步骤1 随机产生N个个体形成初始种群P0,然后对种群进行非劣排序,再对种群执行二元锦标赛选择、交叉和变异操作,得到新的种群Q0,令t=0。
步骤2 形成新的群体Rt=Pt∪Qt,对种群Rt进行非劣排序,得到非劣前段F1,F2,…。
步骤3 对所有Fi按拥挤比较操作进行排序,并选择其中最好的N个个体形成种群Pt+1。
步骤4 对种群Pt+1执行复制、交叉和变异操作,形成种群Qt+1。
步骤5 如果满足终止条件,则结束;否则,t=t+1,转到步骤2。
由上述算法步骤,结合实际问题和本文提出的模型,就可以求出投资者对各资产的最优投资权重。
4. 实证研究
4.1 数据选取及预处理
随机选取深圳证券交易所的n=5支股票:四川九州(000801)、川大智胜(002253)、青岛双星(000599)、经纬纺机(000666)、宜科科技(0020360),数据选取从2010年8月4日至2011年10月13日交易日的原始日收盘价,收益率取每日收盘价的对数,即用ln(pt /pt+1)来表示t+1日的收益率,得到259个观测数据,即T=259。
假设资金量b=100000元,手续费fi=5,佣金率ki=0.2%,li=5,i=1,2,…,n,深圳证券交易所不收取过户费,此时gi=0,mi=0。设种群规模N=50,算法迭代次数D=100,交叉概率pc=0.9,变异概率pm=0.1。4.2 计算结果及分析说明模型(II)的Pareto解比较密集,解的收敛性比较好;同时,在相同的风险水平下,模型(I)的收益率较小,这说明不考虑交易费用将会造成组合价值的损失。模型(II)收益/风险的最大值与均值都较大,这一点与图1相吻合,因此模型中考虑交易费用具有一定的实际价值。
在考虑交易费用的条件下,我们用NSGA-II算法求解风险函数为VaR时的投资组合优化模型,并与均值-BVaR模型的结果对比。
(1)在收益率一定的情况下,BVaR总是大于VaR,这一点与文献[8]的结论也是一致的,因为许多实证研究都认为金融资产收益率的分布是尖峰厚尾的,用正态分布计算其VaR会低估组合所面临的风险,可以说明BVaR虽比VaR保守,但却能更准确地反映市场实际的风险状况。
(2)均值-VaR模型得到的收益率数值较大,但同时其波动也较大,因VaR低估了实际面临的风险,故基于此风险度量指标的模型得到的收益也不准确;均值-BVaR模型得到的收益率虽小,但是波动也小,能够反映实际的收益情况。
5. 结论
本文改进了传统的均值-VaR投资组合模型,依据贝叶斯风险价值,建立新的多目标投资组合选择模型,引入交易费用这一现实因素,并给出求解该模型的算法步骤,最后利用我国金融市场的历史数据进行实证分析,结果表明,算法是有效的,模型中若不考虑交易费用将会影响投资者做出正确的决策。本文提出的模型更能反映实际交易市场数据的波动,对投资者进行组合投资具有一定的参考价值和实际意义。
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1王艳彩,女,2009级应用数学专业。
2高岳林,男,博士,教授,博士研究生导师。
Optimality for Multi-objective Programming involving
Arcwise Connected d-Type-I Functions
Yu Guolin1,Wang Min2
Abstract:This paper deals with the optimality conditions and dual theory of multi-objective programming problems involving generalized convexity.New classes of generalized type-I functions are introduced for arcwise connected functions,and examples are given to show the existence of these functions.By utilizing the new concepts,several sufficient optimality conditions and Mond-Weir type duality results are proposed for non-differentiable multi-objective programming problem.
Key words:Multi-objective programming,Pareto efficient solution,Arcwise connected d-type-I functions,Optimality conditions,Duality
摘要:这篇文章介绍了含有广义凸的多目标规划问题的最优性条件和对偶理论:结合弧连通函数,我们得出了一类新的广义型-I 函数,并给出例子表明了这些函数的存在性。利用新的概念,我们给出了非可微多目标优化问题的最优性条件和对偶理论。
关键词:多目标规划;Pareto有效解;弧连通d-型I 函数;最优性条件;对偶性
1.Introduction
Investigation on sufficiency and duality has been one of the most attraction topics in the theory of multi-objective problems.It is well known that the concept of convexity and its various generalizations play an important role in deriving sufficient optimality conditions and duality results for multi-objective programming problems.The concept of type-I functions was first introduced by Hanson and Mond [1]as a generalization of convexity.With and without differentiability,the type-I functions were extended to several classes of generalized type-I functions by many researchers,and sufficient optimality criteria and duality results are established for multi-objective programming (vector optimization)problems involving these functions (see [1-12]).Another meaningful generalization of convex functions is the introduction of arcwise connected functions,which was given by Avriel and Zang [13].Singh [14]and Mukherjee and Yadav [15]discussed some properties of arcwise connected sets and functions.Bhatia and Mehara [16]investigated some properties of arcwise connected functions in terms of their directional derivatives and established optimality conditions for scalar-valued nonlinear programming problems involving these functions.Mehar and Bhatia [17]and Davar and Mehra [18]studied optimality conditions and duality results for minmax problems and fractional programming problems involving arcwise connected and generalized arcwise connected functions,respectively.