Multi-objective portfolio optimization method considering asset value

Abstract

The invention discloses a multi-target investment portfolio optimization method considering asset value, and the method comprises the steps: generating a high-quality initial population through an offline mechanism, extracting related information from a knee point set through a learning strategy based on a Markov chain and a multi-dimensional normal probability model, predicting a new population based on the learned information, and carrying out the optimization of the new population. And generating another new population through a non-dominated sorting method, and finally combining the two populations through a cooperation improvement mechanism to generate a progeny high-quality non-dominated solution set, thereby providing diversified selection and decision support for managers. According to the method, through experimental comparison with examples of an advanced algorithm under multiple scales, excellent algorithm performance is shown on each index, a group of high-quality non-dominated solutions which are well balanced between value and cost can be provided for managers, and important guidance is provided for formulation of enterprise investment schemes.

Description

Technical Field

[0001] The present invention relates to the technical field of intelligent optimization and investment portfolio optimization, and in particular to a multi-objective investment portfolio optimization method considering asset value. Background Art

[0002] In modern financial markets, portfolio allocation is one of the main problems faced by investors. In traditional portfolio optimization problems, investors need to select several assets from a large number of assets to form a single portfolio in order to maximize returns and minimize risks at the same time. Existing technical solutions to this type of problem include a multi-objective evolutionary algorithm based on decomposition (MOEA / D), which decomposes the multi-objective optimization problem into a series of single-objective sub-problems and uses information interaction between adjacent sub-problems to collaboratively solve them; and a multi-objective optimization method (MOGLS) that combines genetic algorithms with local search strategies. It uses the global search capabilities of genetic algorithms to cast a wide net in the solution space through operations such as crossover and mutation to generate a diverse initial population. These individuals can be regarded as potential investment portfolio solutions, and they have shown a certain ability to find the best solution in small-scale portfolio optimization tasks.

[0003] However, the above methods have shortcomings that cannot be ignored. The MOEA / D algorithm is sensitive to the distribution of weight vectors. If the weight vector is set unreasonably, some areas will be over-searched during the decomposition of sub-problems, while other areas will be under-explored, making the final portfolio solution set biased towards specific areas, unable to fully cover the Pareto frontier, and missing some high-quality portfolios under different risk preferences; at the same time, the algorithm has poor adaptability when dealing with investment scenarios with dynamically changing number of objective functions, and needs to readjust the weight vector and sub-problem structure, consuming a lot of computing resources and time. In the process of solving the problem, the MOGLS algorithm is highly dependent on the initial solution. If the initial solution is not selected properly, it is very easy to fall into the local optimal state, and it is difficult to jump out of the local trap to explore a wider portfolio space, resulting in the final solution not being the global optimal state; and as the size of the portfolio increases and the complexity of the objective function increases, the search efficiency of the algorithm drops sharply. Because its search strategy is relatively fixed, it cannot flexibly respond to changes in the high-dimensional and complex investment environment. Therefore, there is an urgent need for a method to solve the problems of weight vector sensitivity, poor adaptability, initial solution dependence and low search efficiency faced by multi-objective optimization algorithms such as MOEA / D and MOGLS when solving investment portfolio problems, so as to help investors build investment portfolios accurately and efficiently in complex financial markets. Summary of the invention

[0004] In order to solve the above technical problems, the present invention provides a multi-objective investment portfolio optimization method considering asset value.

[0005] To achieve the above object, the present invention provides the following technical solution: a multi-objective investment portfolio optimization method considering asset value, comprising the following steps:

[0006] S1. Construct a mathematical model of a multi-objective portfolio optimization method that considers asset value and set constraints on the model; the expression is as follows:

[0007]

[0008] Where S represents the number of assets; σ ij represents the covariance between the first asset i and the second asset j, where i∈S, j∈S; w i represents the investment proportion of the first asset i; w j represents the investment proportion of the second asset j; p i represents the value of the first asset i; K represents the expected number of assets; ε i represents the minimum investment ratio of the first asset i; δ i represents the maximum investment proportion of the first asset i; z i Indicates whether the first asset i is held, z i 1 means the first asset i is held, z i 0 means that the first asset i is not held; w i represents the investment proportion of the first asset i;

[0009] S2, constructing an evolutionary algorithm based on Markov chain and multi-dimensional normal probability learning model to solve the mathematical model of the multi-objective portfolio optimization method considering asset value in step S1;

[0010] The solution steps are as follows:

[0011] S2.1, using hybrid coding to solve the problem of double decision variables;

[0012] The present invention has both continuous real numbers and integers, which belongs to a double decision variable problem, so a hybrid coding method is adopted;

[0013] Specifically, the solution vector in the dual decision variables consists of discrete decision variables and continuous decision variables;

[0014] The expression of the solution vector is:

[0015]

[0016] The discrete decision variable expression is:

[0017] z=[z1,…,z S ]

[0018] The continuous decision variable expression is:

[0019] w=[w1,…,w S ]

[0020] In the formula, z indicates whether the asset is held, and the value is 0 or 1; w indicates the investment ratio of the held asset; S indicates the number of assets;

[0021] S2.2, use offline mechanism to initialize the population and set key parameters;

[0022] The specific steps of the offline mechanism are as follows:

[0023] S2.2.1. For each example in the original data set, run the fast elite multi-objective genetic algorithm (NSGA-II) for 50 iterations and record the running results of each generation in the database olcDS;

[0024] S2.2.2. Use the non-dominated sorting method to divide the solutions in olcDS into KL layers; the number of layers is automatically generated by the computer;

[0025] S2.2.3, identify the same number of KL knee point solutions as the number of layers from the KL layers by the non-dominated sorting method according to the method of S2.2.2;

[0026] S2.2.4. If the population size is PS, then select the first PS solutions from the KL solutions as the initial population;

[0027] S2.3. Genetic operations are performed using binary tournaments to generate new populations. If the generated new population is an infeasible solution, it needs to be repaired.

[0028] Genetic operations include: crossover, mutation, and selection;

[0029] Infeasible solutions include:

[0030] (1) The number of "1"s in the number of assets in z is not K;

[0031] (2) The sum of the asset ratios in w, i.e. the real values, is not "1";

[0032] The repair process is:

[0033] (1) In the decision variable, i.e., whether the asset is held z, if the number of infeasible solutions “1” is M, then |MK| “1”s are randomly selected and set to “0”;

[0034] (2) In the decision variable, i.e., the investment proportion w of the held assets, the Min-Max normalization method is used to reset the real value;

[0035] S2.4, generate population P by fast non-dominated sorting method;

[0036] S2.5, determine whether the learning conditions are met, if so, execute step S2.6, if not, jump to step S2.3;

[0037] The learning condition is whether the approximate frontier has reached the maximum number of non-updates. In this embodiment, the maximum number of non-updates is set to 50;

[0038] S2.6. Extract the knee solution set of population P from population P by non-dominated sorting method as the training set for probability learning;

[0039] S2.7. Use the probability learning mechanism based on Markov chain and multidimensional normal distribution to update the probability matrix and predict the generation of a new population P1. If an infeasible solution is generated, it needs to be repaired. The specific steps are as follows:

[0040] S2.7.1. Generate a random sampling path l, expressed as follows:

[0041] l=[l1,…,l S ]

[0042] Where S represents the number of assets; l1,…,l S is a random sequence of l;

[0043] The random sampling path changes the Markov chain from a chain structure to a more random mesh structure, which increases the possibility of searching for more high-quality solutions. For example, when S=6, the sampling path of the chain structure is l=[1,2,3,4,5,6], and the mesh structure is l=[1,2,3,4,5,6], l=[2,5,6,1,4,3], l=[1,4,5,6,3,2] or l=[1,3,2,4,6,5]; the mesh structure is generated by the random sampling path according to the computer;

[0044] S2.7.2. Based on the sampling path and the training set generated in S2.6, a new population is generated using the Markov and multidimensional normal learning models. If a solution in the generated population is an infeasible solution, it needs to be repaired. The steps are as follows:

[0045] S2.7.2.1. The learning mechanism of the Markov learning model is as follows:

[0046] According to the discrete decision variable value in the double decision variable, the one-step transition probability matrix OSTPM(gen) is calculated, and the expression is as follows:

[0047]

[0048] In the formula, Respectively represent the assets l in the sampling path l s-1 → Sample asset l in path l sis the probability of "0→0", "0→1", "1→1", "1→0"; therefore, there are (S-1) one-step transition probability matrices, namely (l1,l2), (l2,l3),…, (l S-1 ,l S ).

[0049] For (l1,l2),(l2,l3),…,(l S -1,l S ), respectively calculate the joint probability in the current generation (gen) The expression is as follows:

[0050]

[0051]

[0052] Where KL represents the number of knee point solutions in the training set, is the asset l in the sampling path l in the ii-th knee point in the gen-1 generation s-1 The value of; m represents the first path; n represents the second path;

[0053] Randomly generate the first discrete value of the random sequence l1, and generate the subsequent l2, l3, ..., l in sequence according to conditional probability s The discrete value of , thus forming a discrete vector solution, repeating this step to generate population size solutions;

[0054] S2.7.2.2 The learning mechanism of the multidimensional normal learning model is as follows:

[0055] Calculate the mean of the current generation (gen) of the multidimensional normal probability matrix based on the continuous decision variable values ​​in the double decision variables The expression is as follows:

[0056]

[0057] In the formula, represents the asset l in the sampling path l in the i'th knee solution of the current generation s investment ratio;

[0058] Calculates the variance of the current generation (gen) of the multidimensional normal probability matrix The expression is as follows:

[0059]

[0060] according to and Update the probability matrix and use multidimensional normal sampling to generate a continuous vector solution of population size, where the probability matrix is ​​as follows:

[0061]

[0062] S3. Verify the effectiveness of the method;

[0063] In order to verify the effectiveness of the evolutionary algorithm based on Markov chain and multidimensional normal probability learning model proposed in this paper, the algorithm is compared with two currently mainstream advanced algorithms MOGLS and MOEA / D for solving examples of different scales;

[0064] In order to verify the comprehensive quality of the approximate frontier finally obtained by the above algorithm, the effectiveness evaluation indicators are: convergence index (γ), diversity index (Δ) and dominance index (Ω).

[0065] Beneficial effects of the present invention:

[0066] The technology adopted by the present invention aims to optimize the structure of MOEA, so as to achieve the purpose of improving the efficiency of large-scale example search. The present invention combines the evolutionary algorithm based on Markov chain and multidimensional normal probability learning model to find promising search areas. First, an offline collection method is proposed to generate a high-quality initial population to accelerate the algorithm convergence process. Then, the learning strategy based on Markov chain and multidimensional normal probability model is used to extract relevant information from the "knee point" solution training, and a new population is predicted based on this information. Another new population is generated by the non-dominated sorting method, and finally the two populations are merged by the improvement mechanism based on cooperation to generate offspring solutions. Through experimental comparison, the method of the present invention has advantages in examples of different scales, and can provide investors with a set of high-quality solutions that balance value and risk, which is of great significance in the financial field. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 is a flow chart of the steps of the present invention;

[0068] Figure 2 It is the overall flow chart of the algorithm of the present invention;

[0069] Figure 3 The coding method and restoration diagram of the present invention are shown below;

[0070] Figure 4 Schematic diagram of the random sampling path of the present invention. DETAILED DESCRIPTION

[0071] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative work are within the scope of protection of the present invention.

[0072] In this embodiment, the objective function is calculated using the original data set;

[0073] The original data set contains five scale examples: 31, 85, 89, 98, and 225;

[0074] Each scale contains 10 examples, so the original data set has a total of 50 examples;

[0075] Each calculation example includes: the number of assets, the covariance value between assets, and the asset value.

[0076] like Figure 1 and Figure 2 As shown in Figure 1, a multi-objective portfolio optimization method considering asset value is shown in Figure 1. The steps are as follows:

[0077] S1. Construct a mathematical model of a multi-objective portfolio optimization method that considers asset value and set constraints on the model; the expression is as follows:

[0078]

[0079]

[0080] In the formula, S represents the number of assets; σ ij represents the covariance between the first asset i and the second asset j, where i∈S, j∈S; w i represents the investment proportion of the first asset i; w j represents the investment proportion of the second asset j; p i represents the value of the first asset i; K represents the expected number of assets, K∈S; ε i represents the minimum investment ratio of the first asset i; δ i represents the maximum investment proportion of the first asset i; z i Indicates whether the first asset i is held, z i 1 means the first asset i is held, z i 0 means that the first asset i is not held; w i represents the investment proportion of the first asset i;

[0081] S2, such as Figure 3 As shown, an evolutionary algorithm based on Markov chain and multi-dimensional normal probability learning model is constructed to solve the mathematical model of the multi-objective portfolio optimization method considering asset value in step S1;

[0082] The solution steps are as follows:

[0083] S2.1, using hybrid coding to solve the problem of double decision variables;

[0084] The present invention has both continuous real numbers and integers, which belongs to a double decision variable problem, so a hybrid coding method is adopted;

[0085] Specifically, the solution vector in the dual decision variables consists of discrete decision variables and continuous decision variables;

[0086] The expression of the solution vector is:

[0087]

[0088] The discrete decision variable expression is:

[0089] z=[z1,…,z S ]

[0090] The continuous decision variable expression is:

[0091] w=[w1,…,w S ]

[0092] In the formula, z indicates whether the asset is held, and the value is 0 or 1; w indicates the investment ratio of the held asset; S indicates the number of assets;

[0093] For example, the solution vector It means that assets 1, 3, 5, and 6 are held, and the holding ratios are 0.1, 0.44, 0.38, and 0.08 respectively;

[0094] S2.2, use offline mechanism to initialize the population and set key parameters;

[0095] According to experience, the key parameters in this embodiment are: population size is 150, crossover rate is 0.6, mutation rate is 0.1, maximum number of unupdated times is 40, and probability matrix is ​​150*S full 0 matrix; the population of the present invention is the solution set of the objective function;

[0096] The specific steps of the offline mechanism are as follows:

[0097] S2.2.1. For each example in the original data set, run the fast elite multi-objective genetic algorithm (NSGA-II) for 50 iterations and record the running results of each generation in the database olcDS;

[0098] S2.2.2. Use the non-dominated sorting method to divide the solutions in olcDS into KL layers; the number of layers is automatically generated by the computer;

[0099] S2.2.3, identify the same number of KL knee point solutions as the number of layers from the KL layers by the non-dominated sorting method according to the method of S2.2.2;

[0100] S2.2.4. If the population size is PS, the first PS solutions are selected from the KL solutions as the initialization population; in this embodiment, PS is 150;

[0101] S2.3. Genetic operations are performed using binary tournaments to generate new populations. If the generated new population is an infeasible solution, it needs to be repaired.

[0102] Genetic operations include: crossover, mutation, and selection;

[0103] Infeasible solutions include:

[0104] (1) The number of "1"s in the number of assets in z is not K. In this embodiment, K is 10;

[0105] (2) The sum of the asset ratios in w, i.e. the real values, is not "1";

[0106] The repair process is:

[0107] (1) In the decision variable, i.e., whether the asset is held z, if the number of infeasible solutions “1” is M, where M is generated by a computer, then |MK| “1”s are randomly selected and set to “0”;

[0108] (2) In the decision variable, i.e., the investment proportion w of the held assets, the Min-Max normalization method is used to reset the real value;

[0109] S2.4, generate population P by fast non-dominated sorting method;

[0110] S2.5, determine whether the learning conditions are met, if so, execute step S2.6, if not, jump to step S2.3;

[0111] The learning condition is whether the approximate frontier has reached the maximum number of non-updates. In this embodiment, the maximum number of non-updates is set to 50;

[0112] S2.6. Extract the knee solution set of population P from population P by non-dominated sorting method as the training set for probability learning;

[0113] S2.7. Use the probability learning mechanism based on Markov chain and multidimensional normal distribution to update the probability matrix and predict the generation of a new population P1. If an infeasible solution is generated, it needs to be repaired. The specific steps are as follows:

[0114] S2.7.1, if Figure 4 As shown, a random sampling path l is generated, and the expression is as follows:

[0115] l=[l1,…,l S ]

[0116] Where S represents the number of assets; l1,…,l S is a random sequence of l;

[0117] The random sampling path changes the Markov chain from a chain structure to a more random mesh structure, which increases the possibility of searching for more high-quality solutions. For example, when S=6, the sampling path of the chain structure is l=[1,2,3,4,5,6], and the mesh structure is l=[1,2,3,4,5,6], l=[2,5,6,1,4,3], l=[1,4,5,6,3,2] or l=[1,3,2,4,6,5]; the mesh structure is generated by the random sampling path according to the computer;

[0118] S2.7.2. Based on the sampling path and the training set generated in S2.6, a new population is generated using the Markov and multidimensional normal learning models. If a solution in the generated population is an infeasible solution, it needs to be repaired. The steps are as follows:

[0119] S2.7.2.1. The learning mechanism of the Markov learning model is as follows:

[0120] According to the discrete decision variable value in the double decision variable, the one-step transition probability matrix OSTPM(gen) is calculated, and the expression is as follows:

[0121]

[0122] In the formula, Respectively represent the assets l in the sampling path l s-1 → Sample asset l in path l s is the probability of "0→0", "0→1", "1→1", "1→0"; therefore, there are (S-1) one-step transition probability matrices, namely (l1,l2), (l2,l3),…, (l S-1 ,l S ).

[0123] For (l1,l2),(l2,l3),…,(l S -1,l S ), respectively calculate the joint probability in the current generation (gen) The expression is as follows:

[0124]

[0125] Where KL represents the number of knee point solutions in the training set, is the asset l in the sampling path l in the ii-th knee point in gen-1 s-1 The value of; m represents the first path; n represents the second path;

[0126] Randomly generate the first discrete value of the random sequence l1, and generate the subsequent l2, l3, ..., l in sequence according to conditional probability s The discrete value of , thus forming a discrete vector solution, repeating this step to generate population size solutions;

[0127] S2.7.2.2 The learning mechanism of the multidimensional normal learning model is as follows:

[0128] Calculate the mean of the current generation (gen) of the multidimensional normal probability matrix based on the continuous decision variable values ​​in the double decision variables The expression is as follows:

[0129]

[0130] In the formula, represents the asset l in the sampling path l in the i'th knee solution of the current generation s investment ratio;

[0131] Calculates the variance of the current generation (gen) of the multidimensional normal probability matrix The expression is as follows:

[0132]

[0133] according to and Update the probability matrix and use multidimensional normal sampling to generate a continuous vector solution of population size, where the probability matrix is ​​as follows:

[0134]

[0135] S3. Verify the effectiveness of the method as follows:

[0136] In order to verify the effectiveness of the evolutionary algorithm based on Markov chain and multidimensional normal probability learning model proposed in this paper, the algorithm is compared with two currently mainstream advanced algorithms MOGLS and MOEA / D for solving examples of different scales;

[0137] The case sizes are 31, 85, 89, 98, and 225, respectively. Each size has 10 different cases, for a total of 50 cases. To ensure the fairness of the comparison, all algorithms are independently repeated 21 times on each case, and 3000 generations are used as the termination condition of the present invention, and the running time is used as the termination condition of MOGLS and MOEA / D;

[0138] All algorithms were run on an Intel E3-1220 v5 processor (3.0 GHz), 8 GB of memory, Linux operating system, and Python programming environment, with a different random number seed used for each iteration;

[0139] In order to verify the comprehensive quality of the approximate frontier finally obtained by the above algorithm, the effectiveness evaluation indicators are: γ, Δ and Ω;

[0140] The smaller the γ value, the better. The calculation formula of γ is:

[0141]

[0142] In the formula, APF is the approximate frontier, APF1,…,APF D is the approximate frontier of D algorithms, APF * For APF1∪…∪APF D The objective function set of non-dominated solutions identified in x (APF d ) is APF d Solve x to APF * The Euclidean distance to the nearest solution in ;

[0143] The smaller Δ is, the better. The calculation formula of Δ is:

[0144]

[0145] Where, d ξ and d η For APF d The extreme value to APF * Euclidean distance to the extreme value, d x For APF d The Euclidean distance from x to x+1, d y For APF d The average distance of all solutions in ;

[0146] The larger the Ω, the better. Its calculation formula is:

[0147]

[0148] The experimental results are shown in Tables 1 to 3, which are the statistical results of convergence, diversity and dominance indicators of the non-dominated solutions obtained by the three algorithms of the present invention, MOGLS and MOEA / D for solving 50 examples. The bold indicates the optimal value corresponding to each example, and NB indicates the number of examples with the optimal value among the 50 examples.

[0149] Table 1 Comparison results of the convergence index (γ) of the present invention, MOGLS and MOEA / D

[0150]

[0151]

[0152] It can be seen from Table 1 that among the 50 examples, the NB values ​​of the worst value, the best value and the average value of the convergence index of the present invention are 50, 50, and 50 respectively, while the NB values ​​of MOGLS are all 0, the NB value of the best value of MOEA / D is 7, and the rest are all 0. Therefore, the present invention is superior to MOGLS and MOEA / D in terms of convergence index.

[0153] Table 2 Comparison results of diversity index (Δ) of the present invention, MOGLS and MOEA / D

[0154]

[0155]

[0156] It can be seen from Table 2 that among the 50 examples, the NB values ​​of the worst value, the best value, and the average value of the diversity index of the present invention are 49, 23, and 47 respectively, while the NB values ​​of MOGLS are all 0, and the worst value, the best value, and the average value of MOEA / D are 1, 29, and 4 respectively. Except that the best value is slightly worse than MOEA / D, the present invention is superior to MOGLS and MOEA / D in terms of diversity index.

[0157] Table 3 Comparison results of the dominance index (Ω) of the present invention, MOGLS and MOEA / D

[0158]

[0159]

[0160] It can be seen from Table 3 that among the 50 examples, the NB values ​​of the worst value, the best value and the average value of the present invention on the dominant index are 50, 50, and 50 respectively, while the NB values ​​of MOGLS are all 0, the NB value of the best value of MOEA / D is 5, and the rest are all 0. Therefore, the present invention is superior to MOGLS and MOEA / D in the dominant index.

[0161] In summary, the present invention is superior to MOGLS and MOEA / D in convergence, diversity and dominance, which proves the superiority of the present invention in diversity.

[0162] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A multi-objective portfolio optimization method considering asset value, characterized by: The following steps are involved: S1. Construct a mathematical model for the portfolio problem that takes into account asset value and set constraints on the model; S2, constructing an evolutionary algorithm based on Markov chain and multidimensional normal probability learning model to solve the investment portfolio problem in step S1; S3. Verify the effectiveness of the method; The effectiveness evaluation indicators are: convergence index γ, diversity index Δ and dominance index Ω.

2. The multi-objective investment portfolio optimization method considering asset value according to claim 1, characterized in that: The mathematical model for constructing the portfolio problem considering the asset value and setting constraints on the model are as follows: st In the formula, S represents the number of assets; σ ij represents the covariance between the first asset i and the second asset j, where i∈S, j∈S; w i represents the investment proportion of the first asset i; w j represents the investment proportion of the second asset j; p i represents the value of the first asset i; K represents the expected number of assets; ε i represents the minimum investment ratio of the first asset i; δ i represents the maximum investment proportion of the first asset i; z i Indicates whether the first asset i is held, z i 1 means the first asset i is held, z i 0 means that the first asset i is not held; w i Represents the investment proportion of the first asset i.

3. The multi-objective investment portfolio optimization method considering asset value according to claim 1, characterized in that: The steps of constructing an evolutionary algorithm based on Markov chain and multidimensional normal probability learning model to solve the investment portfolio problem in step S1 are as follows: S2.1, using hybrid coding to solve the problem of double decision variables; Specifically, the solution vector in the dual decision variables consists of discrete decision variables and continuous decision variables; The expression of the solution vector is: The discrete decision variable expression is: from=[from1,…,from S ] The continuous decision variable expression is: in=[in1,…,in S ] In the formula, z indicates whether the asset is held, and the value is 0 or 1; w indicates the investment ratio of the held asset; S indicates the number of assets; S2.2, use offline mechanism to initialize the population and set key parameters; The specific steps of the offline mechanism are as follows: S2.2.

1. For each example in the original data set, run the fast elite multi-objective genetic algorithm for 50 iterations and record the running results of each generation in the database olcDS; S2.2.

2. Use the non-dominated sorting method to divide the solutions in olcDS into KL layers; the number of layers is automatically generated by the computer; S2.2.3, identify the same number of KL knee point solutions as the number of layers from the KL layers by the non-dominated sorting method according to the method of S2.2.2; S2.2.

4. If the population size is PS, then select the first PS solutions from the KL solutions as the initial population; S2.

3. Genetic operations are performed using binary tournaments to generate new populations. If the generated new population is an infeasible solution, it needs to be repaired. Genetic operations include: crossover, mutation, and selection; Infeasible solutions include: (1) The number of "1"s in the number of assets in z is not K; (2) The sum of the asset ratios in w, i.e. the real values, is not "1"; The repair process is: (1) In the decision variable, i.e., whether the asset is held z, if the number of infeasible solutions "1" is M, then |MK| "1"s are randomly selected and set to "0"; (2) In the decision variable, i.e., the investment proportion w of the held assets, the Min-Max normalization method is used to reset the real value; S2.4, generate population P by fast non-dominated sorting method; S2.5, determine whether the learning conditions are met, if so, execute step S2.6, if not, jump to step S2.3; The learning condition is whether the approximate frontier has reached the maximum number of non-updates; S2.

6. Extract the knee solution set of population P from population P by non-dominated sorting method as the training set for probability learning; S2.

7. Use the probability learning mechanism based on Markov chain and multidimensional normal distribution to update the probability matrix and predict the generation of a new population P1. If an infeasible solution is generated, it needs to be repaired. The specific steps are as follows: S2.7.

1. Generate a random sampling path l, expressed as follows: l＝[l1,…,l S ] Where S represents the number of assets; l1,…,l S is a random sequence of l; S2.7.

2. Based on the sampling path and the training set generated in S2.6, a new population is generated using the Markov and multidimensional normal learning models. If a solution in the generated population is an infeasible solution, it needs to be repaired. The steps are as follows: S2.7.2.

1. The learning mechanism of the Markov learning model is as follows: According to the discrete decision variable value in the double decision variable, the one-step transition probability matrix OSTPM(gen) is calculated, and the expression is as follows: In the formula, Respectively represent the assets l in the sampling path l s-1 → Sample asset l in path l s is the probability of "0→0", "0→1", "1→1", "1→0"; therefore, there are (S-1) one-step transition probability matrices, namely (l1,l2), (l2,l3),…, (l S-1 ,l S ); For (l1,l2),(l2,l3),…,(l S-1 ,l S ), respectively calculate the joint probability in the current generation (gen) The expression is as follows: Where KL represents the number of knee point solutions in the training set, is the asset l in the sampling path l in the ii-th knee point in gen-1 s-1 The value of; m represents the first path; n represents the second path; Randomly generate the first discrete value of the random sequence l1, and generate the subsequent l2, l3, ..., l in sequence according to conditional probability s The discrete value of , thus forming a discrete vector solution, repeating this step to generate population size solutions; S2.7.2.2 The learning mechanism of the multidimensional normal learning model is as follows: Calculate the mean of the current generation (gen) of the multidimensional normal probability matrix based on the continuous decision variable values ​​in the double decision variables The expression is as follows: In the formula, represents the asset l in the sampling path l in the i'th knee solution of the current generation s investment ratio; Calculates the variance of the current generation (gen) of the multidimensional normal probability matrix The expression is as follows: according to and Update the probability matrix and use multidimensional normal sampling to generate a continuous vector solution of population size, where the probability matrix is ​​as follows:

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