Chapter 11. Supplemental Text Material S11-1. The Method of Steepest Ascent The method of steepest ascent can be derived as follows. Suppose that we have fit a first-order model ±±±yxiiik=+=∑ββ01and we wish to use this model to determine a path leading from the center of the design region x= 0that increases the predicted response most quickly. Since the first–order model is an unbounded function, we cannot just find the values of the x’s that maximize the predicted response. Suppose that instead we find the x’s that maximize the predicted response at a point on a hypersphere of radius r. That is Max subject to±±yxxriikiiik=+===∑∑ββ01212The can be formulated as Max Gxxiikiiik=+−−rLNMOQP==∑∑ββλ01212±where λis a LaGrange multiplier. Taking the derivatives of Gyields ∂∂=−−∂∂= −−LNMOQP=∑GxxikGxriiiiik±, ,,βλλ21 2212"Equating these derivatives to zero results in xixriiiik====∑±, ,,βλ21 2212"kNow the first of these equations shows that the coordinates of the point on the hypersphere are proportional to the signs and magnitudes of the regression coefficients (the quantity 2λis a constant that just fixes the radius of the hypersphere). The second equation just states that the point satisfies the constraint. Therefore, the heuristic description of the method of steepest ascent can be justified from a more formal perspective.
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