# RSM and steepest ascent for new $x'$ coordinates

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.