if the matrix norm is consistent and $\mu_i \neq 0$ and $\mu_i \neq 1$ , $i=1,2$. Based on the literature on optimal designs [@Fredov1972; @Goodwin1977; @Atkinson1982; @Pronzato1985; @Berger1994; @Pukelsheim1993], the covariance matrix satisfies the following inequality $$\mbox{cov}(\upsilon) \geq \mathcal{F}^{-1}(\upsilon)$$ where, $\mbox{cov}$ denotes the covariance matrix, $\upsilon$ is the parameter vector and $\mathcal{F}$ denotes the Fisher information matrix. In this case, $\mathcal{F}^{-1}$ is the lower bound of the covariance of estimations (known as the Cram' er-Rao bound [@Cramer1946; @Rao1945]). It is a common approach that in order to reduce the covariance of estimations, the the Cram' er-Rao bound is reduced by maximizing the Fisher information matrix.

In this section, we obtain an upper bound of the norm of $M$ and $m$. We then derive a sub-optimal solution, which is based on the maximization of the upper bound of $M$ and $m$. First, we need the matrix norm to be consistent.

([@Lyche2012], §8.1.1, p. 178) A matrix norm is [*consistent*]{} if it is defined on ${\ensuremath{\mathbb{C}}}^{q\times r}$ for all $q,r\in {\ensuremath{\mathbb{N}}}$ and the sub-multiplicative property $$\begin{aligned}
\label{ableqab}
\| AB \| \leq \|A\| ~\|B\|\end{aligned}$$ holds for all matrices $A$ and $B$ for which the product $AB$ is defined.

In this paper, the consistent 