Variable Division and Optimization for Constrained Multiobjective Portfolio Problems
aa r X i v : . [ c s . N E ] J a n Variable Division and Optimization for ConstrainedMultiobjective Portfolio Problems
Yi Chen, Aimin Zhou,
Senior Member, IEEE
Abstract —Variable division and optimization (D&O) is afrequently utilized algorithm design paradigm in EvolutionaryAlgorithms (EAs). A D&O EA divides a variable into partialvariables and then optimize them respectively. A complicatedproblem is thus divided into simple subtasks. For example, avariable of portfolio problem can be divided into two partialvariables, i.e. the selection of assets and the allocation of capital.Thereby, we optimize these two partial variables respectively.There is no formal discussion about how are the partial variablesiteratively optimized and why can it work for both single- andmulti-objective problems in D&O. In this paper, this gap isfilled. According to the discussion, an elitist selection methodfor partial variables in multiobjective problems is developed.Then this method is incorporated into the Decomposition-BasedMultiobjective Evolutionary Algorithm (D&O-MOEA/D). Withthe help of a mathematical programming optimizer, it is achievedon the constrained multiobjective portfolio problems. In the em-pirical study, D&O-MOEA/D is implemented for 20 instances andrecent Chinese stock markets. The results show the superiorityand versatility of D&O-MOEA/D on large-scale instances whilethe performance of it on small-scale problems is also not bad.The former targets convergence towards the Pareto front and thelatter helps promote diversity among the non-dominated solutionsduring the search process.
Index Terms —Variable division and optimization, multiob-jective optimization, mixed integer programming, constrainedportfolio optimization.
I. I
NTRODUCTION M AKING the complicated simple is one effective ideafrequently used in the realm of computer science [1,2]. Variable division and optimization (D&O), dividing thevariable into partial variables and then optimizing them, isnaturally inspired by this idea. It is not a new thing forEvolutionary Algorithms (EAs) since it has been frequentlyutilized in Cooperative Coevolution [3, 4], Neuroevolution [5]and constrained portfolio optimization [6]. The original prob-lems are simplified by dividing variables in these works. Onthe other hand, D&O is reminiscent of Dived-and-Conquer,however, they are different because the latter one emphasizessolving subproblems recursively [2].Formally, a minimization problem is given as min x F ( x ) ≡ min x I ,x II F ( x I , x II ) s.t x ∈ Ω , x I ∈ Ω I , x II ∈ Ω II , (1)where x is the decision variable, x I and x II are two partialvariables, and x = ( x I , x II ) . Ω ( · ) is the search space of each Y. Chen and A. Zhou are with Shanghai Key Laboratory of Multidi-mensional Information Processing, and the Department of Computer Scienceand Technology, East China Normal University, Shanghai 200241, China (e-mail: [email protected]; [email protected]). Correspond-ing Author: A. Zhou variable, and Ω I × Ω II = Ω . With D&O, one variable isdivided into partial variables and thereby they can be opti-mized separately, i.e. the right problem in Eq. (1). Thereafter,the divided problem is generally presented as two sequentialsubtasks arg min x I F ( x I , x II ) s.t x I ∈ Ω I , x II ∈ Ω II . (2) arg min x II F ( x II ; x I ) s.t x II ∈ Ω II . (3)After the division has been made, a major challenge inducedby the problem (2) should be tackled when x I is optimized inEAs. This is how to perform an elitist selection among partialvariables x I to optimize them iteratively? Because there is adimensionality mismatch between a partial variable x I and thevariable x of the original problem, making the evaluation ofthe original objective function not possible [7]. The fitness ofa partial variable x I should be assigned as the fitness of itslocal optimal function values. A set of these values are statedas F ( x I ) = min x II ∈ Ω II F ( x II ; x I ) . (4)In this manner, we will never miss the global optimal variableif every local optimum are checked. Concisely, for D&O basedEAs, we can get the a global optimal variable if each partialvariable is evaluated as the performance of its correspondinglocal optima. Figs. 1 and 2 illustrate this for single andmultiobjective problems respectively. Fig. 1 shows the variablespace of x I and x II . The local optimum of each x I is just onepoint, i.e. |F ( x I ) | = 1 . It is easy to determine x is the optimalsolution. Notwithstanding, it is not natural for multiobjectiveproblems. Fig. 2 shows the objective space of a biobjectiveproblem. The local optima of each x I consists of a local Paretofront (PF) where |F ( x I ) | ≥ . We are hindered from extendingthe advantages of D&O to multiobjective problems since nomethod exists for performing an elitist selection among localPFs.In the discussion above, methods, involving D&O, aremainly concentrated in Cooperative Coevolution, Neuroevo-lution, and constrained portfolio optimization. Concerning theways of getting the F ( x I ) , they can also be roughly classifiedas straightforward and approximate methods as the taxonomyin [8, 9]. The related literature will be briefly presented byfields and ways. Note that although D&O looks like bi-leveloptimization [10], they are two different models. BecauseD&O only involves one objective function contrary to bi-leveloptimization. Furthermore, most of the problems introducedbelow merely include one level optimization. Fig. 1: Simplified local optimal solution search for a single objective problem. (cid:0)✁✂✄
Fig. 2: Simplified local optimal solution search for a multiobjective problem.
First, in Cooperative Coevolution, algorithms are specialcases that frequently utilize D&O [3, 4]. For instance, the vari-ables are divided into partial variables (some groups), and eachpartial variable is taken as x I recursively. Then, the F ( x I ) isapproximated via incorporating x I and the partial variablesfrom other groups [11, 12]. Second, in Neurovelution, somemethods utilize EAs for the task of neural architecture search( x I ). Then backpropagation algorithms are implemented forthe training. These hybrid methods are straightforward in abroad definition, because they find the F ( x I ) directly [13, 14].The approximate F ( x I ) is also employed in this field, EAs areemployed to find approximate optimal connection weights ofsmall networks for the non-deep neural networks [15, 16].Nowadays, a new technique, supernet, is implemented toapproximate optimal connection weights efficiently for deepneural networks [17]. Third, in constrained portfolio optimiza-tion, a portfolio problem can be divided into a combinatorial( x I ) and continuous subtask when a cardinality constraint isinvolved. Some methods use EAs to address the combinationaltasks and thereby call exact mathematical approaches for thecontinuous problems. The straightforward F ( x I ) is adopted inthese hybrid methods [6, 18]. All of them can be considered asD&O-based EAs since they do divide the variable into partialvariables and then optimize them separately.In this work, regarding D&O, we propose an algorithmto perform the elitist selection for local PFs of partial vari-ables with a Decomposition-Based Multiobjective Evolution-ary Algorithm (D&O-MOEA/D). This algorithm is achievedon constrained portfolio problems, which are mixed-integerproblems, via calling a mathematical optimizer.In summary, the contributions of this paper are as follows: • For the first time, the concept of D&O is summarized for-mally. We point out that how should the elitist selectionfor partial variables perform. • According to the discussion about D&O, D&O-MOEA/Dis achieved. Conducted simulation experiments on 20instances demonstrate the superiority of D&O-MOEA/D.First, it keeps small gaps with the exact multiobjectivemethod on small-scale problems while it is more universalthan the exact method on large-scale problems. Second,it dominates the implemented EAs when all 20 problemsare considered. • Further, it is applied to recent Chinese stock markets,Shanghai and Shenzhen stock markets. The results con-firm the feasibility and the value in the application of theproposed method.The remainder of this paper is presented as follows. SectionII presents the formulation of the constrained multiobjectiveportfolio problem and emphasizes the emerging challenge forperforming elitist selection among local PFs. In Section III,the basic MOEA/D is introduced. Meanwhile, we discusshow is the division of the variable x made, and how is theoptimization of x I and x II performed. Then, empirical studiesare presented in Section IV. Finally, a conclusion of this workand promising topics constitute Section V.II. E LITIST S ELECTION FOR L OCAL PF S In this section, firstly, the formulation of the constrainedmultiobjective problem is presented. Then we introduce howto perform an elitist selection of the local PFs.
A. Basic Problem Formulation
Portfolio optimization has been instrumental in the de-velopment of financial markets [19]. As portfolio problemsdevelop in practical applications, EAs outperform conventionaloptimization algorithms since they can tackle complex con-straints [20, 21]. In this subsection, multiobjective portfolioproblems with cardinality constraints are introduced, becauseD&O is very suitable for them [6, 18].This constrained multiobjective portfolio problem, which isan extension Mean-Variance [22] model, is a biobjective prob-lem, finding a trade-off between return and risk. Moreover, itinvolves four constraints: (i) cardinality constraint, (ii) floorand ceiling constraint, (iii) pre-assignment constraint, and (iv)round lot constraint [23]. Let Γ be the set of available assets, Π be the set of pre-assigned assets. Define | Γ | = n , | Π | = L . Theparameter µ i and σ i are the expectation and standard deviationof return for asset i , i ∈ Γ . The parameter ρ ij is the correlationcoefficient of the assets i and j , i, j ∈ Γ , and σ ij = ρ ij σ i σ j ,where all σ ij constitute a covariance matrix (risk). Let ǫ i and υ i be the lower and upper bound, and τ be the round lotsize of the assets; z i be the binary parameter representing pre-assignment of the asset i . The problem formulation followsas min f = X i ∈ Γ X j ∈ Γ w i w j σ ij , (5) max f = X i ∈ Γ w i µ i , (6) s.t. X i ∈ Γ w i = 1 , (7) X i ∈ Γ s i = K, (8) ǫ i s i ≤ w i ≤ υ i s i , i ∈ Γ , (9) s i ≤ z i , i ∈ Γ , (10) w i = τ δ i , i ∈ Γ , δ i ∈ Z + , (11) s i ∈ { , } , i ∈ Γ , (12)where w = { w , . . . , w n } T is a portfolio vector, Eqs. (5) and(6) are two respective objectives, minimizing the risk and max-imizing the return, in the portfolio optimization that conflictwith each other. Eq. (7) requires that all the capital should beinvested in a valid portfolio. Eq. (8) is the cardinality constraint(i.e., just K assets are selected) and s = { s , . . . , s n } T isan indicator vector; s i = 1 if the asset i is selected, and s i = 0 otherwise. Eq. (9) is the floor and ceiling constraint.Besides, Eq. (10) represents that the asset i must be includedin a portfolio if z i = 1 . It is a pre-assignment constraint.Thereafter, Eq. (11) defines the round lot constraint and δ i isa multiplier. The round lot sizes τ is set to be a same constant,with which 1 is divisible, since it will be really hard to handlea flexible one. In that case, it will be beyond the scope of themain discussion in this work. Finally, Eq. (12), which is thediscrete constraint, implies that the s i must be binary.According to D&O, a portfolio w can be divided into x I and x II , where x I is the combination of selected assets, i.e.a selection vector s , and x II represents the correspondingweights. Thereby, x I and x II are optimized in the EA andan exact mathematical optimizer, Opt , respectively.
B. Elitist Selection of The Local PFs
As for the local PF of each x I , we find that only in two extreme cases that an elitist selection can be performedwith existing methods, and the discussion is presented in thesupplement. Moreover, it has been proved that the shape ofa local PF in the Mean-Variance model is a convex parabolaand it will be discontinuous when the round lot constraint isincluded [24]. Therefore, no method exists for performing anelitist selection for the local PFs in this problem.To our best knowledge, the only work, including a PF-based(envelope-based) algorithm, proposed by J. Branke et al [18]is one Dominance-based algorithm. Although this PF-basedalgorithm has some limitations, it encourages us to develop amore versatile one. Since it is hard to determine the dominanceamong local PFs, we try to conduct a decomposition approachfor them through exploiting MOEA/D [25]. MOEA/D is adecomposition-based algorithm, who uses aggregation func-tions to decompose the PF into a number of scalar objectivesubproblems [26]. After solving these subproblems, optimalsolutions will construct approximate PFs [27–30]. Severalmethods for constructing aggregation functions can be found in the literature [31]. Without loss of generality, the weightingmethod is employed. It is defined as min g ws ( x ; λ ) = m X i λ i f i ( x ) s.t x ∈ Ω , (13)where λ = ( λ , λ , . . . , λ m ) is a weight vector. Fig. 3illustrates how is the elitist selection for variables performedon a biobjective problem through the weighting method. Everysolution are assigned with different g ws according to differentweight vectors. For instance, regarding the weight vector λ , x is the best, on the contrary, x is the best according to λ .This aggregation function can also be used to evaluate thefitness of each x I , which reduces a local PF F ( x I ) to avariable x . In these cases, a x can be obtained through slightlyaltering the problem (13) and then solving it, the alteration isgiven as min x II g lws ( x II ; x I , λ ) = m X i λ i f i ( x I , x II ) s.t x II ∈ Ω II , (14)where the solution is x II . Consequently, we can evaluate x I by x = ( x I , x II ) . For example, in Fig. 4, x can be obtainedvia settling the problem (14) with the Opt for a partial variable x I and a weight vector λ . Then F ( x I ) is reduced to x and thereby x is the best concerning the weight vector λ ,denoting x I is the best. According to this elitist selectionmethod for local PFs, D&O-MOEA/D is proposed and detailedin the next section. III. D&O-MOEA/DIn this section, the framework of MOEA/D is introducedfirstly and then we develop it for the constrained portfolioproblems.MOEA/D is an iterative algorithm, where a populationof N solutions x , . . . , x N is optimized iteratively. Eachsolution x i corresponds to the subproblem i . Meanwhile,each subproblem i has a T -neighborhood structure, involvingclosest T subproblems according to the weight vector. Assumeeach subproblem is a minimization problem and the objectivefunction of subproblem i is g i ( x ) . The main loop of MOEA/Dworks as follows.For each subproblem i :
1) Mating:
Select some solutions from the T -neighborhoodof subproblem i and generate new solution x inew with theselected ones via reproduction operators.
2) Evaluating:
For each subproblem j in the T -neighborhood of subproblem i , evaluate g j ( x inew ) .
3) Replacing:
For each solution x j in the T -neighborhoodof subproblem i , replace x j with x inew if g j ( x inew ) < g j ( x j ) .We take MOEA/D-DE in [27] as the basic of D&O-MOEA/D. D&O-MOEA/D is achieved via searching the x I (combination of assets) with evolutionary method and opti-mizing x II (weights) with an exact mathematical optimizer, Opt . Specifically, x I is searched in the M atting and x II isoptimized in the Evaluating . (a) (b)Fig. 3: The weighting method for variables on a biobjective problem with two different weight vectors.(a) (b)Fig. 4: The weighting method for partial variables on a biobjective problem with two different weight vectors. Regarding x I , it is the combination of assets. Severalrepresentations and reproduction operators can be chosen for x I , our real-valued representation, CCS [23], is followed inthis work. Specifically, a real-valued n -dimensional vector, in [0 , n , is utilized. Assets which are pre-selected or correspondto the K − L highest values are selected, therefore, real-valuedvectors can represent combinations of selected assets, thecardinality and pre-assignment constraints are handled [32].As for the reproduction, it involves a combination of Dif-ferential Evolution [33, 34] and Polynomial Mutation [35]as in MOEA/D-DE and a swap operator as in [23]. Thecombination operator and the swap operator will occur with asame probability, i.e. 50% for each one.Concerning x II , it is the capital weight. After the x I is de-termined, we are going to settle the subtask, the problem (14)for x II . Specifically, when x I , i.e. variable s , is found by theEA, this subtask is specified as min g lws = λ X i ∈ Γ ′ X j ∈ Γ ′ w i w j σ ij − λ X i ∈ Γ ′ w i µ i s . t X i ∈ Γ ′ w i = 1 ,ǫ i ≤ w i ≤ υ i , i ∈ Γ ′ ,w i = τ δ i , i ∈ Γ ′ , δ i ∈ Z + , (15)where λ = ( λ , λ ) is the weight vector and the Γ ′ is aset of assets, where Γ ′ ⊂ Γ and | Γ ′ | = K . Thus it impliesEqs. (8), (10) and (12) are satisfied. The problem (15) isa quadratic problem with an integer constraint (round lotconstraint). Therefore, a quadratic optimizer of the commercialsolver CPLEX is taken as the Opt and a proved optimal solution of g i ( x ) can be obtained in a very short time, about30 milliseconds, with an ordinary personal computer.IV. E MPIRICAL S TUDIES
In the empirical studies, D&O-MOEA/D is compared withthree methods, two state-of-the-art MOEAs, and an exactmultiobjective method AUGMECON2&CPLEX, on 20 con-strained portfolio instances. In the field of optimization, bothexact methods and heuristics take important positions. Wewould like to analyze the different performances among theexact mathematical programming method, EAs, and the hybridone since they are supposed to have their advantages. Thissection is hence composed of two parts. First, the descriptionabout implementation details, including instances, a constraintset, employed algorithms, and parameter settings. Besides, theperformance metrics. Second, the performance analysis.
A. Implementation Details
All of the problems consist of instances from OR-Libraryand NGINX. The first one, OR-Library, contains 5 classicinstances, and the second one, NGINX, includes 15 instancesestablished by the historical stock data from Yahoo Financewebsite. The instances are sorted with the number of availableassets and are shown in Table I . Concerning the constraintset, it follows one set from [23] as cardinality K = 10 , floor ǫ = 0 . , ceiling υ = 1 . , pre-assignment z = 1 and roundlot τ = 0 . . All the data is available at https://github.com/CYLOL2019/Portfolio-Instances
TABLE I: Twenty Used Instances.
Instance Origin Name
D13 Australia All ordinaries 264D14 USA NASDAQ Bank 380D15 USA NASDQ Computer 417D16 USA S&P 500 469D17 Korea KOSPI Composite 562D18 USA NASDQ Industrial 808D19 USA AMEX Composite 1893D20 USA NASDAQ 2235
Three algorithms are implemented for comparison, they arelisted below .
1) MODEwAwL [36]:
It is a learning guided MOEA. Itrepresents w with binary and real-valued vectors. The searchof the binary vector, representing the combination, is guidedby a learning mechanism, and the search of the real-valuedvector, representing the weights, is mainly based on DE.
2) CCS/MOEA/D [23, 37]:
It incorporates a compressedcoding scheme into MOEA/D. It utilizes only one real-valuedvector to represent both the combination and weights, makinguse of the dependence among combinations and weights.
3) AUGMECON2&CPLEX [38, 39]:
An exact mathemat-ical multiobjective algorithm based on ε -constraint method(AUGMECON2) is employed. It is incorporated with aquadratic optimizer of CPLEX for the constrained multiob-jective portfolio problems and the number of grids is set asthe size of the population in the EAs.All of the implemented algorithms involve some parameters.There are common parameters, such as population size andparameters for common generating operators. In the meantime,there are also parameters that are independent of the algo-rithms, namely, parameters for the neighborhood in MOEA/Dand the number of grids in AUGMECON2. They are all shownin Table II. Stop criteria are non-trivial for these methods,however, they are always different [40]. Therefore a maximumfitness evaluation and a time budget are both used for thetermination detection. 20 times independent executions forevery method with two different stop criteria are carried outon a server computer. This server computer is constructed with2 64 cores AMD EPYC CPUs and 256GB memories.Furthermore, two performance metrics are applied, Hy-pervolume (HV) [41] and Generational Distance (GD) [42],both of them are easy to use in PlatEMO [43]. HV prefers MODEwAwL and CCS/MOEA/D are available athttps://github.com/CYLOL2019/SEC-CCS, and AUGMECON2&CPLEX,D&O-MOEA/D, and related files have been uploaded tohttps://github.com/CYLOL2019/D&O-MOEAD. TABLE II: The parameter settings for all algorithms.
Common parametersPopulation size N F CR η m p m N p Parameters for MOEA/DNeighborhood size T m n r p δ heuristics because it takes the diversity of solutions intoaccount. Heuristics, like EAs, always pay much attention to thediversity of solutions while exact methods hardly do it [31, 44].It is given as HV = volume( ∪ | Q | i =1 hc i ) , (16)where Q is the set of obtained solutions of algorithms, and hc i is the hypercube bounded by solution i and the referencepoint r . Better solutions lead to higher HV. On the other hand,GD evaluates the convergence ability of algorithms. In thisproblem, it tends to prefer exact mathematical programmingapproaches since exact methods are going to find the optimumif possible yet heuristics will search for approximations. It isdefined as GD = qP Qi d i | Q | , (17)where Q is the set of obtained solutions of algorithms and d i is the shortest Euclidean distance among solution i and therepresentatives of PF. Better solutions lead to lower GD.Note that the implementations of HV and GD requirethe true PFs or reference points. The PFs are obtained byimplementing AUGMECON2&CPLEX on the original modelwith 2000 grids. However, they are unavailable while runningthe exact method after 7 days for some large-scale problemsin this work. For these cases, the best known unconstrainedPareto fronts (UCPFs) are considered with the following mod-ification. They are truncated by the highest return solutions(since the function of return is linear, it is easy to be solved).This truncated version of the UCPF is referred to as TUCPF.TUCPFs are used for instances including more than 225 assets.Concerning the reference point, it is recommended to set r slightly dominated by the nadir point of the true PF or TUCPF,specifically, r is set as (1 . , . . B. Simulation Experiments
This subsection comprises three sets of simulation experi-ments. The first two are about two different stop criteria andthe third one is about applying the proposed method to thereal stock markets in China.We put the conclusion in advance since some readers areprobably concerned about the conclusion yet do not havethe patience to go through the analysis. Concisely, withmaximum fitness evaluations as 1000, D&O-MOEA/D shows prominent superiority. With a 2000 seconds time budget,AUGMECON2&CPLEX is the best method for small-scaleproblems while D&O-MOEA/D is the best method if all20 instances are involved. Furthermore, the application forChinese stock markets also demonstrates the feasibility andpotential of D&O-MOEA/D.
1) Maximum Number of Fitness Evaluations:
First, D&O-MOEA/D, MODEwAwL, and CCS-MOEA/D are imple-mented and 1000 times fitness evaluations are set as thestop criterion. For these two EAs, the function evaluation isthe fitness evaluation. Notwithstanding, for hybrid methodsabout EAs, sometimes function evaluations are called manytimes, but one time of fitness evaluation is counted whena final solution is obtained [14, 17, 45]. In this case, theresults obtained are shown in Table III. The rank of eachalgorithm on all instances is listed in parenthesis, and thetotal and final ranks of every algorithm are aggregated atthe bottom. Furthermore, the symbol “ + , − , ≈ ” indicates thecorresponding algorithm is significantly better than, worsethan and similar to D&O-MOEA/D in terms of Wilcoxonrank-sum test at 5% significant level. For brevity, ‘A’, ‘B’and ‘C’ represent the methods D&O-MOEA/D, MODEwAwLand CCS-MOEA/D respectively. From D to D , the PFsare obtained by the exact method. As for D to D , thesame method does not get results after 7 days. The demandedcomputational resources will exponentially grow when thenumber of assets increases since these problems are NP-hard [6]. TUCPFs are hence applied. Table III shows D&O-MOEA/D outperforms two EAs dramatically. It wins all thefirst places on 20 instances. Furthermore, the superiorityof D&O-MOEA/D is still significant regarding the obtainedsolutions in the objective spaces. Fig. 5 only illustrates theobtained fronts of every method on D due to the space limit.Solutions from D&O-MOEA/D almost lie on the PF while thesolutions of the other two are unsatisfactory. In this case, oneis reasonable to comment that although these methods takethe same times of fitness evaluations, D&O-MOEA/D takesmuch more computational resources in fact. More simulationexperiments are hence carried out.
2) Maximum Running Time:
Second, a time budget shouldbe an alternative fair stop criterion for these methods, althoughit is hardly used in the realm of EAs. As in [23, 36], maximumfitness evaluations for these two EAs are set as 10e5. There-fore, we reimplement the two algorithms independently 20times on all instances to estimate the required times. Further,AUGMECON2&CPLEX with 100 grids is also carried outwhile the time budget is 1 day. Fig. 6 shows the average timesof three methods on every problem. The running times of thetwo EAs are similar and most of them vary from 500 to 1000seconds. For the largest two instances, they are about 3000seconds. Regarding, the exact method, we can only get thesolutions for D to D in 1 day. 65.27 seconds is the medianof the running times among 12 problems. The running timeon D is 3136.59 seconds, it should be a special case that thedata of D slows down the convergence of the used quadraticoptimizer. In summary, a 2000 seconds execution time is set asthe stop criterion for the EAs and D&O-MOEA/D, with whichmost of them can converge. Then AUGMECON2&CPLEX is TABLE III: Results on D − D concerning HV with 1000 times of fitnessevaluations. Algo. A B C D Mean 8.03e-01[1] 5.67e-01[3] 7.00e-01[2]Std 9.42e-04 2.79e-02 5.60e-02 D Mean 8.93e-01[1] 6.90e-01[2] 6.24e-01[3]Std 3.23e-03 3.60e-02 1.32e-01 D Mean 8.05e-01[1] 6.71e-01[2] 5.23e-01[3]Std 2.50e-03 1.63e-02 1.15e-01 D Mean 9.20e-01[1] 8.24e-01[2] 6.63e-01[3]Std 5.33e-03 1.74e-02 1.09e-01 D Mean 8.49e-01[1] 6.88e-01[2] 5.68e-01[3]Std 4.05e-03 2.18e-02 1.06e-01 D Mean 8.58e-01[1] 7.74e-01[2] 5.79e-01[3]Std 2.40e-03 2.23e-02 8.89e-02 D Mean 7.60e-01[1] 4.70e-01[3] 6.09e-01[2]Std 4.80e-04 5.43e-02 1.31e-01 D Mean 8.23e-01[1] 6.45e-01[2] 5.87e-01[3]Std 3.86e-03 1.96e-02 1.11e-01 D Mean 8.24e-01[1] 5.91e-01[2] 3.94e-01[3]Std 2.27e-03 2.67e-02 9.57e-02 D Mean 8.90e-01[1] 8.12e-01[2] 4.06e-01[3]Std 8.52e-03 1.65e-02 8.80e-02 D Mean 8.03e-01[1] 6.26e-01[2] 4.19e-01[3]Std 2.86e-03 2.00e-02 8.64e-02 D Mean 8.36e-01[1] 8.07e-01[2] 3.40e-01[3]Std 9.14e-03 9.55e-03 1.09e-01 D Mean 8.65e-01[1] 7.11e-01[2] 5.60e-01[3]Std 4.27e-03 3.13e-02 1.24e-01 D Mean 8.28e-01[1] 6.49e-01[2] 5.09e-01[3]Std 1.47e-03 3.33e-02 1.30e-01 D Mean 8.07e-01[1] 6.41e-01[2] 4.25e-01[3]Std 2.54e-03 2.18e-02 8.48e-02 D Mean 9.04e-01[1] 8.63e-01[2] 5.76e-01[3]Std 4.83e-03 4.65e-03 7.51e-02 D Mean 7.65e-01[1] 5.36e-01[2] 4.35e-01[3]Std 1.56e-03 4.16e-02 1.33e-01 D Mean 8.63e-01[1] 7.13e-01[2] 5.44e-01[3]Std 2.37e-03 2.34e-02 7.59e-02 D Mean 7.29e-01[1] 4.48e-01[2] 3.63e-01[3]Std 4.03e-04 4.84e-02 1.69e-01 D Mean 8.57e-01[1] 6.89e-01[2] 4.86e-01[3]Std 3.42e-03 2.68e-02 4.48e-02Total ∗
20 42 58Final Rank ∗ + , − , ≈ - 0/20/0 0/20/0 completely executed on the first 12 instances.The statistics of HV and GD obtained by these four al-gorithms are shown in Tables IV and V. There are somedifferences from Tables IV and V to Table III. Firstly, AUG-MECON2&CPLEX is added and is represented by ‘D’. Twosets of statistical indicators such as ‘Total ∗ ’ and ‘Total’ areused, because the counts are made twice since AUGME-CON2&CPLEX with 100 grids can only be completed onproblems including no more 225 assets with the time budgetof 1 day. The statistical indicators with a star (*) indicatethe results based on 4 algorithms for 12 instances, and thelatter ones indicate the results based on 3 algorithms for 20instances. All the comparisons consist of two parts: (i) theperformance of all methods on D to D and (ii) the resultsof two EAs and D&O-MOEA/D for all problems.In terms of HV, Table IV illustrates AUGME-CON2&CPLEX is undoubtedly the best method for smallinstances, from D to D . It gets the smallest HVs on 11problems and has a very small gap with MODEwAwL on D , HVs of both of them are 7.65e-01. D&O-MOEA/Doutperforms two EAs slightly with regards to the totalranks on these 12 instances, which are 34, 38 and 35respectively. Concerning the Wilcoxon rank-sum test, thedominance of D&O-MOEA/D is also not obvious. Although Risk -3 R e t u r n -3 D PFICDV-MOEA/D (a)
Risk -3 R e t u r n -3 D PFMODEwAwL (b)
Risk -3 R e t u r n -3 D PFCCS/MOEA/D (c)Fig. 5: The efficient fronts with the best HVs obtained by D&O-MOEA/D, MODEwAwL and CCS/MOEA/D with 1000 times of fitness evaluations.Fig. 6: The implementation times for every methods on all problems. the statistic of MODEwAwL is ‘2/9/1’, it of CCS-MOEA/Dis ‘5/4/3’. On problems including more than 225 assets,AUGMECON2&CPLEX can not be completed in 1 day.Notwithstanding, concerning the Wilcoxon rank-sum test,the superiority of D&O-MOEA/D becomes prominent whenresults on all 20 instances are included. These statistics ofMODEwAwL and CCS-MOEA/D are ‘7/12/1’ and ‘6/10/4’.They indicate D&O-MOEA/D wins on more problemsthan the compared peer algorithms do. Fig 7 illustrates theconvergence curves of every method on D , D , D and D . D and D are solvable for AUGMECON2&CPLEX,so its HVs are also drawn. The HV lines start whenAUGMECON2&CPLEX finishes. In Figs. 7 (a) and 7 (b),the results imply AUGMECON2&CPLEX is the best choiceamong these four methods for small-scale problems, itconverges fast and well. Since the exact method is sufficient,so we do not need heuristics like D&O-MOEA/D and theEAs for small-scale problems. In Fig. 7 (c) and 7 (d), D&O-MOEA/D converges faster and the final HV of it is muchbetter than the others on D . It implies that D&O-MOEA/Dis probably a good method for large-scale problems.To be honest, we can not demonstrate that D&O-MOEA/Dis much better than two EAs just based on HVs. BecauseD&O-MOEA/D can not always obtain perfectly distributedsolutions since it involves conventional optimization methodsfor MOPs, which pay much less attention to the diversity of so-lutions than EAs [31, 38]. Furthermore, HV should not be theabsolutely correct indicator of portfolio optimization. Becausethe diversity of solutions on portfolio problems are not alwayssignificant. For example, in Figs. 8 (a)-(d), EAs get better TABLE IV: Results on D − D concerning HV with executions for 2000seconds. Algo. A B C D D Mean 8.06e-01[2] 8.03e-01[4] 8.06e-01[3]Std 1.63e-04 8.65e-04 6.41e-04 8.08e-01[1] D Mean 9.03e-01[2] 9.00e-01[3] 8.99e-01[4]Std 6.12e-04 3.23e-03 2.10e-03 9.05e-01[1] D Mean 8.16e-01[3] 8.14e-01[4] 8.18e-01[2]Std 1.41e-03 5.49e-04 9.05e-04 8.21e-01[1] D Mean 9.38e-01[2] 9.36e-01[3] 9.28e-01[4]Std 1.01e-03 8.20e-04 4.05e-03 9.39e-01[1] D Mean 8.63e-01[3] 8.60e-01[4] 8.65e-01[2]Std 1.49e-03 1.82e-03 1.45e-03 8.68e-01[1] D Mean 8.71e-01[3] 8.66e-01[4] 8.73e-01[2]Std 1.51e-03 3.30e-03 1.52e-03 8.75e-01[1] D Mean 7.62e-01[4] 7.65e-01[1] 7.63e-01[3]Std 4.95e-06 3.10e-04 1.31e-03 7.65e-01[2] D Mean 8.38e-01[3] 8.37e-01[4] 8.42e-01[2]Std 1.48e-03 8.14e-04 8.82e-04 8.44e-01[1] D Mean 8.33e-01[3] 8.33e-01[2] 8.32e-01[4]Std 1.33e-04 2.84e-04 1.72e-03 8.34e-01[1] D Mean 9.19e-01[2] 9.17e-01[3] 9.13e-01[4]Std 3.47e-03 1.76e-03 7.65e-03 9.25e-01[1] D Mean 8.17e-01[3] 8.16e-01[4] 8.19e-01[2]Std 9.44e-04 1.72e-03 3.64e-03 8.23e-01[1] D Mean 8.80e-01[4] 8.82e-01[2] 8.81e-01[3]Std 4.22e-03 5.18e-04 3.74e-03 8.87e-01[1]Total ∗
34 38 35 13Final Rank ∗ + , − , ≈ ∗ - 2/9/1 5/4/3 12/0/0 D Mean 8.79e-01[2] 8.80e-01[1] 8.74e-01[3]Std 6.40e-05 1.73e-04 2.63e-03 - D Mean 8.35e-01[2] 8.38e-01[1] 8.32e-01[3]Std 7.06e-05 2.41e-04 4.21e-03 - D Mean 8.23e-01[2] 8.22e-01[3] 8.24e-01[1]Std 7.81e-04 9.82e-04 3.85e-03 - D Mean 9.26e-01[3] 9.29e-01[1] 9.27e-01[2]Std 1.45e-03 1.32e-03 3.55e-03 - D Mean 7.70e-01[2] 7.70e-01[1] 7.64e-01[3]Std 4.64e-06 8.77e-04 3.54e-03 - D Mean 8.77e-01[1] 8.73e-01[2] 8.73e-01[3]Std 4.98e-04 5.36e-03 1.38e-03 - D Mean 7.30e-01[2] 7.34e-01[1] 7.26e-01[3]Std 4.11e-05 4.86e-04 2.95e-03 - D Mean 8.69e-01[1] 8.08e-01[3] 8.23e-01[2]Std 4.91e-04 3.44e-02 5.55e-02 -Total 37 40 43 -Final Rank 1 2 3 - + , − , ≈ - 7/12/1 6/10/4 - diversities when risk is more than 8e-4. Notwithstanding, arethose solutions important? They are perhaps not. Because thereturn increases from 3.5e-3 to 3.6e-3 while the risk doubles,from 8e-4 to 1.6e-3. It is irrational to take such a large risk witha little improvement of return. In our opinion, the employedperformance metric should not pay much attention to diversityon portfolio problems since they consist of objectives withpreference. GD is hence introduced when HV is not definitelysuitable sometimes. CPU Times H V D1 ICDV-MOEA/DMODEwAwLCCS/MOEA/DAUGMECON2&CPLEX (a)
200 400 600 800 1000 1200 1400 1600 1800 2000
CPU Times H V D12
ICDV-MOEA/DMODEwAwLCCS/MOEA/DAUGMECON2&CPLEX (b)
200 400 600 800 1000 1200 1400 1600 1800 2000
CPU Times H V D19
ICDV-MOEA/DMODEwAwLCCS/MOEA/D (c)
200 400 600 800 1000 1200 1400 1600 1800 2000
CPU Times H V D20
ICDV-MOEA/DMODEwAwLCCS/MOEA/D (d)
CPU Times -7 -6 -5 -4 G D D1 ICDV-MOEA/DMODEwAwLCCS/MOEA/DAUGMECON2&CPLEX (e)
200 400 600 800 1000 1200 1400 1600 1800 2000
CPU Times -7 -6 -5 -4 G D D12
ICDV-MOEA/DMODEwAwLCCS/MOEA/DAUGMECON2&CPLEX (f)
200 400 600 800 1000 1200 1400 1600 1800 2000
CPU Times -3 -2 G D D19
ICDV-MOEA/DMODEwAwLCCS/MOEA/D (g)
200 400 600 800 1000 1200 1400 1600 1800 2000
CPU Times G D -3 D20
ICDV-MOEA/DMODEwAwLCCS/MOEA/D (h)Fig. 7: The convergence curves of every method on different problems.
Concerning GD, in Table V, the results still reveal theobvious advantage of AUGMECON2&CPLEX on small-scaleinstances. Although the GDs are supposed to be 0 sinceall the obtained solutions are optima, some numerical errorsoccur due to different numbers of grids and the calculationaccuracy of employed tools. Further, the performance ofD&O-MOEA/D closely follows AUGMECON2&CPLEX onsmall-scale instances and it outperforms two EAs on mostproblems. Regarding the Wilcoxon rank-sum test, the statis-tics are ‘1/18/1’ and ‘3/15/2’ for MODEwAwL and CCS-MOEA/D. It indicates the superiority of D&O-MOEA/D isoutstanding. Moreover, Figs. 7 (e)-(h) show that two EAsfall into local optima although they converge early. D&O-MOEA/D converges better to the contrary since it gets lowerGDs. Further, Figs. 8 (e)-(j) illustrate solutions from D&O-MOEA/D are closer to the PF, although they do not distributeuniformly.
3) Application for Chinese Stock Markets:
In order to testthe proposed method in practice, data from Shanghai andShenzhen stock markets are collected. The data consists ofthe close prices of 4510 funds and the trade date ranges fromJune 13, 2017 to June 12, 2020. Arbitrarily, we determine thedata from June 13, 2017 to June 13, 2019 (2 years) as thetraining data and the data from June 13, 2019 to June 12,2020 (1 year) as the test data. We normalize the close priceswhile setting the close prices at June 13, 2017 as the baselines.It is given as follows: prof it i,t = price i,t − price i, price i, , (18)where price i,t is the close price of asset i at time t , prof it i,t is the normalized return and t = 0 is the start date. Thenthe expected return µ i and the risk σ ij can be obtained fromthese normalized returns [19]. This data is from a financialdata platform Tushare and is also uploaded with the source hhttps://tushare.pro/register?reg=405855 TABLE V: Results on D − D concerning GD with executions for 2000seconds. Algo. A B C D D Mean 2.05e-07[2] 1.85e-06[4] 8.13e-07[3]Std 5.08e-08 3.23e-07 2.65e-07 2.02e-07[1] D Mean 2.90e-07[2] 1.35e-06[3] 3.22e-06[4]Std 7.04e-08 1.73e-06 3.36e-06 2.47e-07[1] D Mean 1.76e-06[3] 1.55e-06[2] 2.54e-06[4]Std 2.49e-07 1.92e-07 9.70e-07 1.24e-06[1] D Mean 1.32e-05[2] 1.95e-05[3] 3.23e-05[4]Std 9.91e-06 6.17e-06 2.27e-05 0.00e+00[1] D Mean 5.40e-06[2] 8.60e-06[4] 5.46e-06[3]Std 2.18e-06 2.46e-06 5.86e-06 1.53e-12[1] D Mean 1.55e-06[1] 2.19e-06[3] 2.61e-06[4]Std 4.36e-07 3.11e-07 1.21e-06 1.62e-06[2] D Mean 3.27e-05[2] 8.28e-05[3] 1.33e-04[4]Std 9.67e-06 1.53e-05 4.25e-05 0.00e+00[1] D Mean 2.97e-05[3] 4.00e-05[4] 1.33e-05[2]Std 1.06e-05 5.52e-06 7.91e-06 0.00e+00[1] D Mean 1.01e-04[2] 1.97e-04[4] 1.70e-04[3]Std 1.98e-05 2.36e-05 4.00e-05 0.00e+00[1] D Mean 1.05e-05[2] 1.72e-05[3] 1.82e-05[4]Std 7.63e-06 6.18e-06 1.11e-05 0.00e+00[1] D Mean 1.08e-05[3] 2.24e-05[4] 9.94e-06[2]Std 3.42e-06 5.25e-06 9.40e-06 0.00e+00[1] D Mean 1.16e-06[2] 1.25e-06[3] 1.51e-06[4]Std 4.98e-07 4.78e-07 8.06e-07 4.40e-07[1]Total ∗
26 40 41 13Final Rank ∗ + , − , ≈ ∗ - 1/10/1 2/8/2 9/0/3 D Mean 4.28e-04[1] 4.79e-04[2] 5.29e-04[3]Std 7.47e-07 6.19e-06 2.28e-05 - D Mean 9.50e-05[1] 1.11e-04[2] 1.40e-04[3]Std 4.70e-06 4.67e-06 3.32e-05 - D Mean 2.09e-04[2] 2.78e-04[3] 1.92e-04[1]Std 2.35e-05 2.75e-05 5.03e-05 - D Mean 6.11e-05[1] 6.77e-05[3] 6.71e-05[2]Std 8.76e-06 5.01e-06 1.01e-05 - D Mean 1.79e-02[1] 2.48e-02[2] 2.81e-02[3]Std 1.22e-05 1.43e-03 1.47e-03 - D Mean 6.01e-04[1] 8.64e-04[3] 8.20e-04[2]Std 1.49e-05 4.40e-05 2.79e-05 - D Mean 5.46e-03[1] 6.80e-03[2] 7.55e-03[3]Std 4.96e-04 3.11e-04 4.20e-04 - D Mean 8.70e-04[1] 1.20e-03[2] 1.82e-03[3]Std 2.09e-05 9.67e-05 2.05e-04 -Total 24 47 49 -Final Rank 1 2 3 - + , − , ≈ - 1/18/1 3/15/2 - Risk -4 R e t u r n -3 D PFICDV-MOEA/D (a)
Risk -4 R e t u r n -3 D PFMODEwAwL (b)
Risk -4 -0.500.511.522.533.54 R e t u r n -3 D PFCCS/MOEA/D (c)
Risk -4 R e t u r n -3 D PFAUGMECON2&CPLEX (d)
Risk R e t u r n D TUCPFICDV-MOEA/D (e)
Risk R e t u r n D TUCPFMODEwAwL (f)
Risk R e t u r n D TUCPFCCS/MOEA/D (g)
Risk R e t u r n D TUCPFICDV-MOEA/D (h)
Risk R e t u r n D TUCPFMODEwAwL (i)
Risk R e t u r n D TUCPFCCS/MOEA/D (j)Fig. 8: The efficient fronts with the best HVs obtained by D&O-MOEA/D, MODEwAwL, CCS/MOEA/D and AUGMECON2&CPLEX with executions for2000 seconds. code of D&O-MOEA/D. D&O-MOEA/D with N = 100 and1000 fitness evaluations is implemented on this data whileall the constraints, except the pre-assignment, are involved.In Fig. 9, with the Mean-Variance model, points representedwith ‘o’ are the performance of every portfolio and the leftvertical coordinate is the return. This PF is regular and thereturn ranges from 0 to 1.6. Meanwhile, the performance ofevery corresponding portfolio is illustrated as ‘x’ and the rightvertical coordinate is the expected return during the investmentperiod, from June 13, 2019 to June 12, 2020. These pointsshow good consistency and prominent expected return rates ofthese investments achieve about 13%. Further, a portfolio withthe highest return is picked out. We compare its profit rateswith stock market trends, i.e. China Securities Index (CSI300).CSI300 reflects the general picture of the stock price changesin both Shanghai and Shenzhen stock markets. Fig 10 showsthe profit rates of this portfolio are higher than the CSI300most of the time. Moreover, the ups and downs of this portfolioand CSI300 are very consistent. The profit rate of this portfoliohas reached about 23% at the end. Whatever, this is just avery simple application and it is just utilized to confirm thefeasibility and the values in application of our method. Thereal financial market should be much more complicated. Risk R e t u r n E x pe c t ed R e t u r n D u r i ng I n v e s t m en t Mean-Variance Model
Fig. 9: Portfolio obtained by D&O-MOEA/D for Shanghai and ShenzhenA-shares.
C. Without A Perfect Optimizer
The existence of a perfect optimizer, which can settle thesubtask quickly and accurately, is a strong assumption indeed.Hence, some simulation experiments (black problems in [27])are conducted and a simple yet effective EA, JADE [46], isan alternative to the perfect
Opt for D&O-MOEA/D. SinceMOEA/D-DE is simultaneously proposed with these problemsand it performs well, we make a comparison between D&O-MOEA/D and MOEA/D-DE on these simple experiments.The parameters are the same as in [27] for D&O-MOEA/Dand MOEA/D-DE while the maximum fitness evaluation is Workdays from June 13, 2019 to June 12, 2020 -0.0500.050.10.150.20.25 R e t u r n /I nde x The highest return portfolio and Stock Market Indices
Highest Return PortfolioCSI300 Index
Fig. 10: Profit and Securities IndexTABLE VI: Summary statistics on F − F . Metrics HV GD + , − , ≈ + , − , ≈ ” denotes D&O-MOEA/D is significantly better than,worse than and similar to MOEA/D-DE. More details arepresented in the supplement. The results of MOEA/D-DE areworse than D&O-MOEA/D, however, the running times raiseto about an hour from several seconds. Apparently, exchangingmassive computational resources with limited performance im-provement is not rational. Therefore, a perfect optimizer Opt should be found first when implementing D&O-MOEA/D.V. C
ONCLUSION AND F UTURE W ORK
This work is the first one that discusses a concept of variabledivision and optimization (D&O), to our best knowledge. Itshould be one potential research realm since this techniqueis frequently used in the practices of EAs. More detailed andin-depth discussions will not only help us clearly understandthe nature of this technique but also assist us to develop newtheories and methods for EAs. According to the discussions,D&O-MOEA/D, in which the variable of the original problemis divided and the partial variables are optimized respectively,is proposed. Then, it is achieved via integrating with aquadratic optimizer of CPLEX for constrained portfolio prob-lems, mixed-integer problems. D&O-MOEA/D, representingthe hybrid method, is compared to one exact method andtwo EAs. The empirical studies show that different kindsof methods have their own advantages. The exact methodshows its superiority on small-scale instances. On the otherhand, the hybrid one performs well on small-scale problemswhile it shows its superiority and versatility for large-scaleinstances. Further, we apply the proposed method to the realstock markets, the results show the feasibility and potential ofD&O-MOEA/D although this application is simple. Since thestock markets are dynamic, we will extend the adopted singleperiod MV model to multi-period in the future and therebythe requirement of new methods will emerge. Further, whilefew works discuss D&O, a huge gap is waiting to be filled.We will not stop exploring this realm. R
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Evolutionary multiob-jective optimization , pp. 105–145, Springer, 2005. This supplement discusses various local Pareto fronts (PFs) and presentsthe results of the supplementary experiment. There should be many differentlocal PFs, however, only two cases of them can utilize the existing method todetermine which local PF is better. First, when the local PF of x I is just onepoint. Second, there is always at least one point in a local PF that dominatesany point from the other one. A D&O based multiobjective minimizationproblem is stated as follows min F ( x ) = ( f ( x I , x II ) , · · · , f m ( x I , x II )) s.t x ∈ Ω , x I ∈ Ω I , x II ∈ Ω II , where F is a multiobjective function consists of m real-valued objectivefunction, x is the decision variable, x I and x II are two partial variables, and x = ( x I , x II ) . Ω ( · ) is the search space of each variable, and Ω I × Ω II = Ω .Formal discussions are presented below.
1) Case One:
When the local PF of x I is just one point if the followingproposition holds. Proposition 1. ∀ x I ∈ Ω I , ∀ x II ∈ Ω II there is a ( or some ) x ∗ II ∈ Ω II that F ( x I , x ∗ II ) (cid:22) F ( x I , x II ) .For example, an alteration of DTLZ2 [47] is given as min f ( x ) = (1 + X i =2 , ( x i − . ) cos ( π x ) min f ( x ) = (1 + X i =2 , ( x i − . ) sin ( π x ) s.t x ∈ [0 , , (19)where the Pareto optimal solutions are x = ( · , . , . . Let x I = ( x , x ) and x II = x . The unique Pareto optimal solution of x I = ( x , x ) isalways x = ( x , x , . , i.e. x = 0 . . It is shown in Fig. 11. Existingpoint-based methods can rank these local PFs since they are points.
2) Case Two:
There is always at least one point in a local PF,