Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 379734 | https://doi.org/10.1155/2015/379734

Xuewen Mu, Yaling Zhang, "An Alternating Direction Method for Convex Quadratic Second-Order Cone Programming with Bounded Constraints", Mathematical Problems in Engineering, vol. 2015, Article ID 379734, 10 pages, 2015. https://doi.org/10.1155/2015/379734

An Alternating Direction Method for Convex Quadratic Second-Order Cone Programming with Bounded Constraints

Academic Editor: Gerhard-Wilhelm Weber
Received20 Oct 2014
Revised26 Mar 2015
Accepted30 Mar 2015
Published30 Apr 2015

Abstract

An alternating direction method is proposed for convex quadratic second-order cone programming problems with bounded constraints. In the algorithm, the primal problem is equivalent to a separate structure convex quadratic programming over second-order cones and a bounded set. At each iteration, we only need to compute the metric projection onto the second-order cones and the projection onto the bound set. The result of convergence is given. Numerical results demonstrate that our method is efficient for the convex quadratic second-order cone programming problems with bounded constraints.

1. Introduction

In this paper, we consider a convex quadratic second-order cone programming (CQSOCP) problem with bounded constraints which is defined by minimizing a convex quadratic function over the intersection of an affine set, a bounded set, and the product of second-order cones. The primal convex quadratic second-order cone programming problem is defined aswhere is a bounded set, is an symmetric positive semidefinite matrix, , , , and is viewed as a column vector in with . In addition, and , where is the -dimensional second-order cone given bywhere is the standard Euclidean norm.

Convex quadratic second-order cone programming problem with bounded constraints is a nonlinear programming problem, which can be seen as a trust region subproblem in the trust region method for the nonlinear second-order cone programming [1, 2]. Since is symmetric positive semidefinite, we can compute its positive semidefinite square root by the Cholesky method. Then problem (1) can be equivalently transformed as the following mix linear and second-order cone programming (MLSOCP) [3]:In paper [1, 2], the authors use those well developed and publicly available softwares, based on interior-point methods, such as SeDuMi [4] and SDPT3 [5] to solve the equivalent MLSOCP (3).

Interior-point methods have been well developed for linear symmetric cone programming [68]. However, at each iteration these solvers require to formulate and solve a dense Schur complement matrix, which for the CQSOCP problem with bounded constraints amounts to a linear system of dimension . In addition, the transformed method needs to compute the square root of semidefinite matrix . When is large, because of the very large size and ill-conditioning of the linear system of equations, interior-point methods are difficult to solve the transformed MLSOCP problem efficiently [3].

The alternating direction method (ADM) has been an effective first-order approach for solving large optimization problems, such as linear programming [9], linear semidefinite programming (LSDP) [10, 11], nonlinear convex optimization [12], and nonsmooth minimization arising from compressive sensing [13, 14]. A modified alternating direction method is proposed for convex quadratically constrained quadratic semidefinite programs in paper [15]. In the thesis [3], a semismooth Newton-CG augmented Lagrangian method is proposed for large scale convex quadratic symmetric cone programming. In paper [16], an alternating direction dual augmented Lagrangian method for solving linear semidefinite programming problems in standard form is presented and extended to the SDP with inequality constraints and positivity constraints.

In the paper, an alternating direction method for the CQSOCP problem with bounded constraints is proposed. Firstly, the primal problem is equivalent to a separate structure convex quadratic programming over second-order cones and a bounded set. Then the alternating direction method is proposed to solve the separate structure convex quadratic programming. In the alternating direction method, we only need to compute the metric projection onto the second-order cones and projection onto the bounded set at each iteration. We also give the convergence results and the numerical results.

2. The Projection on the Second-Order Cone and the Bounded Set

In this section, we will give the projection results on the second-order cones and the bounded set.

Let for ; then the spectral decomposition of associate with second-order cone can be described as [1719]wherewith if and any vector in satisfying if .

Next we introduce the projection lemma over the second-order cone [1719].

Lemma 1 (see [1719]). For any , let be the projection of onto the second-order cone ; then we havewhere for .

Let ; then the projection of over the cone is described as

Let ; then the projection on the bounded set is easy to carry out, namely, through an element by element method:

3. An Alternating Direction Method for CQSOCP Problems with Bounded Constraints

In this section, we give an alternating direction method for convex quadratic second-order cone programming problems with bounded constraints.

Firstly, we give an equivalent separate structure convex quadratic programming over second-order cone and bounded set as follows:

The Lagrangian function for the separate structure convex quadratic programming problem is written aswhere .

Under mild constraint qualifications (e.g., Slater condition), strong duality holds for problem (9), and hence, is an optimal solution of (9) if and only if there exists satisfying the following KKT system in variational inequality form:

The augmented Lagrangian function for the the separate structure convex quadratic programming problem is defined aswhere .

The variational inequality form of alternating direction method for (12) is as follows.

3.1. The Original Alternating Direction Method

Given , and . For , then consider the following.

Step 1. Consider ; we compute , which satisfies

Step 2. Consider ; we compute , which satisfies

Step 3. Consider ; update the Lagrange multiplier by

Step 4. Consider ; update the Lagrange multiplier by

In Steps 1 and 2, we should solve variational inequalities. In the following analysis, we will convert them to simple projection operations.

Lemma 2 (see [20]). Let be a closed convex set in a Hilbert space and let be the projection of onto . Then

Taking and in (17), we see that (13) is equivalent to the following nonlinear equation:where can be any positive number.

Taking and in (17), we see that (14) is equivalent to the following nonlinear equation:where can be any positive number.

Due to the existence of the terms and in (18), we can not compute directly. We therefore use the following approximate approach which is similar to the one in paper [15]. For certain constants and , letbe the residual between and their linearization at , respectively.

Instead of computing (18), we computeWe choose so that , where and are the largest eigenvalues of and , respectively.

Settingin (21), we have which will be used as an approximation to the solution of variational inequality (13).

Let in (19); we have

In summary, the modified alternating direction method is given as follows.

3.2. The Modified Alternating Direction Method

Given , and . For , then consider the following.

Step 1. Consider ; we compute , which satisfies

Step 2. Consider ; we compute , which satisfies

Step 3. Consider ; update the Lagrange multiplier by

Step 4. Consider ; update the Lagrange multiplier by

From Steps 1 and 2, the modified alternating direction method only needs to compute the metric projection of vectors onto and . From Steps 3 and 4, we could interpret and as the dual stepsizes. Therefore, the iteration of our method is simple and fast.

4. The Convergence Result

In this section, we extended and modified the convergence results of the alternating direction methods for convex quadratically constrained quadratic semidefinite programs in paper [15] and then give the convergence analysis of the alternating direction method for CQSOCP problems with bounded constraints.

Lemma 3. The sequence generated by the modified alternating direction method satisfieswhere is a KKT point of system (11).

Proof. Let in the second inequality in system (11); we haveLet in (14), and coupled with (16), we haveAdding (30) and (31) together, we haveIn addition, from (14) and (16), we haveAdding the two inequality above, we haveNote that (21) can be written equivalently asSetting , we have Let in the first inequality in system (11); we have Adding (36) and (37) together, we haveFrom first part at the left side of (38) and the third equation in system (11), we haveFrom (16), (36), the last equation in system (11), and the second part at the left side of (38), we haveIn addition, from the third part at the left side of (38), we haveIt follows from (32)-(34) and (38)–(41) that

Now, we give the convergent conclusion.

Theorem 4. The sequence generated by the modified alternating direction method converges to a solution point of problem (9).

Proof. We denotewhere denotes the -dimensional unit matrix and is positive definite. Here, we define the -inner product of and asand the associated -norm aswhere .
Observe that, by Lemma 2, solving the optimal condition (11) for problem (9) is equivalent to finding a zero point of the residual function:From (15), (16), and the first equation in (21), we have thatFrom (19) and (16), we have Based on (47)-(48), (15)-(16), and the nonexpansion property of the projection operator, we havewhere is a positive constant depending on parameters , and the largest eigenvalue of and , for example, settingFrom Lemma 3, we can write (29) aswhich implies that Thus From the above inequality, we have That is, the sequence is bounded. Thus there exists at least one cluster point of .
It also follows from (53) thatand thusLet be a cluster point of and the subsequence converges to . We haveso satisfies system (11). Setting , we haveThe sequence satisfies

5. Simulation Experiments

In this section we present computational results by comparing the modified alternating direction method with the interior-point method. The interior-point method is used to solve the transformed mix linear and second-order cone programming problems (3). All the algorithms are run in the MATLAB 7.0 environment on an Inter Core processor 1.80 GHz personal computer with 2.00 GB of Ram.

The test problems are formulated by random method as follows:(1)Given the values of , with .(2)Generate a random matrix , and set . At the same time, generate a random matrix with full row rank.(3)Set , where is a vector whose components are all ones.(4)Given , then generate the random vector and make it an interior point of second-order cone for .(5)We obtain by computing .

The first set of test problems includes 16 small scale CQSOCP problems with bounded constraints, which is shown in Table 1. In Tables 1 and 3, an entry of the form “” in the “SOC” column means that there are 20 5-dimensional second-order cones, and the “ratio” denotes the ratio between the number of the second-order cones and the value of .


Problems SOC Ratio

P01 40 100 1.00%
P02 40 100 21.00%
P03 40 100 20.00%
P04 40 100 33.00%

P05 120 200 0.50%
P06 120 200 17.00%
P07 120 200 20.00%
P08 120 200 33.00%

P09 200 400 0.25%
P10 200 400 16.75%
P11 200 400 20.00%
P12 200 400 33.25%

P13 300 600 0.16%
P14 300 600 11.16%
P15 300 600 20.00%
P16 300 600 33.33%

As is known to all, the interior-point methods have proved to be one of the most efficient class of methods for SOCP. Here the Matlab program codes for the interior-point method are designed from the software package by SeDuMi [4]. In the SeDuMi software, we set the desired accuracy parameter .

Let , where . In the alternating direction method, we stop our algorithm whenfor . Here we set and . We choose the initial point , and , where is the -dimensional vector of ones.

For the first set of test problems, the iteration number and average CPU time are used to evaluate the performances of the modified alternating direction method and the interior-point method by SeDuMi. The test results are shown in Table 2. In the Tables 2 and 4, “Time” represents the average CPU time (in seconds), and “Iter.” denotes the average number of iteration. In addition, “MADM” represents the modified alternating direction method. In Table 4, “/” denotes that the method does not work in our personal computer because the method is “out of memory.”


Problems MADM SeDuMi
Iter. Time Value Iter. Time Value

P01 216 0.14 6.8296723 9 0.34 6.8296742
P02 245 0.25 71.9788074 9 0.31 71.9788099
P03 264 0.30 76.2533942 11 0.39 76.2533927
P04 261 0.34 143.5300081 11 0.44 143.5300023

P05 269 0.33 47.1664568 10 0.81 47.1664516
P06 271 0.56 164.0914031 12 1.16 164.0914052
P07 293 0.66 467.8962158 13 1.28 467.8962123
P08 320 0.89 671.0620751 13 1.34 671.0620812

P09 322 1.53 71.2857981 12 5.19 71.2857926
P10 337 2.28 85.4731057 13 7.41 85.4731061
P11 304 2.14 1085.3976583 15 7.94 1085.3976475
P12 316 2.62 2256.4765633 16 10.11 2256.4765646

P13 351 4.17 89.3228874 11 15.03 89.3228832
P14 377 5.14 113.5586948 12 16.16 113.5586923
P15 327 5.13 2198.2136742 16 27.53 2198.2136739
P16 326 5.81 2727.7797204 16 32.67 2727.7797233


Problems SOC Ratio

P21 400 1000 10.00%
P22 400 1000 16.10%
P23 400 1000 33.30%

P24 600 2000 2.5%
P25 600 2000 26.70%
P26 600 2000 33.33%

P27 800 3000 3.33%
P28 800 3000 26.70%
P29 800 3000 33.33%

P30 1000 4000 2.50%
P31 1000 4000 19.02%
P32 1000 4000 33.32%

P33 2000 5000 2.00%
P34 2000 5000 18.42%
P45 2000 5000 33.32%


Problems MADM SeDuMi
Iter. Time Value Iter. Time Value

P21 326 10.88 1382.4709177 17 136.73 1382.4709168
P22 330 11.61 206.0023854 15 130.14 206.0023835
P23 378 16.78 7714.2171166 17 178.88 7714.2171243

P24 429 58.73 1678.7355144 19 1013.98 1678.7355313
P25 449 65.80 312.2865058 18 1596.53 312.2865057
P26 520 64.86 26103.1426757 19 1640.63 26103.1426826

P27 448 109.83 2882.0125678 / / /
P28 536 141.35 414.5181029 / / /
P29 566 150.15 35810.2685515 / / /

P30 425 190.92 6190.8005238 / / /
P31 506 239.69 801.0388729 / / /
P32 599 289.32 53966.4593065 / / /

P33 348 273.73 10471.3329318 / / /
P34 383 310.31 1145.2045565 / / /
P45 507 413.41 170350.7785126 / / /

Table 2 shows that the modified alternating direction method costs less CPU time than the interior-point method by SeDuMi. But, the iteration number of the interior-point method is less than that of the modified alternating direction method.

In addition, Table 1 gives different kinds of test problem, including the problems with only one large second-order cone, such as P01, P05, P09, and P13, the problems with many small second-order cones, such as P04, P08, P12, and P16, and the problems with one large second-order cone and some small second-order cones, such as P02, P06, P10, and P14. The test results in Table 2 show that the modified alternating direction method can solve different kinds of convex quadratic second-order cone programming problems within appropriate CPU time and accuracy.

The second set of test problems includes 15 medium scale problems, which is shown in Table 3. For the second set of test problems, the test results are shown in Table 4.

The results in Table 4 show the interior point method by SeDuMi does not work for the transformed problem (3) because of being “out of memory” in our personal computer when , but the modified alternating direction method is still efficient because the modified alternating direction method needs less memory space than the interior-point method.

In addition, we add test results of P04 and P12 in smaller criteria and with random initial points. The smaller criteria of our method is . In addition, we do one hundred experiments with the random initial point. The test results are shown in Table 5. In the SeDuMi software, we set the desired accuracy parameter .


Problems MADM SeDuMi
Iter. Time Value Iter. Time Value

P04 (, random point) 275 0.37 169.24049123 21 0.88 169.24049003
P04 (, fixed point) 287 0.41 169.24049234 21 0.88 169.24049003
P04 (, random point) 485 0.62 169.24049007 21 0.88 169.24049003
P04 (, fixed point) 494 0.67 169.24049007 21 0.88 169.24049003

P12 (, random point) 355 2.55 1904.587401 27 20.52 1904.587416
P12 (, fixed point) 367 2.95 1904.587402 27 20.52 1904.587416
P12 (, random point) 628 4.37 1904.587409 27 20.52 1904.587416
P12 (, fixed point) 642 5.20 1904.587408 27 20.52 1904.587416

Table 5 shows that the performances of MADM with random initial points are a bit better than that of MADM with fixed initial points in two different stop criteria. In addition, the number of iteration of MADM with is more than that of MADM with , and the CPU time of MADM with is longer than that of MADM with .

6. Conclusion

In the paper, a modified alternating direction method is proposed for solving convex quadratic second-order cone programming problems with bounded constraints. The proposed method does not require solving subvariational inequality problems over the second cones and the bounded set. At each iteration, we only need to compute the metric projection onto the second-order cones and a projection onto the bounded set. The proposed modified method does not require second-order information and it is easy to implement. The random simulation results show that our method can efficiently solve some convex quadratic second-order cone programming problems of vector size up to 5000 within reasonable time and accuracy by using a desktop computer.

Disclosure

This work was conducted while Xuewen Mu has been visiting Ohio University, Department of Mathematics.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work is supported by China Scholarship Council (CSC). This work was also supported by the National Science Foundations for Young Scientists of China (11101320, 61201297) and the Fundamental Research Funds for the Central Universities (JB150713).

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Copyright © 2015 Xuewen Mu and Yaling Zhang. 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.


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