Algorithm for inequality-constrained least squares problems
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Algorithm for inequality-constrained least squares problems Jing-Jing Peng1 · An-Ping Liao1
Received: 3 July 2014 / Revised: 6 March 2015 / Accepted: 20 March 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract In this paper, we investigate the following inequality-constrained least squares problem min AX B − C s.t. E X F ≥ D,
X symmetric.
The necessary and sufficient conditions for the solvability of the problem are established. An iteration method to compute the solution of the problem is presented, and some convergence results of the algorithm are proved. Numerical experiments to illustrate the effectiveness of the algorithm are given. Keywords
Matrix equation · Matrix inequality · Iterative method · Nonnegative matrix
Mathematics Subject Classification
15A24 · 15A39 · 65F30
1 Introduction Let m, n, p, q, t be positive integers. The symbols R m×n and S R n×n denote, respectively, the set of m × n real matrices and the set of n × n real symmetric matrices. The inequality A ≥ B, for any two real n × n matrices, means that Ai j ≥ Bi j , for all 1 ≥ i, j ≤ n. We use
Communicated by Ernesto G. Birgin. Research is supported by National Natural Science Foundation of China (11271117,11261014,11301107).
B
Jing-Jing Peng [email protected] An-Ping Liao [email protected]
1
Hunan University, Changsha 410082, People’s Republic of China
123
J.-J. Peng, A.-P. Liao
A denotes the Frobenius norm of the matrix A, defined as A2 = A, A =
n
Ai2j
i, j=1
where the inner product is given by A, B = trace(A T B). In this paper, we are concerned with the following problem: Problem I Given matrices A ∈ R m×n , B ∈ R n× p , C ∈ R m× p , E ∈ R q×n , F ∈ R n×t and D ∈ R q×t , find X such that 1 AX B − C2 2 subject to E X F ≥ D, X ∈ S R n×n . minimize F(X ) =
The interest that we consider this problem stems from the following reasons: first, solutions (or least squares solutions) X to the matrix equation AX B = C with special structures have been widely studied. For example, different kinds of the conjugate gradient-type methods to compute symmetric solutions were considered in Deng et al. (2006), Huang and Nong (2010), Liao and Lei (2007), Peng et al. (2005), and to compute least squares symmetric solutions were considered in Lei and Liao (2007), Peng (2005), Qiu et al. (2007). Ding et al. (2008, 2010) used the extended Jacobi and Gauss–seidel iterative methods to compute general solutions, and Huang et al. (2008) used the conjugate gradient-type method to compute skew-symmetric solutions. Using matrix general inverse and various kinds of matrix decompositions, the necessary sufficient conditions for the existence of and the expression for the reflexive solutions, the general reflexive solutions and the symmetric solutions were given by Cvetkovic-Ilic (2006), Yuan and Dai (2008) and Hua (1990), respectively. The Hermitian nonnegative definite solutions were investigated by Zhang (2004), and the Re-nonnegative definite solutions were investigated by Cvetkovic-Ilic (2008), Wang
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