Let $\mathbf{A}$ and $\mathbf{U}$ be arbitrary complex $M\times N$ and $N\times M$ matrices, respectively. Let denote superscript $(\cdot)^{\dagger}$ and $(\cdot)^{\mathrm{H}}$ as pseudo-inverse and conjugate-transpose operations. Then, is there any close-form solution for the following minimax optimization problem: $$\min_{\mathbf{U}}\max_n [(\mathbf{A}^{\dagger}+\mathbf{B}\mathbf{U})(\mathbf{A}^{\dagger}+\mathbf{B}\mathbf{U})^{\mathrm{H}}]_{n,n},$$ where $\mathbf{B}=\mathbf{I}-\mathbf{A}^{\dagger}\mathbf{A}$ and denotes the $n$th diagonal entries of matrix, and $\mathbf{I}$ is the identity matrix.
Is there any propose for choosing $\mathbf{U}$ to have $$\max_n [(\mathbf{A}^{\dagger}+\mathbf{B}\mathbf{U})(\mathbf{A}^{\dagger}+\mathbf{B}\mathbf{U})^{\mathrm{H}}]_{n,n}<\max_n [\mathbf{A}^{\dagger}(\mathbf{A}^{\dagger})^{\mathrm{H}}]_{n,n}.$$
