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.
翻译:我们显示,美元-BXC\rVert$的全球最低解决方案可以在单值分解和通用单值分解的封闭式模式中找到,这些封闭式解决方案涉及对美元的各种限制,包括等级、规范、对称、双面产品和规定的egenvaly。这把弗里德兰-托罗赫蒂解决普遍等级限制的近似问题的办法扩大到其他制约因素,并为单值分解的等级限制提供了替代解决方案。对于涉及美元较复杂的限制,如托普利茨、汉克尔、比尔库兰、非强化性、静态、正半确定性、处方能等结构,我们证明简单的迭代方法是直线性的,而且在全球范围内与全球最低解决方案趋同。