5. Process Improvement
5.5.3. How do you optimize a process?
5.5.3.1. Single response case

## Single response: Optimization subject to experimental region constraints

Optimal operating conditions may fall outside region where experiment conducted Sometimes the optimal operating conditions x* simply fall outside the region where the experiment was conducted. In these cases, constrained optimization techniques can be used to find the solution x* that optimizes $$\hat{Y}(x)$$ without leaving the region in the factor space where the experiment took place.
Ridge analysis is a method for finding optimal factor settings that satisfy certain constraints "Ridge Analysis", as developed by Hoerl (1959), Hoerl (1964) and Draper (1963), is an optimization technique that finds factor settings x* such that they
$$\mbox{optimize} \hspace{.5in} \hat{Y}(x) = b_{0} + b'x + x'Bx$$

$$\mbox{subject to} \hspace{.37in} x'x = \rho^{2}$$

The solution x* to this problem provides operating conditions that yield an estimated absolute maximum or minimum response on a sphere of radius ρ. Different solutions can be obtained by trying different values of ρ.
Solve with non-linear programming software The original formulation of Ridge Analysis was based on the eigenvalues of a stationarity system. With the wide availability of non-linear programming codes, Ridge Analysis problems can be solved without recourse to eigenvalue analysis.