 0 0 122 0 286 99 954 7.664 4 8.388 6 6.059 1 1.0 AGA 2 20 70 33 283 99 822 7.191 1 7.831 1 7.191 1 FRABC 0 0 122 0 286 99 954 7.664 4 8.388 6 7.664 4 注:表中AGA的计算结果引自文献[2]

w为0.4,0.5,0.6时，5种算法分别独立运行50次，平均最优目标函数值进化曲线如图 1~3所示，优化性能指标见表 2。其中，FbFwFav分别表示最好、最差和平均最优目标函数值，表示最优目标函数值的标准方差。由表 2可知，对于每个w，FRABC算法除指标Fb与FRPSO算法相当外，其他指标均好于4种对比算法。与FRPSO算法相比，FRBABC算法的前期收敛速度较快，但后期的精细搜索能力较差。而FRABC算法既具有较快收敛速度、又具有较强精细搜索能力，表明其蜂群寻找新蜜源时，同时改变原蜜源的多维分量以扩大搜索范围，可明显提高优化性能。同时，FRABC算法的FbFwFav较接近，很小，表明该算法具有良好稳健性。

 图 1 5种算法的平均进化曲线(w=0.4) Fig. 1 The average evolution curve of five algorithms(w=0.4)
 图 2 5种算法的平均进化曲线(w=0.5) Fig. 2 The average evolution curve of five algorithms(w=0.5)
 图 3 5种算法的平均进化曲线(w=0.6) Fig. 3 The average evolution curve of five algorithms(w=0.6)

 风险偏好因子w 算法 最好值Fb 最差值Fw 平均值Fav 标准方差σF2 0.4 FRGA1 7.939×10-3 6.147×10-3 7.098×10-3 3.427×10-4 FRGA2 8.009×10-3 6.699×10-3 7.176×10-3 2.672×10-4 FRPSO 8.287×10-3 6.962×10-3 8.096×10-3 2.715×10-4 FRBABC 8.087×10-3 7.263×10-3 7.761×10-3 2.178×10-4 FRABC 8.287×10-3 8.270×10-3 8.283×10-3 6.931×10-6 0.5 FRGA1 1.304×10-2 1.131×10-2 1.208×10-2 3.995×10-4 FRGA2 1.271×10-2 1.153×10-2 1.209×10-2 2.849×10-4 FRPSO 1.347×10-2 1.192×10-2 1.325×10-2 2.776×10-4 FRBABC 1.324×10-2 1.219×10-2 1.271×10-2 2.623×10-4 FRABC 1.347×10-2 1.345×10-2 1.347×10-2 4.575×10-6 0.6 FRGA1 1.923×10-2 1.718×10-2 1.820×10-2 4.555×10-4 FRGA2 2.011×10-2 1.785×10-2 1.851×10-2 4.195×10-4 FRPSO 2.050×10-2 1.833×10-2 2.012×10-2 4.318×10-4 FRBABC 2.013×10-2 1.841×10-2 1.932×10-2 3.605×10-4 FRABC 2.051×10-2 2.031×10-2 2.049×10-2 3.462×10-5

5 结论

1)给出包含交易费用基于投资者风险偏好的最佳证券投资组合模型；针对其约束条件，定义了约束违反度函数，进而引入求解约束优化问题的可行性规则。

2)新型智能算法ABC在求解非线性优化问题中具有很强的全局寻优能力和很好的实用性，本文应用ABC算法求解最佳证券投资组合模型，形成面向该类问题的FRABC算法。

3) FRABC算法的计算复杂度为立方阶复杂度，与基本算法FRBABC一致。还应用Markov链分析其种群状态的一步转移概率，证明了该算法的全局收敛性。

4)应用MATLAB编写FRABC算法的计算程序，通过实例验证了该算法具有很强的寻优性能和良好的稳健性，且结果优于AGA。在相同函数评价次数的条件下，FRABC算法的各项优化指标均好于FRGA、FRPSO和FRBABC等对比算法，表明了本文方法的有效性和实用性。

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DOI: 10.3969/j.issn.1673-4785.201308047

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#### 文章信息

LIU Yongbo

An artificial bee colony algorithm with the feasibility rule for portfolio investment optimizations

CAAI Transactions on Intelligent Systems, 2014, 9(2): 491-498
http://dx.doi.org/10.3969/j.issn.1673-4785.201308047