In Myers, Montgomery, & Anderson-Cook, C. M. (2016)$^1$ the authors provide (p. 235) a general methodology to find new $x'$ coordinates in the path of steepest ascent based on constrained optimization of the form
\begin{equation} \beta_0 + \beta_1 x_1 + \dots + \beta_k x_k \\ \text{s.t.} \sum_{i=1}^k x_i ^2 - r^2 = 0 \end{equation}
On which $r$ is the distance from the origin in the cube or hypercube. The authors solve it using Lagrange multipliers
$$ \max L = \mathbf{b'}\mathbf{x} - 2\lambda (\mathbf{x}'\mathbf{x} - r^2) $$
Which they found with $\partial L/\partial x = 0$ to be
$$ x_j = \frac{\beta_j}{2\lambda} $$
However, later on, they provide an algorithm (p. 239) on which new coordinates can be found by computing the step size with $$ \Delta x_j = \frac{\beta_i}{\beta_i/\Delta x_i},\quad i \neq j $$
So that the new point in the path of steepest ascent is just $\text{base} + \Delta x_k$. Should I interpret that $\beta_i/\Delta x_i = 1/2\lambda$?
Furthermore, in the Engineering Statistics Handbook (Technical Appendix 5A), the solution is
$$ x^* = \rho\frac{b}{\lVert b \rVert} $$
On which (I interpret) $\rho$ is the same as $r$ (the distance from the origin).
Are all those solutions equivalent? I interpret this as that the length is less important than the direction (?).
$^1$Myers, R. H., Montgomery, D. C., & Anderson-Cook, C. M. (2016). Response surface methodology: process and product optimization using designed experiments. John Wiley & Sons.
