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 inexact method for solving SCSDPs where 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.
In this paper, we propose a class of optimization problems called stochastic convex semidefinite programs (SCSDPs):
SCSDPs may be viewed as an extension of the following stochastic models:
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 (
Alternatively, SCSDPs may be viewed as an extension of the following deterministic convex semidefinite programs ([
There are several methods available for solving determinate deterministic convex semidefinite programs and their special cases, which include the accelerated proximal gradient method [
The function
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 [
The main purpose of this paper is to design an efficient algorithm to solve the general problems (
The rest of the paper is organized as follows. In Section
For more generality, we first consider the following stochastic convex optimization problem:
Let
Next we make the following assumptions (also in [
It is possible to generate an independent identically distributed i.i.d sample
There is an oracle, which, for a given input point
There is a positive number
Assumption (A2) is different from the assumption used in [
The
Throughout the paper, we use the following notations.
In the following, we discuss theory of the inexact SA approach to the minimization problem (
The inexact SA algorithm solves (
Give the initial point
Choose suitable step sizes
Find an approximation solution of
Set
In order to prove the convergence of Algorithm
Let
We now state the main convergence theorem.
Suppose that the stochastic optimization problem (
Note that the iterate
Denote
Since
Since
Since assumption (A3) and
It is in contradiction with (
Sometimes, the error at each iteration
Give the initial point
Choose suitable step sizes
Find an approximation solution of
Set
Suppose that the stochastic optimization problem (
Like the proof of Theorem
It is in contradiction with (
Suppose further that the expectation function
Next we will discuss the complexity of the above algorithms; we take the second algorithm, for instance.
Suppose that
Note that strong convexity of
Therefore, it follows from (
For some constant
For
When
Suppose that
Due to (
and hence, setting
By convexity of
Minimizing the righthand side of (
With constant step size policy (
To illustrate the advantage of inexact SA method over classical SA method, we apply our method to SCSDPs in this section. Problem (
Given the initial point
Choose suitable step sizes
Solve the following equation:
Set
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.
In our first example, we take a simple case of stochastic convex SDP
Objective function is random matrix.
Radius  

CPU time (s)  

(1)  0.5 


0.2260 
(2)  1 


0.2868 
(3)  1.5 


0.5979 
(4)  2 


0.7178 
(5)  2.5 


0.8770 
(6)  3 


0.9087 
(7)  3.5 


1.1569 
(8)  4 


1.2725 
(9)  4.5 


1.3046 
(10)  5 


1.3068 
(11)  6 


1.5898 
Table
Discrete map.
In this example, we tested the inexact SA method on the linearly constrained SCQSDPs:
We show the elementary numerical results in Table
Objective function is stochastic quadratic.



CPU time (s)  Iter  Pobj  

(1)  100 


17.46  18 

(2)  200 


35.11  17 

(3)  300 


90.36  16 

(4)  500 


230.89  16 

(5)  800 


756.32  20 

(6)  1000 


1757.73  14 

(7)  1200 


3939.16  17 

(8)  1500 


7500.49  18 

Our numerical results are reported in Table
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
Table
Closing price of smallcap stocks.
CP  002001  002003  002004  002005  002006  002007  002008  002009  002010  002011 

6.3  18.140  9.550  14.500  11.330  8.760  26.000  13.500  9.620  9.380  12.380 
6.4  18.260  9.250  13.870  10.940  9.430  25.430  13.250  9.600  9.440  12.270 
6.5  18.300  9.240  14.010  11.310  9.090  25.300  13.360  9.680  9.600  12.160 
6.6  17.680  9.070  13.990  10.950  8.520  25.800  12.770  9.500  9.560  11.930 
6.7  17.130  8.950  13.890  10.950  8.070  25.680  12.250  9.240  9.530  11.720 


6.13  16.820  8.750  13.370  10.950  8.270  24.500  12.170  8.940  9.380  11.400 
6.14  17.060  8.800  13.630  11.460  8.220  24.450  12.790  9.120  9.620  11.860 
6.17  17.370  8.770  14.090  11.570  8.280  24.260  12.710  9.300  9.620  11.780 
6.18  17.150  8.810  14.450  11.360  8.230  24.800  12.770  9.260  9.860  11.540 
6.19  17.280  8.810  14.300  11.160  8.110  25.350  12.990  9.120  9.590  11.500 


6.20  16.750  8.620  13.960  10.620  7.750  24.580  12.300  8.650  9.200  10.740 
6.21  16.670  8.480  13.990  10.350  7.900  24.560  11.950  8.730  8.760  10.330 
6.24  15.640  8.200  13.500  9.3200  7.110  22.900  10.850  8.210  8.350  9.900 
6.25  15.160  8.200  13.740  9.190  7.240  23.010  11.290  8.500  8.430  10.310 
6.26  15.080  8.180  14.190  9.960  7.310  23.400  11.470  8.770  8.380  10.590 


6.27  14.890  8.180  13.850  9.650  7.020  22.890  11.340  8.620  8.000  10.280 
6.28  14.740  8.200  14.090  9.300  6.960  22.580  11.190  9.030  8.090  10.140 
7.1  14.650  8.250  14.150  9.600  7.050  22.750  11.500  9.490  8.160  10.360 
7.2  15.120  8.240  13.920  10.150  7.130  23.320  11.740  9.540  8.200  10.800 
7.3  14.750  8.130  14.040  10.230  7.400  23.700  11.570  9.590  8.000  10.680 
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 (
Prediction correlation matrix.
CC  002001  002003  002004  002005  002006  002007  002008  002009  002010  002011 

002001  1  0.9597  0.0690  0.8547  0.9343  0.9042  0.9003  0.4002  0.9128  0.8682 
002003  0.9597  1  0.0681  0.8080  0.9129  0.9003  0.9110  0.6429  0.8139  0.9120 
002004  0.0690  0.0681  1  0.2535  0.1418  0.2719  0.2700  0.4429  0.1797  0.1379 
002005  0.8547  0.8080  0.2535  1  0.8080  0.8139  0.8896  0.5691  0.9100  0.8741 
002006  0.9343  0.9129  0.1418  0.8080  1  0.8517  0.8964  0.5837  0.8255  0.8888 
002007  0.9042  0.9003  0.2719  0.8139  0.8517  1  0.8624  0.5235  0.8381  0.8430 
002008  0.9003  0.9110  0.2700  0.8896  0.8964  0.8624  1  0.6828  0.8517  0.9168 
002009  0.4002  0.6429  0.4429  0.5691  0.5837  0.5235  0.6828  1  0.4691  0.7216 
002010  0.9128  0.8139  0.1797  0.9100  0.8255  0.8381  0.8517  0.4691  1  0.8430 
002011  0.8682  0.9120  0.1379  0.8741  0.8886  0.8430  0.9168  0.7216  0.8430  1 
Now we give the true correlation coefficient of the stocks from 2017.6.4 to 2017.7.4 for comparison.
From Tables
True correlation matrix.
CC  002001  002003  002004  002005  002006  002007  002008  002009  002010  002011 

002001  1  0.966  0.016  0.852  0.948  0.905  0.917  0.342  0.928  0.871 
002003  0.966  1  0.043  0.833  0.957  0.908  0.931  0.493  0.908  0.929 
002004  0.016  0.043  1  0.172  0.016  0.133  0.236  0.388  0.106  0.09 
002005  0.852  0.833  0.172  1  0.821  0.832  0.916  0.512  0.913  0.899 
002006  0.948  0.957  0.016  0.821  1  0.868  0.923  0.496  0.848  0.911 
002007  0.905  0.908  0.133  0.832  0.868  1  0.87  0.437  0.864  0.849 
002008  0.917  0.931  0.236  0.916  0.923  0.87  1  0.583  0.898  0.935 
002009  0.342  0.493  0.388  0.512  0.496  0.437  0.583  1  0.339  0.647 
002010  0.928  0.908  0.106  0.913  0.848  0.864  0.898  0.339  1  0.875 
002011  0.871  0.929  0.09  0.899  0.911  0.849  0.935  0.647  0.875  1 
Table
Closing price of bigcap stocks.
CP  002001  002003  002004  002005  002006  002007  002008  002009  002010  002011 

6.3  9.840  4.790  10.71  10.38  4.81  6.740  12.990  9.280  13.770  12.350 
6.4  9.740  4.730  10.70  10.28  4.77  6.730  12.740  9.160  13.750  12.180 
6.5  9.650  4.770  10.59  10.10  4.76  6.710  12.880  9.060  13.360  12.150 
6.6  9.450  4.700  10.39  10.00  4.71  6.710  12.720  9.040  13.180  11.760 
6.7  9.350  4.560  10.26  9.990  4.66  6.600  12.120  8.840  13.170  11.520 


6.13  9.020  4.300  9.990  9.880  4.53  6.390  11.350  8.340  12.320  11.000 
6.14  9.020  4.500  10.06  9.960  4.55  6.290  11.340  8.310  12.210  11.070 
6.17  9.010  4.620  10.03  9.910  4.51  6.350  11.120  8.160  12.100  10.970 
6.18  9.070  4.590  10.10  9.950  4.33  6.350  11.220  8.140  12.200  11.120 
6.19  8.890  4.560  9.960  9.750  4.28  4.680  11.150  8.110  11.980  10.940 


6.20  8.420  4.340  9.520  9.290  4.17  4.500  10.820  7.940  11.470  10.610 
6.21  8.280  4.370  9.480  9.450  4.15  4.450  11.050  7.730  11.710  10.590 
6.24  7.520  4.060  8.690  8.510  4.05  4.210  10.090  7.480  10.930  9.5300 
6.25  7.800  3.960  8.690  8.440  4.03  4.190  10.000  7.540  10.920  9.3000 
6.26  7.770  3.930  8.580  8.300  3.92  4.100  9.9900  7.490  10.660  9.3100 


6.27  7.880  4.230  8.660  8.210  3.92  4.110  9.8400  7.450  10.920  9.3300 
6.28  8.280  4.100  9.020  8.570  3.93  4.180  10.130  7.510  11.600  9.9100 
7.1  8.170  4.030  8.920  8.590  3.91  4.270  10.180  7.420  11.380  9.9900 
7.2  8.150  3.990  8.880  8.480  3.92  4.240  10.190  7.280  11.210  9.8700 
7.3  8.040  3.940  8.680  8.440  3.94  4.250  9.9700  7.060  11.350  9.6600 
Like in the smallcap stocks, we calculate the correlation matrix as shown in Table
Prediction correlation matrix.
CC  600000  600010  600015  600016  600019  600028  600030  600031  600036  600048 

600000  1  0.9119  0.9776  0.9497  0.9388  0.9656  0.9447  0.9427  0.9766  0.9736 
600010  0.9119  1  0.9477  0.9318  0.9209  0.9298  0.9020  0.8960  0.9030  0.9318 
600015  0.9776  0.9477  1  0.9825  0.9646  0.9835  0.9467  0.9408  0.9716  0.9825 
600016  0.9497  0.9318  0.9825  1  0.9437  0.9666  0.9050  0.8940  0.9547  0.9597 
600019  0.9388  0.9209  0.9646  0.9437  1  0.9766  0.9587  0.9716  0.9358  0.9527 
600028  0.9656  0.9298  0.9835  0.9666  0.9766  1  0.9597  0.9557  0.9567  0.9756 
600030  0.9447  0.9020  0.9467  0.9050  0.9587  0.9597  1  0.9776  0.9408  0.9696 
600031  0.9427  0.8960  0.9408  0.8940  0.9716  0.9557  0.9776  1  0.9318  0.9487 
600036  0.9766  0.9030  0.9716  0.9547  0.9358  0.9567  0.9408  0.9318  1  0.9746 
600048  0.9736  0.9318  0.9825  0.9597  0.9527  0.9756  0.9696  0.9487  0.9746  1 
Now we also give the true correlation coefficient of the stocks from 2017.6.4 to 2017.7.4 for comparison.
From Tables
True correlation matrix.
CC  600000  600010  600015  600016  600019  600028  600030  600031  600036  600048 

600000  1  0.911  0.979  0.952  0.937  0.97  0.943  0.932  0.979  0.976 
600010  0.911  1  0.951  0.932  0.92  0.925  0.901  0.9  0.887  0.93 
600015  0.979  0.951  1  0.987  0.968  0.985  0.951  0.944  0.968  0.988 
600016  0.952  0.932  0.987  1  0.946  0.97  0.907  0.893  0.955  0.965 
600019  0.937  0.92  0.968  0.946  1  0.981  0.962  0.972  0.929  0.954 
600028  0.97  0.925  0.985  0.97  0.981  1  0.965  0.952  0.96  0.982 
600030  0.943  0.901  0.951  0.907  0.962  0.965  1  0.975  0.934  0.973 
600031  0.932  0.9  0.944  0.893  0.972  0.952  0.975  1  0.909  0.942 
600036  0.979  0.887  0.968  0.955  0.929  0.96  0.934  0.909  1  0.973 
600048  0.976  0.93  0.988  0.965  0.954  0.982  0.973  0.942  0.973  1 
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 bigcap stocks.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The work is partially supported by the Natural Science Foundation of China, Grants 11626051, 11626052, 11701061, and 11501074.