论文标题
广义矩阵接近性问题
Generalized matrix nearness problems
论文作者
论文摘要
我们表明，$ \ lvert a-bxc \ rvert $的全球最低解决方案可以在封闭形式中找到具有奇异值分解的封闭形式，并且对于$ x $涉及等级，标准，对称性，对称性，两侧产品和规定的特定eigenvalue的$ x $的$ x $限制的广义奇异值分解。这将Friedland--torokhti的解决方案扩展到了广义的等级约束近似问题到其他约束，并为奇异值分解提供了秩约束的替代解决方案。对于涉及诸如Toeplitz，Hankel，循环系统，非负性，随机性，积极的半足以，处方的特征向量等结构的$ x $的更为复杂的限制，我们证明了一种简单的迭代方法是线性的，全球收敛到全球最小的解决方案。
We show that the global minimum solution of $\lVert A - BXC \rVert$ can be found in closed-form with singular value decompositions and generalized singular value decompositions for a variety of constraints on $X$ involving rank, norm, symmetry, two-sided product, and prescribed eigenvalue. This extends the solution of Friedland--Torokhti for the generalized rank-constrained approximation problem to other constraints as well as provides an alternative solution for rank constraint in terms of singular value decompositions. For more complicated constraints on $X$ involving structures such as Toeplitz, Hankel, circulant, nonnegativity, stochasticity, positive semidefiniteness, prescribed eigenvector, etc, we prove that a simple iterative method is linearly and globally convergent to the global minimum solution.
