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Why steepest descent gives a wrong direction search?

  1. May 12, 2015 #1
    1. The problem statement, all variables and given/known data
    I have to minimize the function (x1-1)2+x23+x1x2 by the steepest descent method. The initial point is [1,1]T

    2. Relevant equations


    3. The attempt at a solution
    The gradient of this function is ∇ƒ(x1,x2)=[2(x1-1)-x2 3x22-x1]. 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(x0-α∇ƒ(x0)) in order to find the value of α. So ƒ(x0-α∇ƒ(x0))=-5α+15α2-8α3 and ƒ'(x0-α∇ƒ(x0))=-5+30α-24α2. This function has extreme points in α1=0.95061 and α2=5.094. In order to be a minimum of this curve ƒ''(x0-α∇ƒ(x0))=30-48α has to be positive. This is my problem ƒ''(x0-α∇ƒ(x0) evaluated at both α values is negative so they don´t minimize the direction. So what I am doing wrong?
     
  2. jcsd
  3. May 13, 2015 #2

    SteamKing

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    Just inspecting the gradient of the original function f(x1, x2), something doesn't look right.

    If you take ∂f / ∂x1, how did you obtain [2(x1-1)-x2], specifically, the ' - x2' part? I'm confused, because there were no negative signs between terms in the original definition of f(x1, x2). A similar question arises in what you show to be ∂f / ∂x2.
     
  4. May 13, 2015 #3

    Ray Vickson

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    As SteamKing has pointed out, your gradient formula is incorrect, and your initial steepest-descent direction is wrong. However, when you correct these errors, you will obtain a function ##\phi(\alpha) = f(x_0 - \alpha \nabla f(x_0))## that has no stationary points at all. What does that tell you?
     
  5. May 13, 2015 #4
    Oh! Excuse me! You are right! However I made a mistake when I wrote the original problem. Let me write it again. I have to minimize the function ƒ(x1,x2)=(x1-1)2+x23-x1x2. The initial point is [1,1]T.
     
  6. May 13, 2015 #5
    Excuse me all of you. Finally I got the mistake I made solved. It was an incorrect solutions of ƒ'(x0-α∇ƒ(x0))=-5+30α-24α2 .
     
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