Abstract
We propose a feasible and effective iteration method to find solutions to the matrix equationAXB=Csubject to a matrix inequality constraintDXE≥F, whereDXE≥Fmeans that the matrixDXE-Fis nonnegative. And the global convergence results are obtained. Some numerical results are reported to illustrate the applicability of the method.
Highlights
In this paper, we consider the following problem: AXB = C, (1)DXE ≥ F, where A ∈ Rp×m, B ∈ Rn×q, C ∈ Rp×q, D ∈ Rs×m, E ∈ Rn×t, and F ∈ Rs×t are known constant matrixes and X ∈ Rm×n is unknown matrix.The solutions X to the linear matrix equation with special structures have been widely studied, for example, symmetric solutions, R-symmetric solutions, (R, S)symmetric solutions, bisymmetric solutions, centrosymmetric solutions, and other general solutions
The smallest nonnegative deviation problem (8) is equivalent to the following optimization problem: minimize P (G, Y, Z) = ‖Y‖2 subject to D (G − A+AGBB+) E + Y − Z = F, (11)
We propose Algorithm 8 to find solutions to the matrix equation AXB = C subject to a matrix inequality constraint DXE ≥ F
Summary
DXE ≥ F, where A ∈ Rp×m, B ∈ Rn×q, C ∈ Rp×q, D ∈ Rs×m, E ∈ Rn×t, and F ∈ Rs×t are known constant matrixes and X ∈ Rm×n is unknown matrix. In 2012, Peng et al (see [32]) proposed a feasible and effective algorithm to find solutions to the matrix equation AX = B subjected to a matrix inequality constraint CXD ≥ E based on the polar decomposition in Hilbert space. Motivated and inspired by the work mentioned above, in this paper, we consider the solutions of the matrix equation AXB = C over linear inequality DXE ≥ F constraint. We use the theory on the analytical solution of matrix equation AXB = C to transform the problem into a matrix inequality smallest nonnegative deviation problem.
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