界约束下算子方程最小二乘问题的条件梯度法

CONDITIONAL GRADIENT ALGORITHM FOR SOLVING OPERATOR EQUATION LEAST SQUARES PROBLEM UNDER THE BOUND CONSTRAINTS

研究如下界约束下算子方程最小二乘问题:min x∈Ω‖L(X:A_1,…,At;B_1,…,B_t)-T‖~2,其中‖.‖为Frobenius范数,L(X:A_1…A_t;B_1,…,B_t)为关于X的线性矩阵算子(或齐次线性变换),Ai∈R^(p×m),B_j∈R^(n×q)i,j=1,…,n为算子L的系数矩阵,丁为右端矩阵,ΩR^(m×n)为界约束凸集合.提出了求解问题的条件梯度迭代算法及其简要收敛性分析,并给出条件梯度算法的几类加速形式.随机数据和图像恢复模型数据的实验结果表明说明算法是可行高效的.

In this paper we discuss the operator equation least squares problem under the bound constraints with the form min min x∈Ω｜｜￡（X：A1,…,At;B1,…,Bt）-T｜｜^2, where ｜｜.｜｜ denotes the Frobenius norm,￡（X：A1,…,At;B1,…,Bt） denotes the linear matrix operator or homo- geneous linear transform to the unknown matrix X ∈ R^m×n,Ai∈R^p×m,Bj∈Rn×qi,j=1,…,nn are the coefficient matrices of the operator ￡, T is the right hand matrix, ΩCRRm×n is the bound constrained sets. A type of conditional gradient algorithm is pro- posed for solving this problem and its convergence analysis are then proved. We also describe some new improvement of the conditional gradient algorithm. Numerical experiments are performed to illustrate the feasibility and efficiency of the proposed algorithm, including when the algorithm is tested with randomly generated data and on some image restoration problems.