Optimal selection of manufacturing enterprise financialization based on simulated annealing investment portfolio theory

2.1 SA algorithm for finding the optimal solution

Markov chain is a stochastic process in probability theory and mathematical statistics that has Markov property and exists in discrete exponential set and state space. SA can solve the problem of high computational complexity in traditional optimization algorithms. Traditional optimization algorithms, such as exhaustive search and dynamic programming, usually need to traverse the entire search space, and the computational amount increases exponentially with the increase of the problem scale, resulting in high computational complexity.

SA algorithm is an optimization method based on the annealing principle of solid metal, which has strong local optimization ability. It uses the ergodicity theory of Markov chain to find the global optimum of the objective function. During the search process, it can accept optimized solutions and can accept deteriorating solutions to a limited extent. In portfolio theory, various investment instruments involved in the investment process of manufacturing enterprises can include the following: stocks, bonds, options and futures.

Investment requires both profit and risk taking. Investment portfolios can find a balance between expected returns and investment risks to achieve higher returns and effectively avoid or reduce risks [13, 14]. The expected return on asset portfolio is used to determine an effective investment portfolio, measure the return on investor assets, and describe the investment risks faced by investors [15]. Portfolio min ⁡ f ⁢ ( x ) can be represented as:

min ⁡ f ⁢ ( x ) = X T ⁢ V ⁢ X(1)

Under the condition of limited expected returns, the optimal investment rate with minimal risk is obtained using the optimal solution method. From an economic perspective, under the predetermined expected return constraints of investors, by setting the weights of various assets in the investment portfolio, the overall investment risk is reduced. Based on the maximum expected return level, the corresponding investment portfolio with the smallest variance is selected, which is called the “effective investment portfolio”.

Value at risk refers to the maximum possible loss of a financial asset or portfolio under normal market fluctuations, that is, the maximum possible loss of a specific financial asset or portfolio at a specific time in the future, which can be expressed as:

𝑝𝑟𝑜𝑏 ( Δ h ⩽ 𝑉𝑎𝑅 ) = 1 - y(2)

Among them, y represents the investor’s confidence level, and Δ ⁢ h represents the amount of change in the asset value of the investment portfolio during the holding period. In an asset portfolio, variance can only reflect the deviation of the asset portfolio from risk, and cannot quantitatively reflect the risks and losses brought about by the decline in asset prices.

Genetic algorithms are usually searched and optimized through iteration. Each iteration is called a generation, and in each generation, the genetic algorithm evolves the current solution using operations such as selection, crossover, and mutation. The number of iterations refers to the algebra performed by the genetic algorithm. In general, the higher the number of iterations, the higher the probability of convergence of the algorithm, but it also increases the computation time. The biggest difficulty in finding the optimal solution is how to find a maximum problem with multiple constraints. Most nonlinear programming problems have multi-peak characteristics and multiple local optimal solutions. Therefore, when using the SA algorithm to solve the above problems, the algorithm itself relies on a series of parameters to control its process, and the selection of parameters is closely related to the effectiveness of the algorithm. The significance of portfolio optimization in asset weight adjustment mainly includes the following aspects. Risk control: Portfolio optimization can help investors minimize risks in asset allocation. Through the optimization algorithm, the risk of different assets can be modeled and measured based on historical data or risk models, and then the optimal asset weight allocation can be calculated mathematically to reduce the volatility of the overall portfolio. Return maximization: Portfolio optimization can not only control risks, but also pursue the maximization of returns. By choosing appropriate optimization objectives, such as maximizing expected returns and minimizing the tradeoff between risks and returns, investors can find the optimal asset weight allocation under given constraints, so as to maximize returns.

Both GA and SA can be applied to solve portfolio optimization problems in investment markets and use random search to overcome the problem of local minima. Portfolio optimization problem is to realize the optimal portfolio by adjusting the weight allocation of different assets given a set of available investment assets and related constraints. The setting of initial stability has a significant impact on the solving performance of SA. The higher the initial temperature, the greater the probability of obtaining the global optimal solution. However, the higher the number of iterations, the lower the feasibility of the algorithm. Conversely, it affects the algorithm’s global optimal solution ability. Therefore, the purpose of determining the initial temperature is to improve the quality of the final result t 0 as much as possible within a certain time range:

t 0 = Δ ⁢ f * I ⁢ n ⁢ m 1 m 2 ⁢ ¯ ⁢ λ(3)

In the formula, m 1 and m 2 denote the number of decreases and increases of the objective function in the experiment. Δ ⁢ f * represents the average increment of the objective function, and ¯ ⁢ λ represents the initial acceptance rate of the new solution. The SA algorithm gradually converges to the overall optimal solution of the problem while the temperature slowly decreases. Only when the ending temperature is small enough can the algorithm obtain high-quality optimal solutions. However, in order to reduce computational complexity, the end temperature should not be too high. Therefore, a certain temperature decay function should be taken as the condition to ensure that the algorithm can recover to a quasi equilibrium state at all temperatures. The meaning of the temperature decay function is that the temperature first decays violently and then slows down. If the temperature decay function has been selected, its selection should follow the principle of restoring quasi equilibrium at each value of the control parameter, and the length of the Markov chain is also the main factor affecting time.

Adjacent state is the state that can be reached through a single movement from the current state, which is a new solution to the problem. The generation of new states is usually achieved by using a random number generator to randomly select an initial state and then scale it. Assuming a certain state, the generation function X i k + 1 of adjacent states of X i k is:

X i k + 1 = X i k ± α ⁢ ( b - a )(4)

SA uses Markov chain traversal theory to determine the global optimal solution of the objective function. Specifically, the SA algorithm performs a random search in the solution space by accepting the differential solution strategy, and accepts the solution that is worse than the current solution with a certain probability. This probability of accepting the difference is based on changes in the temperature parameter and the value of the objective function. Since the temperature decay function is used to describe the free cooling process of solid materials, when the cooling rate is very small, its internal energy is also very small, so its essence is to slowly cool the system until all states reach equilibrium, and finally return to the ground state of the solid. The heuristic cooling criterion simulates the process of physical annealing. The particles of high-temperature objects can transition from high-energy state to low-energy state, and the transition from low-energy state to high-energy state has a certain degree of randomness. The lower the temperature of an object, the more difficult it is to convert from low energy to high energy, and ultimately all particles are converted to the lowest energy state. The heuristic cooling criterion is adopted:

T k = T 0 k m(5)

Among them, T 0 is the initial temperature, which is a constant, and k m is the temperature after the m -th cooling. The corresponding transfer probability is calculated:

ρ = exp ⁡ ( Δ ⁢ f / T )(6)

Among them, Δ ⁢ f is the change in the value of the objective function. In addition to accepting optimized solutions, SA can also accept deteriorating solutions within a certain range.

SA algorithm has strong local search ability, which is suitable for large-scale combinatorial optimization problems, and does not rely on the initial solution. Although an initial solution is required at the beginning, it is not limited by the initial solution because of its limited probability. The results obtained by this method may not be optimal, but must be suboptimal. The disadvantage of SA algorithm lies in its strong local search ability but weak global grasp ability. Although theoretically speaking, it can converge to the global optimum as long as sufficient computational time is given. In the actual implementation process of the algorithm, it is difficult to guarantee that the obtained result is the global optimum due to the limitations of computational time and speed. Therefore, the optimization effect is not very ideal.

GA is an optimization algorithm based on the theory of biological evolution, which searches for the optimal solution by simulating the operation of heredity, crossover and variation. SA is a random search algorithm based on simulated annealing process, which gradually finds the optimal solution by simulating the metal annealing process. In recent years, GA has been widely used in combinatorial optimization, automatic control, robotics, image processing, gene coding, machine learning and other fields. At present, many scholars have proved the effectiveness of GA in combinatorial optimization, but most of the research is to use GA to solve after building the combinatorial optimization model.

The basic composition of GA includes the following parts: coding mechanism, fitness function, genetic operators and control parameters. The coding mechanism is the foundation of GA, which corresponds to the coding rules of chromosome gene strings in genetics. GA is a encoding mechanism that uniformly encodes individuals into specific strings. The most commonly used encoding mechanism is binary encoding, which utilizes and encodes individuals into a binary string.

The essence of GA optimization is to continuously search for individuals with better fitness values through selection, crossover, and other operations based on the fitness values of each individual in the population, in order to obtain the optimal solution. Fitness function is an important index to measure the quality of individuals in the population, and also the only basis for natural selection. Therefore, the choice of the fitness function has a great impact on the convergence of GA and the ability to find the optimal solution. Individual fitness function is counted:

A = a i ∑ i = 1 n a i(7)

The fitness reflects the survival of the fittest. The higher the fitness, the greater the possibility of reproduction and selection. The lower the individual’s fitness, the lower the possibility of breeding offspring, and the lower the possibility of being selected or even eliminated. The optimal individual generated by this screening method has a high probability, and the investment portfolio obtained through GA solution is P 0 :

P 0 = ( w 11 , w 12 , … , w i ⁢ j , … , w m ⁢ n )(8)

If transaction costs are considered, the investment portfolio needs to be adjusted. Transaction costs can be expressed as:

C i ⁢ j = { k α * Δ ⁢ w i ⁢ j(9)

Among them, C i ⁢ j represents the transaction cost incurred by assets in the investment portfolio during adjustment. The corresponding change in investment income Δ ⁢ E i ⁢ j can be expressed as:

Δ ⁢ E i ⁢ j = Δ ⁢ w i ⁢ j * Δ ⁢ R i ⁢ j(10)

Among them:

Δ ⁢ R i ⁢ j = R i ⁢ j ⁢ ( t ) - R i ⁢ j(11)

Among them, Δ ⁢ R i ⁢ j represents the changes in returns before and after the investment portfolio adjustment, so comparing the changes in investment returns can determine whether to adjust the investment portfolio and ultimately obtain a larger return. In the process of dynamically adjusting investment portfolios, the transaction costs incurred are likely to be higher than the benefits they can bring. Therefore, when optimizing them, it is necessary to consider the relevant transaction costs to ensure that the adjusted results can bring greater benefits. In this way, transaction costs can be added to the objective function to obtain a dynamically optimal asset portfolio.

The advantage of GA is that it does not require constraints on the search space, and does not require conditions such as continuity or differential existence for the objective function. It has strong global optimization ability and inherent parallelism. However, its drawback lies in its cumbersome program design, which requires writing a real problem first and ultimately determining the optimal solution before decoding. GA has a certain dependence on the selection of the initial population, and the implementation of the three operators requires many parameters. However, the selection of many parameters mostly relies on experience. In the early stage of algorithm search, the rapid increase of excellent individuals can easily lead to the loss of population diversity, resulting in the program falling into local optima.

In theory, GA and SA are both optimization algorithms based on probability distribution. The difference is that SA can effectively avoid falling into local minima and ultimately approach global optima, and GA performs the optimal solution in the context of “survival of the fittest”.

The expected return of a portfolio can be weighted by the expected return of each asset. Typically, investors conduct detailed research and analysis of each asset to determine its expected rate of return. This includes a comprehensive consideration of company fundamentals, industry developments, macroeconomic factors and more. The complementarity between GA and SA is mainly manifested in their strong global optimization ability and weak local optimization ability. The SA algorithm is a good local optimization algorithm, so it can combine GA and SA to achieve complementary advantages. The GA-SA algorithm used in this paper focuses on the genetic algorithm, and the SA operation is placed as a separate operator in the course of the genetic algorithm. This means that the SA algorithm is used to perturb the population on a large scale, thereby making GA overcome the problem of premature convergence.

The objective functions and constraints of linear programming problems can be expressed as linear combinations of variables. This means that there is only a linear relationship between the objective function and the variables in the constraint, such as a first-order formula or linear inequality. The number of iterations of GA is used as the annealing time of SA. The initial temperature is T 0 , and the temperature update function corresponding to the number of iterations is T p :

T p = T 0 t l(12)

Among them, l represents the length of the individual encoding string. Real number encoding is adopted. The true value of the variable is used as the encoding of the variable, and the encoding of l variables is connected in a certain order to form an individual encoding string. When decoding, the genetic values of chromosomes only need to be assigned to the corresponding variables. On this basis, new solutions are generated and the corresponding fitness function is calculated:

p α = min ⁡ { 1 , exp ⁡ f ⁢ ( x j ) - f ⁢ ( y j ) α ⋅ T p }(13)

Among them, α represents the constant of the objective function transformation. The sorting selection method is adopted, with the basic idea of sorting individuals in a group according to their fitness value. When seeking the maximum value of the objective function, it is arranged in ascending order, while the minimum value is arranged in descending order. The probability of each individual being selected into the group is:

p i = q ⁢ ( 1 - q ) r - 1 1 - ( 1 - q ) M(14)

Among them, q represents the probability of being selected as the optimal individual; r represents the ranking number; M represents the population size.

When the final temperature is too high, the numerical calculation may be unstable. In numerical calculation, if the temperature value becomes very large, problems such as numerical overflow or rounding errors may occur during the calculation process, resulting in the loss of accuracy and reliability of the calculation results. Adopting the same crossover rate and mutation rate is unfavorable for optimizing GA, especially in the late stage of genetic evolution. Due to the high similarity between populations, the crossover rate should not be fixed and should gradually decrease. On the contrary, in order to create better individuals and develop in a better direction, it is necessary to continuously improve the dynamic crossover rate and mutation rate. The calculation formulas for dynamic crossover rate p c and mutation rate p d are:

p c = η i ⋅ F ′ - F min F - - F min - μ i ⁢ t T(15) p d = η d ⋅ F ′ - F min F - - F min + μ d ⁢ t T(16)

Among them, F represents the average fitness value of the population, and F min represents the minimum fitness value in the population. F ′ represents the smaller fitness value of two individuals participating in the crossover operation. η i and η i represent adjustment coefficients. The objective function is expressed as the desired form of the design variable, so the objective function is a function of the design variable and is a scalar. When the objective function has reached its optimal value or the known algebra has reached its maximum value, the known optimal solution is output and the algorithm operation ends.

Generation functions are functions used in evolutionary algorithms to create new individuals. It generates new individuals by mutating and crossing existing individuals. The design of generating function directly affects the exploration ability and diversity of search space. A good generating function should be able to fully explore the search space during the search process in order to find a better solution. Fitness function is a function used to evaluate the advantages and disadvantages of an individual in evolutionary algorithms. It evaluates each individual and gives a fitness value, which is used to measure the superiority of the individual in the solution space. The design of fitness function is the core of evolutionary algorithm, which directly affects the quality of the solution in the search process. Each gene locus of an individual undergoes variation with a probability of variation, that is, randomly selecting positions based on probability to transform the original values:

v i = ∑ j = 1 m i u i ⁢ j(17)

If v i does not meet the constraint conditions, it would be reset, and the selection, crossover, or mutation process would be repeated until the resulting individual meets the constraint conditions.

The initial temperature is a key parameter of the SA algorithm, which determines whether to conduct large-scale exploration during the search process. Higher starting temperatures can make the search process more global and help escape local optimal solutions, but the search process may be slow. Conversely, a lower starting temperature can converge to the local optimal faster, but may miss the global optimal. Therefore, the starting temperature needs to be set according to the specific problem to balance the tradeoff between global and local search. The SA algorithm is developed on the basis of the solid-state metal annealing mechanism, and can quickly converge to the global optimal solution in theory within a certain computational time. The SA algorithm has strong local search ability, but it cannot understand the entire search space well, making it inconvenient to bring the search process into the optimal desired search area, and resulting in low search efficiency. GA is an adaptive algorithm constructed based on Darwin’s evolutionary theory and gene selection principles, with excellent global optimization ability. However, in practical applications, GA often exhibits a “precocious phenomenon”, which means that at the beginning, excellent individuals rapidly increase, leading to a loss of diversity in the population and ultimately leading to the population falling into local extremum.

3.1 Comparison of the number of optimal solutions

The number of optimal solutions searched for by different algorithms must vary. The comparison of the number of optimal solutions searched for by Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms under different enterprises is shown in Fig. 1.

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As shown in Fig. 1: From Fig. 1a, it was found that in Enterprise 1, the average optimal solutions of GA, SA, and GA-SA algorithms for four stock investment portfolios were 64, 62, and 97, respectively. The average optimal solutions of the three algorithms for 36 stock investment portfolios were 69, 69, and 107, respectively. The highest average optimal solutions for GA, SA, and GA-SA algorithms were 69, 79, and 109, respectively.

From Fig. 1b, it can be seen that the average optimal solutions of GA, SA, and GA-SA algorithms for four stock investment portfolios in Enterprise 2 were 73, 92, and 111, respectively. The average optimal solutions of the three algorithms for 36 stock investment portfolios were 73, 90, and 112, respectively. The highest average optimal solutions of the three algorithms were 79, 92, and 115, respectively.

SA algorithm has strong local search ability by simulating the structural changes of substances during annealing. It can accept the probability of poor solutions, so as to avoid falling into the local optimal solution and being unable to jump out. This gives the SA algorithm a greater chance of finding a better solution in the solution space, thereby reducing the deviation rate.

The comparison of the execution time of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms in different enterprises is shown in Fig. 2.

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As shown in Fig. 2: From Fig. 2a, it can be seen that the execution time for Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms to find the optimal solution for the four stock investment portfolios in Enterprise 1 was 14.36 seconds, 9.81 seconds, and 5.75 seconds, respectively. The execution time for three algorithms to find the optimal solution for 36 stock investment portfolios was 14.44 seconds, 9.48 seconds, and 5.77 seconds, respectively.

From Fig. 2b, it was shown that the execution time for GA, SA, and GA-SA algorithms to search for the optimal solution of four stock investment portfolios in Enterprise 2 was 11.86 seconds, 6.55 seconds, and 4.30 seconds, respectively. The execution time for three algorithms to find the optimal solution for 36 stock investment portfolios was 11.19 s, 8.32 s, and 4.85 s, respectively. Overall, the execution time of GA, SA, and GA-SA algorithms in searching for the optimal solution of the stock investment portfolio in Enterprise 2 was shorter than that in searching for the optimal solution of the stock investment portfolio in Enterprise 1, and the execution time of GA-SA algorithm has always been lower than that of the other two algorithms.

GA-SA algorithm has saved a lot of execution time, especially when solving large-scale combinatorial optimization problems. The larger the scale, the more obvious the advantages. This is because the GA-SA algorithm achieves high result quality and good initial solution robustness after multiple adjustments to the cooling process, and solves the problem of high computational complexity in traditional SA algorithms. In the case of multiple peaks, the GA-SA algorithm can also solve the optimal problem well and achieve a good balance. For a large investment portfolio problem, the analyzed method can obtain a high-quality, global, and nearly optimal result within a polynomial time cycle.

The GA-SA algorithm is a randomized search algorithm used to solve portfolio optimization problems. In the investment market, it can be used to make the optimal combination of investments and predict the deviation rate of investments. Stocks invested by different companies were used as experimental data, and the prediction bias rates of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms in different companies were compared as shown in Fig. 3.

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As shown in Fig. 3: From Fig. 3a, it was found that the Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms had the highest prediction bias rates for stocks purchased by Enterprise 1, which were 0.19%, 0.11%, and 0.06%, respectively. The three algorithms had the lowest prediction bias rates for stocks purchased by Enterprise 1, which were 0.10%, 0.04%, and 0.01%, respectively.

In Fig. 3b, it was observed that the Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms had the highest prediction bias rates for stocks purchased by Enterprise 2, which were 0.11%, 0.10%, and 0.03%, respectively. The three algorithms had the lowest prediction bias rates for stocks purchased by Enterprise 2, which were 0.07%, 0.05%, and 0.01%, respectively.

Compared with single Genetic Algorithm and Simulated Annealing algorithms, GA-SA algorithm had a lower stock prediction bias rate and can better fit the actual situation. In summary, it can be seen that compared to single GA and SA algorithms, the prediction performance of GA-SA algorithm has been improved, achieving the desired effect. As shown in Fig. 3, the prediction bias rate of the GA-SA algorithm was very low, and the prediction bias rates of the non GA and SA algorithms were higher than those of the GA-SA algorithm.

By using the GA-SA algorithm, the weight of each asset can be adjusted to maximize its returns while ensuring a certain level of risk. In the aspect of combinatorial optimization, GA-SA algorithm was used to transform the asset’s weight value into a state, and the initial weight value can be selected randomly in the combinatorial optimization. When the initial conditions are better, the algorithm is easier to obtain the optimal solution.

The iteration times of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms in Enterprise 1 are shown in Table 1.

As shown in Table 1, the GA, SA, and GA-SA algorithms had 163, 152, and 140 iterations to obtain the optimal solution for the investment portfolio of 4 stocks in Enterprise 1, while the three algorithms had 170, 154, and 130 iterations to obtain the optimal solution for the investment portfolio of 36 stocks.

The iteration times of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms in Enterprise 2 are shown in Table 2.

As shown in Table 2, the iteration times for Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms to obtain the optimal solution for the investment portfolio of four stocks in Enterprise 2 were 150, 137, and 127, respectively. The iteration times for obtaining the optimal solution of 36 stocks for investment portfolio using three algorithms were 140, 136, and 125, respectively. It can be seen that the GA-SA algorithm has fast convergence speed and can achieve a stable optimal solution.

The convergence speed of GA and SA algorithms depends on the number of iterations. Compared to GA-SA algorithms, GA and SA algorithms have more iterations. In practical applications, complexity and other factors should be considered and appropriate iteration numbers should be selected. The GA-SA algorithm utilizes a randomized search method to jump out of local minima, and its degree of randomization is related to temperature parameters. Although the parameters of the GA-SA algorithm are variable, its randomness enables it to effectively avoid local minima and has been widely applied in practice.

If the expected return of an investment portfolio performs well, it indicates that it has less risk compared to other investment portfolios, and it is a reasonable investment portfolio. The comparison of expected returns of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms in different enterprises is shown in Fig. 4.

As shown in Fig. 4: In Fig. 4a, the expected returns of GA, SA, and GA-SA algorithms on the four stocks in Enterprise 1 were 0.57, 0.68, and 0.78, respectively. The expected returns of the three algorithms on the 36 stocks were 0.55, 0.69, and 0.86, respectively.

In Fig. 4b, the expected returns of GA, SA, and GA-SA algorithms on the four stocks in Enterprise 2 were 0.63, 0.71, and 0.89, respectively. The expected returns of the three algorithms on the 36 stocks were 0.63, 0.70, and 0.81, respectively. The results showed that the expected return rate of combinatorial optimization obtained by GA-SA algorithm was better than that obtained by GA, SA and algorithm. Considering the transaction costs in the actual investment market, under the same conditions of risk and return, the fewer assets in the portfolio, the less transaction costs. Therefore, the structured portfolio based on GA-SA algorithm is effective.

3.6 Comparison of search accuracy

According to the evolutionary principle of “survival of the fittest”, GA was adopted to solve the optimal solution and improved to effectively avoid falling into local minima when converging to zero. The SA algorithm was introduced for annealing processing. By controlling parameters such as termination conditions and annealing temperature, the weight of each temperature point was fully optimized to achieve the optimal state of each stock. The search accuracy of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms for the optimal solution in Enterprise 1 is shown in Table 3.

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As shown in Table 3, in the case of Enterprise 1, the search accuracy of GA, SA, and GA-SA algorithms for the optimal solution of 8 stock investment portfolios was 78.44%, 87.92%, and 91.70%, respectively. The search accuracy of the three algorithms for the optimal solution of 36 stock investment portfolios was 81.63%, 86.62%, and 90.70%, respectively.

The search accuracy of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms for the optimal solution in Enterprise 2 is shown in Table 4.

As shown in Table 4, in the case of Enterprise 2, the search accuracy of GA, SA, and GA-SA algorithms for the optimal solution of 8 stock investment portfolios was 81.87%, 87.72%, and 94.05%, respectively. The search accuracy of the three algorithms for the optimal solution of 36 stock investment portfolios was 84.52%, 87.04%, and 94.90%, respectively. The search accuracy of the GA-SA algorithm for the optimal solution of the investment portfolio was higher than that of the GA and SA algorithms for the optimal solution of the investment portfolio. This is because the SA algorithm also has strong local search ability.

GA-SA algorithm has good adaptability and robustness. There are no too many restrictions on the selection of initial value and objective function, and it can ensure the convergence of the global optimal solution. In order to obtain the best results, it is necessary to reduce the calculation temperature of the algorithm to a certain extent. Therefore, the setting of termination conditions should also be considered during the calculation.