Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 379734, 10 pages

http://dx.doi.org/10.1155/2015/379734

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

^{1}School of Mathematics and Statistics, Xidian University, Xi’an 710071, China^{2}School of Computer Science, Xi’an Science and Technology University, Xi’an 710054, China

Received 20 October 2014; Revised 26 March 2015; Accepted 30 March 2015

Academic Editor: Gerhard-Wilhelm Weber

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.

#### 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 [6–8]. 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 [17–19]wherewith if and any vector in satisfying if .

Next we introduce the projection lemma over the second-order cone [17–19].

Lemma 1 (see [17–19]). *For any , let be the projection of onto the second-order cone ; then we have**where 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 .*