The Linear and Quadratic Approximations and Newton Raphson

A program about Linear and Quadratic Approximations and Newton Raphson

%In my Optimization and Applications class in 2022, the computer science master students at Xiangtan University expressed their difficulties in making sense of the content of linear and quadratic approximations. Therefore, I compose this code to provide them with the necessary help. The code segment consists of three parts, surrogate1, surrogate2, and Newton-Raphson, to compare the searching trajectories corresponding to each optimization model on the 2-dimensional Rosenbrock benchmark.

In the first part, the first-order derivative is approximated by using central difference formula, PFx_f(x,y) = (f((x,y)+delta(x,y))- f((x,y)+delta(x,y)))/(delta(x))

In the second part, the quadratic model is assumed as, f(x,y) = a+b*x+c*y+d*x^2+e*y^2+ f*x*y+O(error), let X = [a,b,c,d,e,f]

In the third part, a standard Newton-Raphson algorithm is processed.

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The first edition was written on 17 Jan 2019, UoW,AU

The recent edition is revised on 16 Oct 2022, Changsha, China

f(x,y) = 100*(y-x^2)^2+(1-x)^2;