In the above given monograph (1999, 1E) the following parametrized system of equations $R:\mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ is introduced:
$$ R(z,t) := \left[ \begin{array}{c} c(z) - tAd \\ Z^T (z - x^* - td) \end{array} \right] = 0, $$
where $t>0$ with $\lim\limits_{k\rightarrow\infty}t_k = 0$.
(for a definition of the remaining symbols follow [1] and [2]).
Question: How is this motivated?
I can see that $c(z) - tAd$ is some sort of reformulation of the feasible sequence's property to satisfy the constraints (for $z$ being sufficiently close to $x^*$). But what does $Z^T(z-x^*-td)$ refer to? My first attempt was to interpret is as a reformulation of the requirement of $z_k-x^*/||z_k-x^*||$ to approach $d$. But then I realized that $Z^T(z-x^*-td)=0$ does not imply $(z-x^*-td)=0$..
Ok. I should have just read on (my own private Idaho error source no. 1):
The choice of the lower part of $R$ is motivated by the fact that $z = x^*$ and $t > 0$ never show up simultaneously. Since, by then,
$$ R(x^*, t) = t \underbrace{ \left[ \begin{array}{c} A \\ -Z^T \end{array} \right]}_{\bar{R}} d = 0, $$
which would require, due to the regularity of $\bar{R}$, that $d=0$. Whereas $||d||=1$, by proposition, and the following holds:
$$||d||=1 \wedge (z \neq x^* \vee t = 0).$$