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The starting-point for the development of quasi-Newton methods is the Newton equation (Broyden, 1970), which prescribes a condition which the Hessian (evaluated at a specified point) must satisfy. The “Newton Equation”, which may be regarded as a generalization of the “Secant Equation” (Byrd et al., 1988; Dennis and Schnable, 1979), is usually employed in the construction of quasi-Newton methods for optimization. According to Ortize et al. (2004): “Many designed experiments require the simultaneous optimization of multiple responses. A common approach is to use a desirability function combined with an optimization algorithm to find the most desirable settings of the controllable factors” (p. 432).
This work directs attention to problems of the form:
for a twice continuously differentiable function f. Let 
be the objective function, where 
, and let 
and 
denote the gradient and Hessian of 
, respectively. If we define 
to denote a differentiable path in 
, where 
, then, upon applying the Chain Rule to 
in order to determine its derivative with respect to 
, we obtain