New Safe Approximation of Ambiguous Probabilistic Constraints for Financial Optimization Problem

In financial optimization problem, the optimal portfolios usually depend heavily on the distributions of uncertain return rates. When the distributional information about uncertain return rates is partially available, it is important for investors to find a robust solution for immunization against the distribution uncertainty. The main contribution of this paper is to develop an ambiguous value-at-risk (VaR) optimization framework for portfolio selection problems, where the distributions of uncertain return rates are partially available. For tractability consideration, we deal with new safe approximations of ambiguous probabilistic constraints under two types of random perturbation sets and obtain two equivalent tractable formulations of the ambiguous probabilistic constraints. Finally, to demonstrate the potential for solving portfolio optimization problems, we provide a practical example about the Chinese stock market. The advantage of the proposed robust optimization method is also illustrated by comparing it with the existing optimization approach via numerical experiments.


Introduction
The core of modern portfolio theory is how to formulate a tradeoff between the return and risk, specifically to determine an effective investment strategy for allocating capital over a number of available assets so as to maximize the return while minimizing the risk. The first systematic approach to the problem of asset allocation, called mean-variance (MV) one, was developed by Markowitz [1]. To overcome the limitation of variance as a risk measure, some useful realistic risk measures were proposed to separate undesirable downside movements from desirable upside movements. In this respect, the interested reader may also refer to the absolute semi-deviation [2,3], moment [4,5], value-at-risk (VaR) [6][7][8], and conditional value-at-risk (CVaR) [9,10].
The researches mentioned above need the precise estimates of the return and risk based on the explicit distribution extracted from considerable current market data. Due to the noise and outliers in the data, or the imperfect data of newly listed stocks, it is often impossible to determine the exact distributions of future returns [11]. As a result, distributionally robust optimization is a useful approach for decision under uncertainty, where uncertain model data are governed by a family of probability distributions. Using this approach, decision makers can find a robust solution for immunization against the distribution uncertainty.
The first research of distributionally robust optimization was proposed by Scarf [12] to solve a linear inventory problem with a min-max solution. Since then, the approach has been well developed in the literature [13][14][15][16][17]. Among them, Sun et al. [16] developed a distributionally robust joint chance constrained optimization model for a dynamic network design problem, and Wiesemann et al. [17] proposed a unifying framework for modeling and solving a distributionally robust multiproduct newsvendor problem.
In financial field, distributionally robust optimization method has proved useful in constructing the portfolio selection model. Shen and Zhang [18] proposed a robust optimization resolution to stochastic programming based on a scenario tree, with the system parameters being modeled to 2 Discrete Dynamics in Nature and Society reside in ellipsoidal ambiguity sets. Natarajan et al. [19] proposed a computationally tractable approximation method for minimizing the VaR of a portfolio based on robust optimization techniques, where the uncertainty set ranged from polytopes to more advanced conic-representable uncertainty sets. Huang et al. [20] considered the relative robust conditional value-at-risk portfolio selection problem, where the underlying probability distribution of portfolio return belonged to a certain set. Pınar and Paç [21] derived the distributionally robust portfolio rules, addressed potential uncertainty in the mean return vector, and derived a closed-form portfolio rule when the uncertainty in the return vector was modeled via an ellipsoidal uncertainty set. Liu et al. [22] adopted an uncertainty set in distributionally robust portfolio selection problems, where the considered uncertainties are in terms of both the distribution and the first two order moments.
The choice of uncertainty set is important for the tractability of distributionally robust portfolio selection model. Box uncertainty and ellipsoidal uncertainty are the most often used uncertainty structures in robust optimization formulations [23]. The obtained solution based on single uncertainty set like box is usually too conservative. In addition, when chance constraints exist in distributionally robust optimization model, the feasible region is usually nonconvex; thus it is difficult to solve such hard optimization model. In this case, a common technique is to find the safe approximation of ambiguous chance constraint under a given uncertainty set [13,24].
A set of constraints is called a safe approximation of the chance constraint if the feasible set of the safe approximation is a subset of the feasible set of the chance constraint [25]. For the time being, there are several approximation methods for solving chance constraint problems, such as sample approximation [26][27][28], scenario approximation [29,30], Bernstein approximation [31,32], and robust counterpart approximation [33,34]. To get a further understanding about the approximation method of chance constraint, the interested readers may refer to the following references. Ben-Tal and Nemirovski [35] developed a computationally tractable approximation of chance constrained linear matrix inequalities based on measure concentration results. Ding et al. [36] presented a tractable convex programming approximation for distributionally robust individual chance constrained problem under interval sets of mean and covariance information. Yang and Xu [37] derived sufficient conditions to guarantee tractability and investigated tractable approximations of joint probabilistic envelope constraints. Postek et al. [38] presented three new safe tractable approximations of chance constraints under partial information of increasing computational complexity and quality.
Among all the approximation methods of chance constraint, the method on approximating the ambiguous chance constraint with its robust counterpart has received an increasing attention recently [39]. Various robust counterpart optimization formulations are derived based on different types of uncertainty set. Ben-Tal et al. [13] discussed some robust counterpart approximations of chance constraints under box, ellipsoid, and the intersection of box and ellipsoid uncertainty sets. Yuan et al. [39] proposed a two-layer algorithm to investigate the tractable robust optimization approximation framework for solving the joint chance constrained problem under the intersection of interval & polyhedral and interval & ellipsoidal type uncertainty sets. Li et al. [40] investigated different types of uncertainty set for robust optimization, including box, ellipsoidal, polyhedral, interval & ellipsoidal, and interval & polyhedral. To the best of our knowledge, no discussion of the robust counterpart approximation is under ball & budget and ellipsoid & generalized budget uncertainty sets.
In this paper, we will study the distributionally robust portfolio selection problem from a new perspective, where the uncertain model parameters are described by two new combinations of perturbation sets of ball & budget and ellipsoid & generalized budget. The proposed safe approximation method incorporated the robust counterpart, can refine the approximation set, and further improve approximation capacity via changing the intersection mode of perturbation sets. For tractability consideration, the safe approximations of ambiguous chance constraint are also derived under the new combined perturbation sets.
Our contributions in this paper include the following several aspects: (1) We develop an ambiguous VaR optimization framework for distributionally robust linear optimization problem, where we optimize the worst case VaR over an ambiguity set. The obtained robust solution is capable of providing immunization against the distribution uncertainty.
(2) To obtain tractable formulation, we approximate the ambiguous probabilistic constraints under two new combinations of perturbation sets: one is the intersection of ball with budget, and the other is the intersection of ellipsoid with generalized budget. Depending on the choice of two new ambiguity sets, the resulting framework of safe approximation includes a system of efficiently computable convex constraints, which can be solved efficiently by generalpurpose commercial-grade solvers.
(3) We demonstrate our approach for addressing a practical case about the Chinese stock market. To show the advantage of the proposed approach, we compare the proposed robust optimization approach with nominal distribution optimization approach via numerical experiments.
The paper is organized as follows: In Section 2, we develop an ambiguous VaR optimization framework for distributionally robust linear optimization problem. In Section 3, we approximate the ambiguous probabilistic constraints under two new combinations of perturbation sets. In Section 4, we demonstrate our approach for addressing a practical case study. Section 5 briefly concludes the paper. All proofs are given in the appendix.

Stochastic Optimization with Ambiguous Probabilistic Constraints
In this section, we consider a class of uncertain linear optimization problems of the form where ∈ is the vector of decision variables, is a collection of deterministic constraints, ∈ and ∈ in the objective are uncertain, and the problem data ∈ , ∈ , = 1, . . . , are also uncertain. In stochastic optimization, the uncertain model data are assumed to be random. In the simplest case, these random data follow known in advance probability distributions [41,42], while in more advanced settings, these distributions are only partially known. In this case, all we know is that belongs to a prescribed family P of distributions on the space of the data [43,44].
In this section, we deal with uncertain model (1) by introducing risk levels ∈ (0, 1), ∈ (0, 1), = 1, . . . , , and enforcing the constraints in the probability sense. To make model (1) meaningful, we introduce an artificial variable , and we build the following VaR optimization problem of the form max When the distribution of random data is only partially known, we replace problem (2) with the following ambiguous VaR optimization problem max s.t. Pr ( , where the distribution 0 of , belongs to the distribution family P 0 , and the distribution of , belongs to the family P , = 1, . . . , . Model (3) is called an ambiguous probabilistic constraint model because we do not have any information about the probability distribution except for the fact that it belongs to a family P.
Model (3) is a semi-infinite programming problem with infinite probability constraints. It is usually difficult to evaluate with high accuracy the probability in the left hand side of (3), even in the case when is simple [45]. Due to the severe computational difficulties associated with ambiguous probabilistic constraints, a natural course of action is to replace the constraints with their computationally tractable safe approximations.
In the next section, we will deal with the scheme for new safe approximations of the ambiguous probabilistic constraints.

New Safe Approximation Methods
It has been well known that the safe approximations of chance constraints are highly dependent on the choice of the uncertainty set. In this section, we will introduce two kinds of new uncertainty sets.
For the sake of clarity, we assume that the uncertainty set is parameterized in an affine fashion by perturbation vector = [ 1 , . . . , ] varying in a given perturbation set : [ ; ] : ∈ ∈ ) ; (4) Ben-Tal et al. [13] reported that the values , , , of uncertain data are obtained from the nominal values 0 , 0 , 0 , 0 of the data by random perturbation. The affine formula of the uncertainty set has been used in many works [25,34,38].
In the following, we approximate the ambiguous probabilistic constraints of (3) under two new combinations of perturbation sets: one is the intersection of ball with budget, and the other is the intersection of ellipsoid with generalized budget.
Case I (safe approximations under ball and budget perturbation set). We first address the perturbation set in (4), and (5) is the intersection of ‖ • ‖ 2 − and ‖ • ‖ 1 − balls, i.e., where Ω > 0 and > 0 are given parameters. In this case, the perturbation set has the following conic representation: where In the light of Remark 1.3.6 in [13], the semi-infinite constraint (4) can be represented by the system of conic constraints in variables , , = 1, 2: which is an equivalent representation of (4) and (6). On the other hand, the following system is an equivalent representation of (5) and (6). Next we present a scheme for the safe approximation of ambiguous probabilistic constraints. For this purpose, suppose perturbation vectors 1 , . . . , included in (4) and (5) possess the following properties: On the basis of the above analysis, we obtain the following observation.

Theorem 1. e RC of the uncertain linear constraint ( ) with the perturbation set ( ) is equivalent to the system of conic quadratic constraints ( )-( ).
In the case of ( ), the component of every feasible solution to this system satisfies the randomly perturbed inequality ( ) with probability of at least 1 − exp{− 2 /2}, = min{Ω, / √ }.

Theorem 2. e RC of the uncertain linear constraint ( ) with the perturbation set ( ) is equivalent to the system of conic quadratic constraints ( )-( ).
In the case of ( ), the component of every feasible solution to this system satisfies the randomly perturbed inequality ( ) with probability of at least 1 − exp{− 2 /2}, = min{Ω, / √ }.
Case II (safe approximations under ellipsoid and generalized budget perturbation set). Next, consider the perturbation set in (4) and (5) is the intersection of ellipsoid and generalized budget, i.e., where Ω > 0, > 0 and > 0 are given parameters. In this case, the perturbation set has the following conic representation: where In virtue of the conic representation of (4), the semiinfinite constraint (4) can be represented by the following system of conic constraints in variables 1 ∈ , 2 ∈ , 1 ∈ , 2 ∈ , : Discrete Dynamics in Nature and Society 5 Similarly, according to the conic representation of (5), the following system of conic constraints in variables 1 , 2 , 1 , 2 , can be obtained: Eliminating from above two systems the variables 1 , 2 − for every feasible solution to these two systems, we have 1 ≥ which is an equivalent representation of (4) and (16). Furthermore, the following system is an equivalent representation of (5) and (16). Next we focus on a scheme for the safe approximation of ambiguous probabilistic constraints. For this purpose, we assume the perturbation vector affecting (4) and (5) possesses the following properties: P.1 , = 1, . . . , are mutually independent random variables; P.2 the distributions of the components satisfy the following conditions with known constants ≥ 0.
In virtue of the above analysis, we obtain the following observations.

Theorem 3. Let the random perturbations affecting ( ) obey P.1-2; the RC of the uncertain linear constraint ( ) with the perturbation set ( ) is equivalent to the system of conic quadratic constraints ( )-( ).
In addition, the component of every feasible solution to this system satisfies the randomly perturbed inequality ( ) with probability of at least 1 − exp{− 2 /2}, = min{Ω, / √ }.

Theorem 4. Let the random perturbations affecting ( ) obey P.1-2; the RC of the uncertain linear constraint ( ) with the perturbation set ( ) is equivalent to the system of conic quadratic constraints ( )-( ).
In addition, the component of every feasible solution to this system satisfies the randomly perturbed inequality ( ) with probability of at least 1 − exp{− 2 /2}, = min{Ω, / √ }.

Financial Application
Portfolio selection is the problem of allocating capital over a number of available assets in order to maximize the return on the investment while minimizing the risk. In this section, we consider the situation that the distribution of random return is partially known and model the portfolio optimization problem with an ambiguous VaR method.

. . e Equivalent Conic Quadratic Financial Optimization
Model. Suppose there exist stocks that can be chosen by an investor in the financial market. The goal is to determine the capital allocation among the stocks in order to maximize the VaR of the resulting portfolio. Let random vector r = ( 1 , . . . , ) denote uncertain return rates of the stocks, and = ( 1 , . . . , ) denote the amount of the investments in the stocks. Each component of uncertain return rate vector r = ( 1 , . . . , ) is represented as where 0 is the nominal value of , and , = 1, . . . , , are the basic shifts data of . , = 1, . . . , , are independent random perturbations, and = ( 1 , . . . , ) is varying in a given perturbation set . Thus an ambiguous VaR portfolio optimization model can be built as max where P is the family of potential probability distributions of random perturbation vector .
In model (26), the uncertain return rate of each stock is an affine function of bounded and mutually independent perturbation variables . When historical data for stocks are available, some statistical tools such as principal component analysis and linear regression can be used to calibrate the values of from the data [46]. The perturbation set of provides the support information of r, while the moment information of characterizes the random nature of the return r. In this section, we consider the following two cases in our numerical experiments: 6 Discrete Dynamics in Nature and Society (i) Case I. Perturbation vector fluctuates in a ball and budget perturbation set depicted in (6), and the distribution family P of , = 1, . . . , satisfies the case of (15).
(ii) Case II. Perturbation vector fluctuates in an ellipsoid and generalized budget perturbation set depicted in (16), and the distribution family P of , = 1, . . . , satisfies the assumptions P.1-2.
When fluctuates in a ball and budget perturbation set as depicted in Case I, it follows from Theorem 1 that ambiguous VaR model (26) is equivalent to the following conic quadratic optimization model: On the other hand, when fluctuates in an ellipsoid and generalized budget perturbation set as depicted in Case II, by Theorem 3, ambiguous VaR model (26) is equivalent to the following conic quadratic optimization model: The above conic quadratic optimization models (27) and (28) are established optimization formats supported by commercial solvers that scale well with the number of variables in the problem. In our numerical experiments, we employ the LINGO 11.0.0.20 solver to find the robust optimal portfolio. All the experiments are conducted in a personal computer with Inter(R) Core(TM) i5-4200U CPU 1.60GHz and RAM 4.00GB.
. . Data Description. We select ten stocks with different return rates from Chinese stock market on October 24, 2018, as the candidate. Each stock has a daily return rate , = 1, . . . , 10. These data are obtained through stock software, named Straight Flush, which can provide stock market display, analysis, and trading. Since the stock return rates are affected by various factors, the daily return rate is varying in a segment around its daily expected return rate. The daily expected return rate is viewed as the nominal value of the random return rate . For instance, the return rate of stock 601116 takes its value in [−0.03%, 2.32%], where −0.03% and 2.32% represent the minimum value and the maximum value of the return rate, respectively, and its nominal value is 1.20%. Figure 1 shows the fluctuations of the daily return rate of stock 601116, from which it is not difficult to see that daily return rate fluctuates around the nominal value.
The nominal values of ten return rates and the variation ranges of the return rates are summarized in Table 1. In the following, we will conduct some numerical experiments to find the optimal portfolio policy based on these data.
. . e Computational Results under Case I. In order to find the optimal portfolio selection, we solve the distributionally robust portfolio optimization problem under Case I. First of all, we need to analyze the connections between perturbation parameters Ω, and risk level to ensure that the selected values of parameters Ω, are reasonable. From the proofs of Theorems 1 and 3, it is known that the relationship between the risk level and perturbation set parameters is Note that the values of parameters Ω and reflect the relationship between ball and budget perturbation set. When Ω = , the intersection of ball and budget perturbation set is illustrated in Figure 2(a); when Ω = / √ 2, the intersection of ball and budget perturbation set is illustrated in Figure 2(b). Based on this observation, the parametric values should be chosen from the interval / √ 2 < Ω < .
According to the discussion above, we give the corresponding values of perturbations set parameters Ω, and 1− in Table 2. Then, we solve model (27) to find the optimal portfolio strategy under the values of these parameters. Note that the value of perturbation parameter Ω can be different even at the same risk. Taking into account the different values of Ω, we select some of the computational results on portfolio strategy and report them in Table 3. For example, when 1 − is set to be 0.93 and equals 7.293, we take Ω as 5.2, 6.4, and 7.2. In order to better demonstrate the variation of optimal portfolio strategies in Table 3 at different parameter values of , , and Ω, it is plotted in Figures 3(a)-3(j). The table shows the variation of allocation proportion of each stock with respect to 1 − , and the corresponding values of perturbation parameter Ω are marked in Figure 3.
Further, we plot Figure 4 to illustrate the effect of parameter Ω on the value of the objective function. When 1 − equals 0.92 and Ω takes values in the corresponding interval (5.026,7.107), the changeable range of optimal value is smaller.      budget parameters- ball parameters-Ω objective function is larger. Thus the optimal objective value is sensitive to the parameter Ω. The reason why different portfolio strategies and objective values at the same risk level are obtained is that the size of the perturbation sets is different. When the perturbation set is larger, the safe approximation is more conservative. From the computational results, the decentralized investment decisions can be obtained when uncertain return fluctuates under Case I. Taking into account the difference of portfolio under different allocations, the relationship between the optimal objective value and perturbation set parameters is given; see Figure 5.
Here it is required to solve model (28). Similarly, the relationship between risk level and perturbation set parameter Ω, is shown in Table 2, and the computational results on portfolio strategies with different values of parameters are reported in Table 4. Further, to illustrate the variation of optimal portfolio strategies in Table 4, it is depicted in Figures  6(a)-6(j). From these figures, we can know how to change the allocation proportion of each stock at risk parameter 1 − , and the values of perturbation parameter Ω are marked in Figure 6.
From the computational results, the decentralized investment decisions can be obtained when uncertain return fluctuates under Case II. Taking into account the difference of portfolio under different allocations, the relationship between optimal objective value and perturbation set parameters is given in Figure 7.
. . Comparison Study. In this subsection, the distributionally robust portfolio optimization model is compared with nominal distribution optimization model, in which uncertain return vector has a nominal probability distribution.
According to uncertain portfolio optimization model (26), we have the stochastic portfolio optimization model with a nominal distribution of random return. To compare the distributionally robust optimization model with nominal distribution optimization model, we analyze the normal distribution information under Case II.
In Case II, the perturbation vector fluctuates in an ellipsoid and generalized budget perturbation set depicted in (16), the distribution family P in model (26) satisfies assumptions P.1-2, and the mutually independent random variables are only partially known with expectations = 0 and variances 2 ≤ 2 . We next consider the scenario of 2 = 2 , i.e., ∼ (0, 2 ), = 1, . . . , . In this situation, the family of normal distributions in Case II degenerates into a single normal distribution. As a result, model (26) reduces to the following nominal distribution model:  To compare the ambiguous VaR model with nominal distribution model, we analyze the computational results at the same risk level. Based on the data in Table 2, we solve model (31) and obtain the computational results under different risks as shown in Table 5. Further, Figure 8 is plotted to show the evolution of the optimal portfolio values of nominal distribution model at risk parameter 1 − .
Under the same risk level, we select the portfolio of ten stocks with positive return, by using the methods of distributionally robust optimization and stochastic optimization with nominal distribution, and obtain the following observations.
First of all, stochastic optimization method with nominal normal distribution is based on the assumption that the full distribution information of future return can be estimated exactly. When the full information on the probability distribution is unavailable, this optimization method will be invalid. In fact, due to the impacts of the economic environment and political factors, it is usually difficult to obtain the exact distribution information of future return. In such case, portfolio strategy under the nominal distribution cannot be recommended to the investors.
When only partial information on the underlying probability distribution is available, the investor should employ the distributionally robust portfolio optimization method. The distributionally robust optimal portfolio policy is the best choice for investors when the uncertain returns are fluctuating in a given ambiguity set. That is, the robust optimal portfolio provides immunization against the distribution uncertainty.
In order to compare the optimal values of distributionally robust optimization model and nominal distribution model, we firstly transform the values of risk parameter in model (31) to the corresponding values of Ω and and then plot, in Figure 9, a comparison between the optimal values of distributionally robust model under ellipsoid and generalized budget perturbation set and nominal distribution model. In terms of optimal solution, we find that the optimal portfolio policy obtained from distributionally robust optimization model is more decentralized than that from nominal distribution model. For example, if 1 − is 0.98, based on nominal distribution model, the investor should select three stocks with number 8, 9, and 10 as a portfolio. However, solving   distributionally robust model, the investor should allocate his/her money to ten stocks. The results show that the optimal portfolio strategies of the two methods are completely different, and the distributionally robust optimization method can better resist the distributional uncertainty.

Conclusions
In this paper, we studied portfolio optimization problem under distribution uncertainty from a new perspective. The obtained major results include the following three aspects. First, to model the distribution of model parameters varying in a prescribed ambiguity set, we developed an ambiguous VaR optimization framework for distributionally robust linear optimization problem. The objective of the proposed model is to optimize the worst case VaR over an ambiguity set. As a result, the obtained robust solution provides immunization against the distribution uncertainty.
Second, we addressed the tractability of the ambiguous probabilistic constraints under two new combinations of perturbation sets. The first ambiguity set is the intersection of ball with budget, while the second one is the intersection of ellipsoid with generalized budget. Depending on the choice of the two new ambiguity sets, the resulting framework of the safe approximation can be reduced to a system of efficiently computable convex constraints.
Finally, to show the advantage of the proposed approach, we applied our approach to a practical case about the Chinese stock market. By comparison study, we concluded that the optimal solution to the proposed robust optimization approach provides immunization against distribution uncertainty, and even small perturbation of nominal probability distribution cannot be ignored in practical portfolio optimization problems.  The proof of theorem is complete.
Next we prove Theorem 3.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.