On certain optimization problems related to matrix norms.

In this paper, first, we show that the following optimization problem always has an optimal solution which does not depend on the matrix norm, and give a sufficient condition for the uniqueness of optimal solutions:(Section.Display) whereis any unitarily invariant norm,, andor. Then, we characterize the optimal solutions and also give sufficient conditions for the uniqueness and the rank-one property of optimal solutions for the following optimization problem:(Section.Display) where, is the inner product of matrices;,,, and’s,are different arbitrary norms. [ABSTRACT FROM AUTHOR]