Holetaug

2022-07-02

We need to prove that when we apply the Newton-Raphson method to strictly quadratic concave function. It will converge in one step.

How to apply this method to maximization of
$f\left(x\right)=4\cdot {x}_{1}+6\cdot {x}_{2}-2\cdot {x}_{1}^{2}-2\cdot {x}_{1}\cdot {x}_{2}-2\cdot {x}_{2}^{2}$
I did not understand how to apply method to this function? what should be the interval?

Freddy Doyle

Expert

You don't need an interval. You are using Newton-Raphson to find a solution of the system
$\begin{array}{rl}\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_{1}}& =4-4{x}_{1}-2{x}_{2}=0\\ \frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_{2}}& =6-2{x}_{1}-4{x}_{2}=0\end{array}$
This being a linear system, it doesn't matter where your initial point is: Newton-Raphson simply solves the linear system.

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