Portfolio investment based on probabilistic multi-objective optimization and uniform design for experiments with mixtures

Conclusion: This method naturally reflects the essence of the portfolio investment problem and opens a new way of solving the relevant problem.


Introduction
Portfolio investment aims to diversify investment risks in an effective way. Markowitz proposed a decision -making model of portfolio investment in 1952 which is seen as the foundation of the modern portfolio theory (Wang, 2022).
In Markowitz's treatment, the expected rate of return and the variance of the rate of return are employed to evaluate risky securities, with the latter used to reflect risk. The significant consequence of Markowitz's investigation is that investors should invest their funds in several securities instead in only one, so that investment risk could be reduced and appropriate investment returns obtained.
However, Markowitz's algorithm could only deal either with maximizing the expected rate of return and setting the variance of the rate of return as a restraint condition or with minimizing the variance of the rate of return and letting the expected rate of return be a constraint condition once at a time. In other words, such an approach could not handle simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return rationally due to the lack of appropriate methodology for dealing with multi-objective optimization (Sarmas et al, 2020;Oberoi et al, 2020;Nisani & Shelef, 2021).
Recently, Zheng et al (2022a) proposed probability -based multiobjective optimization in viewpoint of system theory, which created a brand new concept of "preferable probability"; furthermore, assessments for probabilitybased multiobjective optimization were put forward from the respects of the probability theory and the set theory. As a rationally novel approach concerning multiple objectives, it could be used in many fields, including energy planning, programming problems, operation research, financial affairs, management programs, material selection, mechanical design, engineering design, etc.
In this article, probability -based multiobjective optimization is combined with uniform design for experiments with mixtures to deal with the portfolio problem, so as to deal with the problem of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return rationally.
Solution of the portfolio problem in the light of probability -based multi -objective optimization methodology and uniform design for experiments with mixtures In this section, probability -based multi -objective optimization and uniform design for experiments with mixtures are organically combined, which establishes a rational method for solving the portfolio investment problem of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return, i.e, a bi-objective problem. The probability -based multi -objective optimization method is used to transfer a bi -objective optimization problem into a singleobjective optimization problem from the perspective of the probability theory naturally; the discretization of uniform design for experiments with mixtures provides an effective discrete sampling to simplify mathematical processing.
The systematic implementation is demonstrated by subsections A), B), and C).

A) Fundamental spirit of probabilistic multi -objective optimization
In the spirit of probability -based multi -objective optimization, each objective can be analogically represented as a single event in a system (Zheng et al, 2022a) and the whole event of multi -objective simultaneous optimization corresponds to the product of all single objectives (events). All performance utility indexes of a candidate are preliminarily classified into two types: i.e., beneficial type and unbeneficial type, in accordance with the role and preference of a candidate in the optimization, respectively. Specifically, the assessment of the preferable probability Pij of both beneficial indicators and unbeneficial indicators can be carried out according to the evaluation procedure in Fig. 1 (Zheng et al, 2022a).
The meanings of the quantities and the factors in Fig. 1 are as follows: Pij indicates the partial preferable probability of the j-th performance utility indicator of the i-th alternative scenario, Xij; n expresses the total number of the alternative scenario; m reflects the total number of the performance (objective); j X represents the arithmetic value of the j-th performance utility indicator; Xjmax and Xjmin show the maximum and minimum values of the j-th performance utility indicator, respectively; j optimization and uniform design for experiments with mixtures, pp.516-528 and j express the normalized factors of the j-th performance utility indicator Xij in the beneficial status and in the unbeneficial status, individually; the beneficial status or the unbeneficial status of the j-th performance utility indicator Xij is determined according to its specific role or preference in the instant problem; and Pi represents the total (overall) preferable probability of the i-th alternative scenario (Zheng et al, 2022a).
Here, as to the portfolio investment problem, a simultaneous maximization of the rate of return and minimization of the variance of the rate of return is a typical bi-objective problem which contains a beneficial indicator and an unbeneficial indicator, respectively. In this bi -objective portfolio investment problem, the total preferable probability is the decisive objective function which needs to be maximized in a high-dimensional space, so the complex data treatment might be involved; thus uniform design for experiments with mixtures (UDEM) can be employed to simplify data processing rationally.
Uniform design for experiments with mixtures (UDEM), based on the good lattice point (GLP), was proposed by Fang et al (2018). The method of UDEM can be used to create a set of effective sampling points for experimental design with the restraint of xl + x2 + x3．．．+ xs =1 for the proportion xi with the total number of s (Fang et al, 2018); therefore, it can be used as an efficient sampling method for the portfolio investment problem here to conduct the simplification of data processing with discretization.
In addition, Fang especially developed uniform design tables and their usage tables for proper application (Fang et al, 2018).
According to Fang et al (2018), the concrete steps of uniform design for experiments with mixtures (UDEM) are generally as follows:

I. Selection of the uniform design table
Given the number of mixtures s and the number of sampling points n, select the corresponding table U*n(n t ) or Un(n t ) and the usage table from the uniform design table provided by Fang et al (2018), and the number of columns of the usage table is selected as s-1 at this time. Give a mark of the original elements in the uniform design table U*n(n t ) or Un(n t ) with {qik}.

II. Constructing a new element cki
For each i, construct its new element cki according to the following formula, (1)

C) Portfolio investment problem
According to Markowitz's study, the return rate function f1 and the risk function f2 could be introduced, and their expressions are, respectively, In Eq. (3), i is the rate of return of the i-th security and s is the total number of the securities. In Eq. (4), i,j is the correlation coefficient between the i-th security and the j-th security; i is the risk of the i-th security.
According to the objective evaluation in probability -based multiobjective optimization methodology (Zheng et al, 2022a), f1 exhibits the bigger the better, which therefore belongs to the beneficial objective, and f2 manifests the smaller the better, which thus belongs to the unbeneficial objective, respectively.
Therefore, the answer to the portfolio problem is the optimization of a bi-objective problem. Furthermore, probability -based multi -objective optimization methodology can be used to assess it reasonably. Of course, all the evaluations in probability -based multi -objective optimization methodology can be applied rationally.

Applications
In this section, an example illustrates the use of the above steps for solving the portfolio investment problem.
Take a combination of four securities, i.e., securities A, B, C, and D as a typical example. The specific optimization process is explained in detail.
Let the expected rate of return of Security A be 1 = 11.29%, and let its standard deviation of return be 1 = 24.53%; let the expected rate of return of Security B be 2 = 18.10%, and let its standard deviation of return be 2 = 19.94%; let the expected rate of return of Security C be 3 = 8.29%, and let its standard deviation of return be 3 = 11.80%; and let the expected rate of return of Security D be 4 = 11.52%, with its standard deviation of return of 4 = 12.75%. Furthermore, it is assumed that the following correlation coefficients between two of the above securities are 1,2 = 0, 1,3 = 0.68, 1,4 = 0, 2,3 = 0, 2,4 = 0, 3,4 = 0. Now we need to make a decision of simultaneous optimization of both the maximization of the rate of return and minimization of the variance of the rate of return on this portfolio investment.

Solution
In this section, the problem of "portfolio" is analyzed based on probability -based multi -objective optimization methodology. This is a typical bi-objective optimization problem.
Let x1, x2, x3 and x4 be the investment percentages of four securities, A, B, C and D, respectively. There is actually a constraint condition for this problem, i.e., x1 + x2 + x3 + x4 = 1; therefore, it has actually three independent variables, namely x1, x2 and x3.
Since the sampling points of this bi-objective optimization problem are positioned in the 4dimensional space, it is necessary to include at least 23 sampling points with the characteristics of the "good lattice point" in the effective region for the discretization of data processing (Yu et al, 2022;Zheng et al, 2022a;Zheng et al, 2022b).
According to Fang et al (2018), this is a "uniform design for experiments with mixtures" problem due to the constraint condition of the four variables. Let us take the uniform  (23 7 ). Because here the number of variables s equals to 4, and n equals to 23, from the above rules, xk1 = 1 -ck1 1/3 , xk2 = ck1 1/3 (1 -ck2 1/2 ), xk3 = ck1 1/3 ck2 1/2 (1-ck3), xk4 = ck1 1/3 ck2 1/2 ck3 (Fang et al, 2018). Furthermore, we can get the values of the rate of return function f1 and the risk function f2, the distribution of their partial preferable probability and their total preferable probability, as well as the ranking at the sampling points (alternative scenario), which are shown in Table 2. Fig. 2 shows the variation of the return rate with respect to risk at the discrete sampling points. The results reflect that the 2nd discrete sampling point gives the maximum total preferable probability closely followed by the 6th sampling point; therefore, they could be taken as the optimal solution to this portfolio problem. optimization and uniform design for experiments with mixtures, pp.516-528 Regarding the 2nd sampling point, the corresponding investment ratio is x1 = 0. 0222, x2 = 0.3494, x3 = 0.2596, x4 = 0.3688, which leads to the rate of return of 12.98% and the risk of 9.09%.

Conclusion
In this paper, the probability-based multi-objective optimization method is combined with uniform design for experiments with mixtures to study the portfolio investment problem, which aims to give a rational approach to handling the problem of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return; the analysis shows that probability -based multi -objective optimization methodology could provide the optimal solution with the characteristics of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return; uniform design for experiments with mixtures could be used to properly conduct discretization for data processing and simplification.