Steepest Ascent for Multiple-Response Applications

Abstract

The path of steepest ascent proposed by Box and Wilson consists of points that maximize the predicted response for a fitted first-order model among all points with the same standard error of prediction. When there are multiple responses or additional constraints (on, e.g., cost or throughput), the standard steepest ascent is of limited use, because it often optimizes one response to the detriment of others. We answer three pertinent questions: (1) For multiple-response applications, what search directions are inferior and can be excluded from consideration?; (2) What graphical tools can assist in our understanding the tradeoffs and choosing a compromise direction for exploration?; and (3) Do compromise directions exist that ensure improvement in all responses? We prove that only search directions that are convex combinations of the paths of steepest ascent/descent need be considered. We also construct confidence bounds for improvement of each response relative to its performance at the center of the design. Whereas the literature often deals with multiple response optimization in the later stages of response surface exploration, our article addresses how to proceed after a successful initial screening experiment.

Compromise ascent directions for multiple‐response applications

“…f is the ith simulated 1×q vector from the posterior predictive distribution given in Equation (6) and N is the number of simulations. This provides the experimenter with a measure of the reliability of y f being in S for a given x f .…”