- #1

ymhiq

- 11

- 0

**1. Homework Statement**

I have to minimize the function (x

_{1}-1)

^{2}+x

_{2}

^{3}+x

_{1}x

_{2}by the steepest descent method. The initial point is [1,1]

^{T}

## Homework Equations

## The Attempt at a Solution

The gradient of this function is ∇ƒ(x

_{1},x

_{2})=[2(x

_{1}-1)-x

_{2}3x

_{2}

^{2}-x

_{1}]. This gradient evaluated in the initial point is ∇ƒ(1,1)=[-1 2]. Following the steepest descent method it is mandatory to minimize the function ƒ(x

_{0}-α∇ƒ(x

_{0})) in order to find the value of α. So ƒ(x

_{0}-α∇ƒ(x

_{0}))=-5α+15α

^{2}-8α

^{3}and ƒ'(x

_{0}-α∇ƒ(x

_{0}))=-5+30α-24α

^{2}. This function has extreme points in α

_{1}=0.95061 and α

_{2}=5.094. In order to be a minimum of this curve ƒ''(x

_{0}-α∇ƒ(x

_{0}))=30-48α has to be positive. This is my problem ƒ''(x

_{0}-α∇ƒ(x

_{0}) evaluated at both α values is negative so they don´t minimize the direction. So what I am doing wrong?