On optimality of designs with three distinct eigenvalues

Let $\D_{v,b,k}$ denote the family of all connected block designs with $v$ treatments and $b$ blocks of size $k$. Let $d\in\D_{v,b,k}$. The replication of a treatment is the number of times it appears in the blocks of $d$. The matrix $C(d)=R(d)-\frac{1}{k}N(d)N(d)^\top$ is called the information matrix of $d$ where $N(d)$ is the incidence matrix of $d$ and $R(d)$ is a diagonal matrix of the replications. Since $d$ is connected, $C(d)$ has $v-1$ nonzero eigenvalues $\mu_1(d),...,\mu_{v-1}(d)$. Let $\D$ be the class of all binary designs of $\D_{v,b,k}$. We prove that if there is a design $d^*\in\D$ such that (i) $C(d^*)$ has three distinct eigenvalues, (ii) $d^*$ minimizes trace of $C(d)^2$ over $d\in\D$, (iii) $d^*$ maximizes the smallest nonzero eigenvalue and the product of the nonzero eigenvalues of $C(d)$ over $d\in\D$, then for all $p>0$, $d^*$ minimizes $(\sum_{i=1}^{v-1}\mu_i(d)^{-p})^{1/p}$ over $d\in\D$. In the context of optimal design theory, this means that if there is a design $d^*\in\D$ such that its information matrix has three distinct eigenvalues satisfying the condition (ii) above and that $d^*$ is E- and D-optimal in $\D$, then $d^*$ is $\Phi_p$-optimal in $\D$ for all $p>0$. As an application, we demonstrate the $\Phi_p$-optimality of certain group divisible designs. Our proof is based on the method of KKT conditions in nonlinear programming.