Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 7175385, 8 pages
https://doi.org/10.1155/2017/7175385

Research Article

Uniqueness of the Minimal -Norm Solution to the Monotone Linear Complementarity Problem

Ting Zhang¹ and Xiaoqin Jiang²

¹Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
²Department of Public Basic, Wuhan Technology and Business University, Wuhan 430065, China

Correspondence should be addressed to Xiaoqin Jiang; jiangxq_xx@sina.com

Received 12 October 2016; Accepted 13 December 2016; Published 4 January 2017

Academic Editor: Wanquan Liu

Copyright © 2017 Ting Zhang and Xiaoqin Jiang. 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.

The linear complementarity problem (LCP) has wide applications in economic equilibrium, operations research, and so on, which attracted a lot of interest of experts. Finding the sparsest solution to the LCP has real applications in the field of portfolio selection and bimatrix game. Motivated by the approach developed in compressive sensing, we may try to solve an -minimization problem to obtain the sparsest solution to the LCP, where an important theoretical problem is to investigate uniqueness of the solution to the concerned -minimization problem. In this paper, we investigate the problem of finding the minimal -norm solution to the monotone LCP and propose a sufficient and necessary condition for the uniqueness of the minimal -norm solution to the monotone LCP, which provides an important theoretical basis for finding the sparsest solution to the monotone LCP via solving the corresponding -minimization problem. Furthermore, several examples are given to confirm our theoretical finding.

1. Introduction

The linear complementarity problem (LCP) can be described as follows; we want to find a vector such that

where and are given. If is positive semidefinite, then problem (1) is called a monotone LCP. Instances of the LCP can date back to 1940, but it became a scientific direction of study until the mid-1960s. This class of problems has many applications in economics and engineering [1–4]. In this paper, we denote (1) by the and the solution set of the by .

Compressive sensing (CS) has attracted plenty of attention in the fields of signal processing and machine learning [5, 6]. The basic model of CS can be described as the problem of seeking the sparsest solution under linear equality constraints; that is,

where , , and denotes the number of nonzero components of the vector . Since (2) is known to be an NP-hard problem, a tractable approximation is the commonly considered -minimization problem:

where is the -norm of , that is, the sum of each component’s absolute value.

Problem (3) is a convex program which can be reformulated as a linear program [7]. In most cases, -minimization problem can find the sparsest solution to the underdetermined linear system effectively from a lot of empirical results. Recently, the theoretical results with respect to the equivalence of -minimization problem and -minimization problem have made significant progress and led to a sharp increase of research in CS [8–10]. In detail, the mutual-coherence-based analysis [11], the null space property (NSP) [12], the restricted isometry property (RIP) [10], and the range space property (RSP) [13] can guarantee the equivalence of problems (2) and (3), where uniqueness of problem (3) plays an important role in the theoretical analysis.

Problem (2) is to find the sparest solution to the system of equalities. However, only the equalities may be not enough for practical problems sometimes. As the deepening of the research, the nonnegative sparsest solution to the underdetermined linear system of equations has attracted people’s interesting, and some theoretical results have been obtained, such as the RIP [14], the NSP [15], and the RSP [16]. Furthermore, Zhang et al. [17] proposed the conditions to guarantee the equivalence between the -norm solution to the system of absolute value equations and its -norm solution, where a key is to characterize uniqueness of the solution to the corresponding -minimization problem.

Seeking the minimal -norm solution of the LCP has many applications in real world, such as portfolio selection [18] and bimatrix game [1]. Chen and Xiang [19] investigated the characterization and computation of sparse solutions and least--norm () solutions to the LCP and provided some conditions on the involved matrix such that a sparse solution can be found by solving a convex minimization problem. Shang et al. [20] studied minimal -norm solutions of the LCP. They investigated the approximation of the concerned problem by using -norm () and proposed a sequential smoothing gradient method to solve it.

It is well-known that the existence and uniqueness of the solution to the play an important role in theory and algorithms for solving the [1–3]. Motivated by the researches mentioned above, in this paper, we consider the following -minimization problem:

where is positive semidefinite and . It is a relaxation of the problem to find the sparsest solution to the LCP:

We will propose a sufficient and necessary condition to guarantee uniqueness of the solution to problem (4), which provides an important theoretical basis in studying the equivalence between problems (4) and (5) so that the sparest solution to the monotone LCP (1) can be found by solving problem (4).

This paper is organized as follows. Some basic concepts and results for the LCP are stated in Section 2. In Section 3, we investigate the necessary conditions for uniqueness of the solution to problem (4) if it has at least a solution. In Section 4, we propose a sufficient and necessary condition for uniqueness of the minimal -norm solution to the monotone LCP. In Section 5, we give two specific examples to confirm the theoretical result. Conclusions and further work are arrived at the last section.

Now, we outline the notations that will be used in this paper. We denote throughout this paper. For any matrix , index sets and , denotes the submatrix of matrix with rows in , denotes the submatrix of with columns in , and is the submatrix with rows in and columns in , where denotes the number of the elements in . For any vector and an index set , denotes the vector with components of in set . For the vector with , we define index sets and by and , respectively. denotes the vector whose components are all one. For simplicity, we denote as for any vectors .

2. Preliminaries

For the (1), the following results are significant in our sequential analysis.

Theorem 1 (Cottle et al. [1]). Let be positive semidefinite and . Then,

(a)if the (1) is feasible, that is, there exists a vector such that and hold, then the (1) is solvable; that is, ;(b)if the has a solution, then is polyhedral and

 where is an arbitrary solution.

Suppose that is positive semidefinite and the has a solution, denoted by ; then by using Theorem 1(b), we have

where and .

Thus, in order to investigate uniqueness of the sparsest solution to the monotone , we consider the following problem:

where is positive semidefinite, , and . Since problem (8) is NP-hard, we consider its relaxed problem:

which can be rewritten as follows:

Suppose that is an optimal point of problem (10). Then, for inequality constraint , we denote the index set of the active constraints at point by and the index set of the inactive constraints by , which means

These notations will be used throughout this paper.

At the end of this section, we recall a classical theory for linear programming (LP). Consider the LP

and its dual

where is a given matrix and and are two given vectors. The following result is significant for our analysis.

Theorem 2. If the LP (12) and its dual (13) are both feasible, then there exists a pair of strictly complementary solutions satisfying .

In the following, we will discuss some conditions for uniqueness of the solution to problem (10), which further leads to a sufficient and necessary conditions for uniqueness of the minimal -norm solution to the monotone .

3. Necessary Conditions

In this section, we give two necessary conditions for uniqueness of the solution to problem (10) under the assumption that it has at least a solution.

Suppose that is an optimal point to problem (10); then problem (10) has a unique optimal solution if and only if

Since and , it follows that and . Thus the above relationship can be written as

So we have the following lemma.

Lemma 3. Suppose that is an optimal point to problem (10); then problem (10) has a unique optimal solution if and only if is the unique optimal point of the following problem:

The dual problem of problem (16) is given by

Furthermore, by introducing slack variables , problem (17) can be converted to the following problem:

Based on problems (16) and (18), by using the dual theory of the LP, we will show the following result.

Theorem 4. If is the unique optimal point of problem (10) with being defined by (11), then there exist and such that

Proof. Since problems (16) and (18) are both feasible, there exists a pair of strictly complementary solutions of (16) and (18) by Theorem 2. Denote the strictly complementary solution pair by and . Then, we have

Since is the unique optimal point of the problem (10), it follows from Lemma 3 that . Thus, if and if and . These imply

That is,

Furthermore, since , it follows that if and if , which imply that

Substituting (23) into (22), we have

Let ; then . So we have

By setting , then we have

and by (25),

These, together with , imply the desired result.

In this paper, condition (19) given in Theorem 4 is called the range space property (RSP).

Now we establish the other necessary condition.

Theorem 5. If is the unique optimal point of problem (10) with being defined by (11), then

has column full rank.

Proof. Define

we first show that the matrix has column full rank. Suppose that does not have column full rank; then there must exist a nonzero vector such that . On the one hand, let be given by , , and ; it is easy to see that is an optimal point to problem (16). On the other hand, let be given by , , and . Since , there must exist a small satisfying and . Thus is also an optimal point to problem (16). However, since , we obtain two different optimal points of problem (16), which contradicts Lemma 3. Thus, the matrix defined by (29) has column full rank.
Since is the unique optimal point of problem (10), it follows from Theorem 4 that condition (19) holds, which further implies that there exist and such that

Thus, we have , which leads to the desired result.

4. Sufficient and Necessary Condition

In this section, we first discuss the sufficient and necessary condition for uniqueness of the solution to problem (10), which is given as follows.

Theorem 6. Suppose that is an optimal point to problem (10) with being defined by (11). Then, is the unique optimal point of problem (10) if and only if the matrix defined by (28) has column full rank and there exist and such that RSP condition (19) holds.

Proof. By Theorems 4 and 5, we only need to prove the sufficiency of Theorem 6. Suppose that there exist and such that RSP condition (19) holds. Then, there exists such that . Since for any and for any , by letting , we have

Let

Clearly, is a feasible point of problem (17); then we prove that is an optimal point of problem (17). In fact, the objective function value of problem (17) at can be calculated as follows:

By weak duality, the maximal value of the dual problem (17) is 0. Therefore, the point is an optimal point to problem (17). Then the slack variables in (18) have the properties which are described in the following:

Suppose that is an arbitrary optimal point of problem (16), in order to prove the uniqueness of the optimal point for the problem (10); we just need to verify by Lemma 3. Using Theorem 2, we have

Using (34), we obtain that

According to the constraints of problem (16), we have

Let , then we have

which mean Since has column full rank, we have ; that is, . Furthermore, we can obtain and .

By using Theorems 1(a) and 6, it is easy to obtain the sufficient and necessary condition for uniqueness of the solution to the monotone , which is given as follows.

Theorem 7. Let be positive semidefinite and . Suppose that there exists a vector such that and hold; then . Furthermore, let with being defined by (11); then is the unique optimal point of problem (4) if and only if the matrix defined by (28) has column full rank, and there exist and such that RSP condition (19) holds.

5. Examples

In this section, we give two examples to verify the obtained sufficient and necessary condition for uniqueness of the minimal -norm solution to the monotone .

Example 8. Consider the , where

It is easy to find that

is the unique minimal -norm solution of the . In the following, we show that all conditions in Theorem 7 hold for this problem.

First, it is easy to see that the matrix is positive semidefinite.

Second, it is easy to obtain that

and hence, the matrix defined by (28) can be written as

Obviously, has column full rank.

Third, there exist

such that , which shows that the RSP conditions (19) hold at .

Example 9. Consider the , where

It is easy to find that

are two minimal -norm solutions of the , so the minimal -norm solution is not unique. In the following, we show that some condition in Theorem 7 does not hold for this problem.

If we take , then

In this case, we show that the RSP conditions (19) do not hold. In fact, if the RSP conditions hold at , then we have with . Denote and ; then from , we can obtain

that is,

which is impossible. So the RSP conditions (19) do not hold in this case.

If we take , then

In this case, the matrix defined by (28) is

Obviously, does not have column full rank.

6. Conclusions

In this paper, we investigated the uniqueness of the minimal -norm solution to the monotone . By using an equivalence reformulation, we proposed a sufficient and necessary condition to guarantee uniqueness of the minimal -norm solution to the monotone LCP, which provide an important theoretical basis for finding the sparsest solution to the monotone LCP.

Some issues are worth studying in our further research. It is meaningful to investigate various recovery conditions and design high-efficiency algorithms to seek the minimal -norm solution to the monotone LCP. Moreover, since the nonconvex relaxation methods have been studied extensively in the last years (see, e.g., [19–23] and references therein), it is also worth investigating theory and algorithms for solving some nonconvex relaxation problems of finding the sparsest solution to the monotone LCP.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was partially supported by National Nature Science Foundation of China (no. 11431002), the Science Fund of Educational Commission of Hubei Province in China (no. B2015335), and the Science Fund of Wuhan Technology and Business University (no. A2014024).

References

R. W. Cottle, J.-S. Pang, and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, Mass, USA, 1992.

F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, NY, USA, 2003.

J. Han, N. H. Xiu, and H. D. Qi, Theory and Methods for Nonlinear Complementarity Problems, Shanghai Science and Technology Press, Shanghai, China, 2006 (Chinese).

M. C. Ferris and J. S. Pang, “Engineering and economic applications of complementarity problems,” SIAM Review, vol. 39, no. 4, pp. 669–713, 1997.

Z. Lei and Q. Chunting, “Application of compressed sensing theory to radar signal processing,” in Proceedings of the 2010 3rd IEEE International Conference on Computer Science and Information Technology (ICCSIT '10), pp. 315–318, July 2010.

Y. C. Eldar and G. Kutyniok, Compressed sensing, Cambridge University Press, 2012.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Journal on Scientific Computing, vol. 20, no. 1, pp. 33–61, 1998.

D. L. Donoho and J. Tanner, “Neighborliness of randomly projected simplices in high dimensions,” Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 27, pp. 9452–9457 (electronic), 2005.

D. L. Donoho, “High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension,” Discrete & Computational Geometry, vol. 35, no. 4, pp. 617–652, 2006.

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4203–4215, 2005.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 489–509, 2006.

A. Cohen, W. Dahmen, and R. DeVore, “Compressed sensing and best k-term approximation,” Journal of the American Mathematical Society, vol. 22, no. 1, pp. 211–231, 2009.

Y.-B. Zhao, “RSP-based analysis for sparsest and least ℓ ₁-norm solutions to underdetermined linear systems,” IEEE Transactions on Signal Processing, vol. 61, no. 22, pp. 5777–5788, 2013.

A. M. Bruckstein, M. Elad, and M. Zibulevsky, “On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations,” IEEE Transactions on Information Theory, vol. 54, no. 11, pp. 4813–4820, 2008.

M. Khajehnejad, A. G. Dimakis, W. Xu, and B. Hassibi, “Sparse recovery of positive signals with minimal expansion,” https://arxiv.org/abs/0902.4045

Y.-B. Zhao, “Equivalence and strong equivalence between the sparsest and least ℓ ₁-norm nonnegative solutions of linear systems and their applications,” Journal of the Operations Research Society of China, vol. 2, no. 2, pp. 171–193, 2014.

M. Zhang, Z.-H. Huang, and Y.-F. Li, “The sparsest solution to the system of absolute value equations,” Journal of the Operations Research Society of China, vol. 3, no. 1, pp. 31–51, 2015.

J. Xie, S. M. He, and S. Z. Zhang, “Randomized portfolio selection with constraints,” Pacific Journal of Optimization, vol. 4, no. 1, pp. 89–112, 2008.

X. Chen and S. Xiang, “Sparse solutions of linear complementarity problems,” Mathematical Programming, vol. 159, no. 1, pp. 539–556, 2016.

M. Shang, C. Zhang, and N. Xiu, “Minimal zero norm solutions of linear complementarity problems,” Journal of Optimization Theory and Applications, vol. 163, no. 3, pp. 795–814, 2014.

R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Processing Letters, vol. 14, no. 10, pp. 707–710, 2007.

Y.-F. Li, Z.-H. Huang, and M. Zhang, “Entropy function-based algorithms for solving a class of nonconvex minimization problems,” Journal of the Operations Research Society of China, vol. 3, no. 4, pp. 441–458, 2015.

M. Zhang, Z.-H. Huang, and Y. Zhang, “Restricted p-isometry properties of nonconvex matrix recovery,” IEEE Transactions on Information Theory, vol. 59, no. 7, pp. 4316–4323, 2013.