Journal of Function Spaces
Volume 2018 (2018), Article ID 3742575, 12 pages
https://doi.org/10.1155/2018/3742575

Research Article

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

Shuang Chen,¹ Li-Ping Pang,² Jian Lv,³ and Zun-Quan Xia²

¹Information and Engineering College, Dalian University, Dalian, China
²School of Mathematical Sciences, Dalian University of Technology, Dalian, China
³School of Finance, Zhejiang University of Finance and Economics, Hangzhou, China

Correspondence should be addressed to Shuang Chen; chenshuang0707@163.com

Received 4 August 2017; Revised 17 November 2017; Accepted 13 December 2017; Published 23 January 2018

Academic Editor: Dhananjay Gopal

Copyright © 2018 Shuang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

In this paper, we propose a class of optimization problems called stochastic convex semidefinite programs (SCSDPs):

where is a smooth convex function for every realization of on , is a random matrix whose probability distribution is supported on set , is a linear map, and 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 .

SCSDPs may be viewed as an extension of the following stochastic models:

(1) Stochastic (Linear) Semidefinite Programs (SLSDPs) ([1–5])

where is the minimum of the problem

where denotes the Frobenius inner product between and .

(2) Stochastic Convex Quadratic Semidefinite Programs (SCQSDPs)

where is given self-adjoint positive semidefinite linear operator for every and .

(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 .

Alternatively, SCSDPs may be viewed as an extension of the following deterministic convex semidefinite programs ([7–11]):

where is a smooth convex function on and is a linear map and and 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 [15–18]. 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 , 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.

2. Algorithm and Convergence Analysis of General Convex Stochastic Matrix Optimization

For more generality, we first consider the following stochastic convex optimization problem:

where is a nonempty closed convex set, is a random matrix whose probability distribution is supported on set , and is a smooth convex stochastic function on . We assume that the expectation

is well defined and finite valued for every . We also assume that the expected value function is continuous and convex on . If for every the function is convex on , then it follows that is convex. With these assumptions, (8) becomes a convex programming problem.

Let be any convex finite at . A matrix is called -subgradient of at (where ) if

The set of all such -subgradients is denoted by .

Next we make the following assumptions (also in [19–21]) which will be used to analyze the convergence of algorithms in this paper:

(A1)It is possible to generate an independent identically distributed i.i.d sample of realizations of random vector .(A2)There is an oracle, which, for a given input point , returns inexact stochastic subgradient-a vector such that is well defined and is a -subgradient of at , that is, .(A3)There is a positive number such that

Remark 1. Assumption   (A2) is different from the assumption used in [19]. We only need to get a -subgradient which is more likely got in practice.

Definition 2. The convex hull of set (denoted by ) is the smallest convex polygon that contains all the points of .

Throughout the paper, we use the following notations. denotes the Frobenius norm, where means the element at the th row and th column of . denotes standard trace inner product. By , we denote the metric projection operator onto the set : that is, . Note that is a nonexpanding operator, that is,

The notation stands for the largest integer less than or equal to . By , we denote the history of the process up to time . Unless stated otherwise, all relations between random variables are supposed to hold almost surely.

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.

Algorithm 3.  

Step 1. Give the initial point and set , iterate the following steps.

Step 2. Choose suitable step sizes and computational error .

Step 3. Find an approximation solution of

Set ; go back to Step  .

In order to prove the convergence of Algorithm 3 we introduce the following lemma.

Lemma 4 (see [22]). Let be an increasing sequence of -algebras and let , , , and be nonnegative random variables adapted to . If , , it holds almost surely that

and then is convergent almost surely and almost surely.

We now state the main convergence theorem.

Theorem 5. Suppose that the stochastic optimization problem (8) has an optimal solution and . Let , where , and be a sequence generated by (14); then converges to .

Proof. Note that the iterate is a function of the history of the generated random process and hence is random.
Denote

Since and hence , we can write

where the second inequality is due to nonexpansionary of projection operator.
Since is independent of , we have

Since assumption (A3) and , 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 . By Lemma 4, we know that is convergent almost surely and

Suppose that and by the convexity of , we have that , when . For , we know that

It is in contradiction with (21); then we have almost surely.

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 and set ; iterate the following steps.

Step 2. Choose suitable step sizes and computational stochastic error .

Step 3. Find an approximation solution of

Set ; go back to Step .

Theorem 7. Suppose that the stochastic optimization problem (8) has an optimal solution . We also supposed that stochastic error and there is a positive number such that . Let , where , and be a sequence generated by (23); then converges to .

Proof. Like the proof of Theorem 5, we obtain

For , we known that . By Lemma 4, we know that is convergent almost surely and

Suppose that and by the convexity of , we have that , when . For , we know that

It is in contradiction with (25); then we have almost surely.

3. Error Estimation

Suppose further that the expectation function is differentiable and strongly convex on ; that is, there is constant 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

which together with (27) implies that . In turn, it follows that for all and and hence

Therefore, it follows from (24) that

For some constant , by (32), we have

Next we will prove that

where

Using mathematical induction, for , we have

which holds naturally. For

we know that / < 1; then we have .
For is Lipschitz continuous, there is constant such that

and hence

When , it follows from (38) that, after iterations, the expected error in terms of the objective value is of order . 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.

Theorem 9. Suppose that is convex; let and . Consider the points , and let

We know that

Proof. Due to (24) and assumption (A3), we know that

It follows that whenever , we have

and hence, setting ,

By convexity of , we have , and by convexity of , we have . Thus, by (43) and in view of and , we get

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 , we arrive at the constant step size policy

Along with the associated efficiency estimate and , we have

With constant step size policy (47) in the above theorem, after iterations, the expected error is of order . 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 .

4. Specialization to the Case Where

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 . 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).

Algorithm 10.  

Step 1. Given the initial point and setting , iterate the following steps.

Step 2. Choose suitable step sizes and computational error .

Step 3. Solve the following equation:

Set ; go back to Step .

5. 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.

5.1. Numerical Results for Random Matrix

In our first example, we take a simple case of stochastic convex SDP

where 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: 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 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 . This means that the variance of random variable may impact approximation; however, this influence does not exceed the tolerance range.

Figure 1: Discrete map.

5.2. Numerical Results for SCQSDPs

In this example, we tested the inexact SA method on the linearly constrained SCQSDPs:

where is random variable, , and is given as follows:

Then and can be generated at random. The MATLAB code for generating the matrix is given by ; . We generated as follows:

We show the elementary numerical results in Table 2.

Table 2: 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 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.

5.3. 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 of company, we can divide stocks into two parts called small-cap stocks and big-cap 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 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.

Table 3: Closing price of small-cap 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 (6), we can forecast the correlation coefficient from 2017.6.4 to 2017.7.4. We list the results in Table 4.

Table 4: 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 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.

Table 5: 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

(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 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.

Table 6: Closing price of big-cap 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 small-cap stocks, we calculate the correlation matrix as shown in Table 7.

Table 7: 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 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.

Table 8: 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

6. 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.

Conflicts of Interest

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

Acknowledgments

The work is partially supported by the Natural Science Foundation of China, Grants 11626051, 11626052, 11701061, and 11501074.

References

K. Ariyawansa and Y. Zhu, “A class of polynomial volumetric barrier decomposition algorithms for stochastic semidefinite programming,” Mathematics of Computation, vol. 80, no. 275, pp. 1639–1661, 2011.

K. A. Ariyawansa and Y. Zhu, “Stochastic semidefinite programming: a new paradigm for stochastic optimization,” 4OR, vol. 4, no. 3, pp. 239–253, 2006.

S. Jin, K. A. Ariyawansa, and Y. Zhu, “Homogeneous self-dual algorithms for stochastic semidefinite programming,” Journal of Optimization Theory and Applications, vol. 155, no. 3, pp. 1073–1083, 2012.

S. Mehrotra and M. G. Gozevin, “Decomposition-based interior point methods for two-stage stochastic semidefinite programming,” SIAM Journal on Optimization, vol. 18, no. 1, pp. 206–222, 2007.

Y. Zhu, J. Zhang, and K. Partel, “Location-aided routing with uncertainty in mobile ad hoc networks: A stochastic semidefinite programming approach,” Mathematical and Computer Modelling, vol. 53, no. 11-12, pp. 2192–2203, 2011.

N. J. Higham, “Computing the nearest correlation matrix---a problem from finance,” IMA Journal of Numerical Analysis (IMAJNA), vol. 22, no. 3, pp. 329–343, 2002.

F. Alizadeh, “Interior point methods in semidefinite programming with applications to combinatorial optimization,” SIAM Journal on Optimization, vol. 5, no. 1, pp. 13–51, 1995.

K. Jiang, D. Sun, and K. C. Toh, “An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP,” SIAM Journal on Optimization, vol. 22, no. 3, pp. 1042–1064, 2012.

M. J. Todd, “Semidefinite optimization,” Acta Numerica, vol. 10, no. 126, pp. 515–560, 2001.

L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Review, vol. 38, no. 1, pp. 49–95, 1996.

H. Wolkowicz, R. Saigal, and L. Vandenberghe, Handbook of semidefinite programming: theory, algorithms, and applications, vol. 27, Kluwer Academic Pub, New York, NY, USA, 2000.

J. Malick, “A dual approach to semidefinite least-squares problems,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 1, pp. 272–284, 2004.

H. Qi and D. Sun, “A quadratically convergent Newton method for computing the nearest correlation matrix,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 2, pp. 360–385, 2006.

K. C. Toh, R. H. Tutuncu, and M. J. Todd, “Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems,” Pacific Journal of Optimization, vol. 3, pp. 135–164, 2007.

S. Ghadimi and G. Lan, “Stochastic first- and zeroth-order methods for nonconvex stochastic programming,” Siam Journal on Optimization, vol. 23, no. 4, pp. 2341–2368, 2013.

G. Lan, A. Nemirovski, and A. Shapiro, “Validation analysis of mirror descent stochastic approximation method,” Mathematical Programming, vol. 134, no. 2, pp. 425–458, 2012.

G. Lan and S. Ghadimi, “Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization i: a generic algorithmic framework,” Siam Journal on Optimization, vol. 22, no. 4, pp. 1469–1492, 2012.

G. Lan and S. Ghadimi, “Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization, ii: shrinking procedures and optimal algorithms,” Siam Journal on Optimization, vol. 23, no. 4, pp. 2061–2089, 2010.

G. Lan, “An optimal method for stochastic composite optimization,” Mathematical Programming, vol. 133, no. 1-2, Ser. A, pp. 365–397, 2012.

A. Nemirovski and D. Yudin, “On Cezaris convergence of the steepest descent method for approximating saddle point of convex-concave functions,” in Soviet Math. Dokl, vol. 19, 1978.

A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, “Robust stochastic approximation approach to stochastic programming,” SIAM Journal on Optimization, vol. 19, no. 4, pp. 1574–1609, 2008.

H. Robbins and D. Siegmund, “A convergence theorem for nonnegative almost supermartingales and some applications,” in J. S. Rustagi, Optimizing Methods in Statistics, pp. 233–257, Academic Press, New York, NY, USA, 1971.

A. S. Nemirovsky and D. B. Yudin, 1983, Problem complexity and method efficiency in optimization.