Maximal determinant over a certain class of matrices and its application to D-optimality of designs

It is known that some optimality criteria of experimental designs are functionals of the eigenvalues of their information matrices. In this context we study the problem of maximizing the determinant of $\alpha {\mathbf{I}}_t-(\mathbf{P}+{\mathbf{P}}^T)$, $\alpha >2$, over the class of t-by-t permutation matrices, and the determinant of $\alpha {\mathbf{I}}_t+\mathbf{P}+{\mathbf{P}}^T$, $\alpha \ge 2.5$, over the class of t-by-t permutation matrices with zero diagonal (derangement matrices). The results are used to characterize D-optimal complete block designs under an interference model.