multivariate production systems optimization - Stanford University

p is the vector of displacement from x,
g is the gradient n-vector of x,
and H is the Hessian matrix of x,
H =
∂ 2 F
∂x1∂x1
∂ 2 F
∂x2∂x1
∂ 2 F
∂x3∂x1
∂ 2 F
∂xn∂x1
p =
g =
∂ 2 F
∂x1∂x2
∂ 2 F
∂x2∂x2
∂ 2 F
∂x3∂x2
∂ 2 F
∂xn∂x2
Δx1
Δx2
Δx3
Δxn
∂F
∂x1
∂F
∂x2
∂F
∂x3
∂F
∂xn
∂ 2 F
∂x1∂x3
∂ 2 F
∂x2∂x3
∂ 2 F
∂x3∂x3
∂ 2 F
∂xn∂x3
64
∂ 2 F
∂x1∂xn
∂ 2 F
∂x2∂xn
∂ 2 F
∂x3∂xn
∂ 2 F
∂xn∂xn
(6.6)
(6.7)
(6.8)
Considering only the first and second order terms of the Taylor’s series expansion
(Equation 6.4) gives a quadratic approximation of the function F in the neighborhood of x
as
Q x+p = Fx + g T p + 1 2 p T H p (6.9)