核范数和谱范数下广义Sylvester方程最小二乘问题的有效算法

本文从数值角度讨论Schatten q-范数下的广义Sylvester方程约束最小二乘问题min x∈s‖N∑i=1A_iXB_i—C‖_q,其中S为闭凸约束集合,Schatten q-范数定义为‖M‖_q~q=∑_(i=1)~nσ_i~q(M),其中σ_i(M)为M∈R~(n×n)的奇异值.该问题的几类特殊情形在图像处理、控制论等领域有广泛的应用.q=2即Frobenius范数下该问题已被充分研究,故本文着重讨论q=1,+∞,即核范数和谱范数下该问题的数值求解.采用的数值方法是非精确标准容易执行的部分非精确交替方向法,并结合奇异值阈值算法,Moreau-Yosida正则化算法,谱投影算法和LSQR算法等求解相应子问题.给出算法的收敛性证明,并用数值算例验证其高效可行性.

AN EFFICIENT METHOD FOR SOLVING GENERALIZED SYLVERSTER EQUATION MINIMIZATION PROBLEM UNDER THE NUCLEAR AND SPECTRAL NORM

In this paper,we are concerned with the following generalized Sylvester equation least squares problem of the form
min_X∈S||∑_i=1^NA_iXB_i-C||_q,
,where||.||stands for the Schatten q-norm,which defined as||M||_q^q=∑_i=1ⁿσ_i^q(M) and σ_i(M)(i=1,...,n) be the singular values of M∈R^n×n,S be the closed convex set.Some special types of this problem can be applied in image processing and control theory.An inexact version of alternating direction method (ADM) with truly implementable inexactness criteria is proposed for solving this problem under the nuclear norm and spectrum norm,namely q=1,+∞,combining with the Singular Value Threshold algorithm,Moreau-Yosida regularization algorithm,Spectral Projection algorithm and LSQR algorithm to deal with the generated subproblems.Numerical experiments are performed to illustrate the feasibility and efficiency of the proposed algorithm with randomly generated data.