Newton's method in optimization

min f(x), x∈Rn

This is the Newton's method in optimization in nonlinear equations.

g(x)=0

x0 is the starting point.

∇g(x0) is linear and non-singular.

x1=x0+d

∴

Therefore, g(x0+d) is almost Zero.

f(x) is the approximation, and T is Taylor expansion.

This is almost g(x0+d). g is the Jacobian matrix, and ∂ is general topology.