Inexact SA Method for Constrained Stochastic Convex SDP and Application in Chinese Stock Market

We propose stochastic convex semidefinite programs (SCSDPs) to handle uncertain data in applications. For these models, we design an efficient inexact stochastic approximation (SA) method and prove the convergence, complexity, and robust treatment of the algorithm.We apply the inexactmethod for solving SCSDPswhere the subproblem in each iteration is only solved approximately and show that it enjoys the similar iteration complexity as the exact counterpart if the subproblems are progressively solved to sufficient accuracy. Numerical experiments show that the method we proposed was effective for uncertain problem.


Introduction
In this paper, we propose a class of optimization problems called stochastic convex semidefinite programs (SCSDPs): min  () = E [ (, )] where  is a smooth convex function for every realization of  on   + ,  is a random matrix whose probability distribution  is supported on set Ω ∈  × , A :   →   is a linear map,  ∈   and  ⪰ 0 mean that  ∈   + , and   is the space of  ×  real symmetric endowed with the standard trace inner product ⟨⋅, ⋅⟩ and Frobenius norm ‖⋅‖  .  + is the set of positive semidefinite matrices in   .
(3) Stochastic Nearest Correlation Matrix.Now we consider an interesting example from finance industry.In stock research, sample correlation matrices (a symmetric positive semidefinite matrix with unit diagonal) constructed from vectors of stock returns are used for predictive purposes.Unfortunately, on any day when an observation is made data is rarely available for all the stocks of interest.Higham [6] proposed a method that is to compute the sample correlations of pairs of stocks using data drawn only from the days on which both stocks have data available.Compute the nearest correlation matrix and use it for the subsequent stock analysis.This is a statistical application that motivates nearest correlation matrix problem where  ∈   is a vector of all ones and  ∈   is given.Furthermore, with the particularity of Chinese stock market (in Chinese stock market which is different from stock markets in other countries, stock price rise or fall during a day does not exceed ten percent), we could more accurately estimate the information of stock price in the future.We consider computing expected correlations of pairs of stocks returns for better response random factors of the stock market.In order to justify the subsequent stock analysis, it is desired to compute the nearest expectation correlation matrix and to use that matrix in the computation.We use this matrix to predict the correlation of these stocks in the future.The problem we consider is an important special case of ( 4 where () is a stochastic matrix from (F, Ω, ) →   .Alternatively, SCSDPs may be viewed as an extension of the following deterministic convex semidefinite programs ( [7][8][9][10][11]): min  () where  is a smooth convex function on   + and A :   →   is a linear map and  ∈   and  ⪰ 0 mean that  ∈   + .There are several methods available for solving determinate deterministic convex semidefinite programs and their special cases, which include the accelerated proximal gradient method [8], the alternating projection method [6], the quasi-Newton method [12], the inexact semismooth Newton-CG method [13], and the inexact interior-point method [14].However, all the methods mentioned above may not be extended to efficiently solve the SCSDPs (1) because it might not be easy to evaluate [(, )] in the following occasions: (a)  is a random vector with a known probability distribution, but calculations of the expected value involve multidimensional integration, which is computationally expensive if not impossible.
(b) The function (, ) is known, but the distribution of  is unknown and the information on  can only be obtained using past data or sampling.
(c) [(, )] is not observable and it must be approximately evaluated through simulation.
Under these circumstances, the existing numerical methods for deterministic convex semidefinite programs are not applicable to SCSDPs and new methods are needed.On the other hand, Ariyawansa and Zhu [1] and Mehrotra and Gozevin [4] apply barrier decomposition algorithms and interior-point methods for solving stochastic (linear) semidefinite programs.However, these methods may not be extended for solving the convex problems.
The main purpose of this paper is to design an efficient algorithm to solve the general problems (1) including all the special cases mentioned above.The algorithm we propose here is based on the classical SA algorithm whose subproblem must be solved exactly to generate the next iterate point.The SA algorithm approach originates from the pioneering work of Robbins and Monro and is discussed in numerous publications [15][16][17][18].In this paper, we design an inexact SA method which overcomes the limitation just mentioned.Specifically, in our inexact SA method, the subproblem is only solved approximately and we allow the error in solving subproblem to be deterministic or stochastic.From the theoretical point of view we analyze the convergence, complexity, and robust treatment of the algorithm.We also give the numerical results for illustrating the effectiveness of our method.
The rest of the paper is organized as follows.In Section 2, we focus on theory of the inexact SA method to solve the more general model-stochastic convex matrix program over abstract set Θ, and the proof of convergence is given whether the error is determinate or stochastic.In Section 3, we provide two error estimations of the algorithm.In Section 4, we apply the theoretical results proved before for solving SCSDPs (1).Numerical experiments will be showed at last.

Algorithm and Convergence Analysis of General Convex Stochastic Matrix Optimization
For more generality, we first consider the following stochastic convex optimization problem: where Θ ⊂  × is a nonempty closed convex set,  is a random matrix whose probability distribution  is supported on set Ω ⊂  × , and  is a smooth convex stochastic function on Θ × Ω.We assume that the expectation is well defined and finite valued for every  ∈ Θ.We also assume that the expected value function The set of all such -subgradients is denoted by   ().
Next we make the following assumptions (also in [19][20][21]) which will be used to analyze the convergence of algorithms in this paper: In the following, we discuss theory of the inexact SA approach to the minimization problem (8).
The inexact SA algorithm solves (8) by mimicking the subgradient descent method where   () ∈ ().These lead to an executable algorithm below.
Step 1. Give the initial point  1 ∈ Θ and set  = 1, iterate the following steps.
Step 2. Choose suitable step sizes   > 0 and computational error   .
Step 3. Find an approximation solution of Set  :=  + 1; go back to Step 2.
In order to prove the convergence of Algorithm 3 we introduce the following lemma.
We now state the main convergence theorem.
Proof.Note that the iterate ) is a function of the history  [−1] = ( 1 , . . .,  −1 ) of the generated random process and hence is random.Denote Since  * ∈ Θ and hence Π Θ ( * ) =  * , we can write where the second inequality is due to nonexpansionary of projection operator.
Since   is independent of   , we have Since assumption (A3) and   → 0, there is a positive number  such that Then, by taking expectation of both sides of (17) and using (18), we obtain For   = /, we known that ∑  2 / 2 < ∞.By Lemma 4, we know that Suppose that   →  ̸ =  * and by the convexity of (), we have that It is in contradiction with ( 21); then we have Sometimes, the error at each iteration  is relevant to the random variable   .In this environment, we consider stochastic inexact SA algorithm.Algorithm 6.
Step 1. Give the initial point  1 ∈ Θ and set  = 1; iterate the following steps.
Step 3. Find an approximation solution of Set  :=  + 1; go back to Step 2.
Theorem 7. Suppose that the stochastic optimization problem (8) has an optimal solution  * .We also supposed that stochastic error   (  ) → 0 and there is a positive number  such that   (  ) ≤ .Let   = /, where  > 0, and   ⊂ Θ be a sequence generated by (23); then   converges to  * .
Proof.Like the proof of Theorem 5, we obtain For   = /, we known that ∑  2 / 2 < ∞.By Lemma 4, we know that Suppose that   →  ̸ =  * and by the convexity of (), we have that It is in contradiction with (25); then we have E[‖  −  * ‖  ] → 0 almost surely.

Error Estimation
Suppose further that the expectation function () is differentiable and strongly convex on Θ; that is, there is constant  > 0 such that Next we will discuss the complexity of the above algorithms; we take the second algorithm, for instance.Theorem 8. Suppose that () is differentiable and strongly convex, ∇() is Lipschitz continuous, and the conditions of Theorem 7 are satisfied; we have that where Proof.Note that strong convexity of () implies that the minimizer  * is unique.As optimality of  * is unique, we have that Therefore, it follows from (24) that For some constant  > 1/(2), by (32), we have Next we will prove that where Using mathematical induction, for  = 1, we have which holds naturally.For and hence When  > 1/2, it follows from (38) that, after  iterations, the expected error in terms of the objective value is of order ( −1 ).But that the result is highly sensitive to a priori information on strongly convex factor .In order to make the SA method applicable to general convex objectives rather than to strongly convex ones, one should replace the classical step sizes   = /, which can be too small to ensure a reasonable rate of convergence.We take appropriate averages of the search points   rather than these points themselves.This processing method go back to [20,23] and we call it "robust treatment."We will show the effectiveness of the above technique in the following theorem.
We know that Proof.Due to (24) and assumption (A3), we know that It follows that whenever 1 ≤  ≤ , we have Now we can develop step size policies along with the associated efficiency estimates based on (44).Assume that the number  of iterations of the method is fixed in advance and that   = .Then (44) becomes Minimizing the right-hand side of (45) over  > 0, we arrive at the constant step size policy Along with the associated efficiency estimate and (E[  (  )]) < (1/), we have With constant step size policy (47) in the above theorem, after  iterations, the expected error is of order ( 1/2 ).This is worse than the rate for the inexact SA algorithm.However, the error bounds (48) are guaranteed independently of any smoothness and/or strong convexity assumptions on .

Specialization to the Case Where Θ = {A(𝑋) = 𝑏, 𝑋 ⪰ 0}
To illustrate the advantage of inexact SA method over classical SA method, we apply our method to SCSDPs in this section.Problem (1) can be expressed in the form ( 8) with Θ = {A() = ,  ⪰ 0,  ∈   }.Subproblem ( 14) then becomes the following constrained minimization problem: The KKT corresponding to problem (49) is written as We can get the algorithm for problem (49).
Step 1.Given the initial point  1 ∈ Θ and setting  = 1, iterate the following steps.
Step 2. Choose suitable step sizes   > 0 and computational error   .
Step 3. Solve the following equation: Set  fl  + 1; go back to Step 2.

Application and Numerical Results
In order to assess from a practical point of view the inexact SA method, we code the Algorithm in MATLAB and ran it on several subcategories of SDP.We implement our algorithm in MATLAB 2009a and use a computer with one 2.20 GHz processor and 2.00 GB RAM.

Numerical Results for Random Matrix.
In our first example, we take a simple case of stochastic convex SDP where  :  + 4 × Θ →  is a convex function for every  and where "∘" means each element of  multiplied by random variable .We let  mean 1 which is uniformly distributed whose interval radius is mutative from 0.5 to 4. Using inexact SA method to solve this problem, we list the results in Table 1.
Table 1 shows that the inexact SA method for solving stochastic convex SDP is effective.The abscissa of Figure 1 represent the interval radius of  and ordinate of Figure 1 represents norm difference of the optimal value between  which is a random variable and  ≡ 1.This means that the variance of random variable may impact approximation; however, this influence does not exceed the tolerance range.

Numerical Results for SCQSDPs.
In this example, we tested the inexact SA method on the linearly constrained SCQSDPs: where  ∼ [0.5, 1.5] is random variable, Q, and A is given as follows: , where  ≻ 0, Then  and  can be generated at random.The MATLAB code for generating the matrix  is given by  = rand(, 1);  = .* .We generated  as follows: randvec = 1 + 9 * rand (, 1) ; )) (56) We show the elementary numerical results in Table 2.
Our numerical results are reported in Table 2, where ,   ,   , iter, and pobj stand for, respectively, the matrix dimension, the primal infeasibility, the dual infeasibility, the number of outer iterations, and the primal objective value.For each instance, our algorithm can solve the convex QSDP problem efficiently with very good primal infeasibility.Experimental result shows that the exact SA algorithm can achieve a high degree of detecting reliability.

Predict Correlation Coefficient in the Chinese Stock Market.
In this example, we discuss how to use our inexact SA method to predict correlation coefficient in the Chinese stock market.This is the main motivation of this paper.Chinese stock market is different from other countries, and the rise or fall in a day is not more than ten percent.This feature could make us estimate some index more accurately.
In this paper, our main concern is the correlation coefficient of stocks.Correlation coefficient is often useful to know if two stocks tend to move together.For a diversified portfolio, you would want stocks that are not closely related.It helps to measure the closeness of the returns.Usually, we use the determinate correlation coefficient based on historical data.This principle which we mentioned in the previous paragraph makes us discover that we can use stochastic factor to estimate the correlation coefficient in the future.
According to the difference of the total market capitalization (= total equity × stock prices) of company, we can divide stocks into two parts called small-cap stocks and bigcap stocks.
(1) Small-Cap Stocks.They refer to stocks with a relatively small market capitalization.The definition of small cap can vary among brokerages, but generally it is a company with a market capitalization of less than ¥0.1 billion.
Table 3 is the closing price of ten stocks of small-cap from 2017-6-3 to 2017-7-3, where CP is logogram of closing price.The second line of Table 3 is code of stocks.
We know that the change of the closing price in 2017.7.4 will not exceed ten percent and seriously up or down is small probability event.According to this and model ( 6), we can forecast the correlation coefficient from 2017.6.4 to 2017.7.4.We list the results in Table 4. Now we give the true correlation coefficient of the stocks from 2017.6.4 to 2017.7.4 for comparison.
From Tables 4 and 5, we know that the correlation matrix we calculated is very close to the true correlation matrix.This mean that our method for predicting correlation matrix is effective.
(2) Big-Cap Stocks.In China, it is a term used by the investment community to refer to companies with a market capitalization value of more than ¥0.1 billion.Large cap is an abbreviation of the term "large market capitalization."Market capitalization is calculated by multiplying the number of a company's shares outstanding by its stock price per share.
Table 6 is the closing price of ten stocks of big-cap from 2017-6-3 to 2017-7-3.
Like in the small-cap stocks, we calculate the correlation matrix as shown in Table 7.Now we also give the true correlation coefficient of the stocks from 2017.6.4 to 2017.7.4 for comparison.
From Tables 7 and 8, we know that the correlation matrix we calculated is even more close to the true correlation matrix compared to the case in small-cap stocks.This means that our method for predicting correlation matrix is more effective in the company whose total market capitalization is huge.

Conclusion
In this paper, we propose stochastic convex semidefinite programs (SCSDPs) to handle data uncertainty in financial problem.An efficient inexact stochastic approximation method is designed.We proved the convergence, complexity, and robust treatment of the algorithm.Numerical experiments show that the method we proposed was effective for SCSDP and also for its special cases.We also numerically demonstrated that the method is more effective in big-cap stocks.

Table 1 :
Objective function is random matrix.

Table 2 :
Objective function is stochastic quadratic.

Table 3 :
Closing price of small-cap stocks.

Table 6 :
Closing price of big-cap stocks.