Steepest descent method algorithm

Question

Luqman Saleem 2019-9-17

2
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/480830-steepest-descent-method-algorithm

评论： Saleh Msaddi 2020-3-9

For practice purpose, I want to find minima of -humps() function.

I have written the following code but it's not giving correct answer

clear; clc;
%function
f = @(x) -humps(x);
dx = 0.1; %step length
x_current = 1; %starting guess
delta = 1e-4; %threshold value
alpha = 0.1; %finding optimal step length
g = inf; %starting gradient
while norm(g) > delta
    
    %gradient by finite difference
    f1 = f(x_current + dx/2); 
    f2 = f(x_current - dx/2);
    g = (f1-f2)/dx;
    
    x_next = x_current-alpha*g; %new solution
    
    x_current = x_next; 
    
    fprintf('%d %d\n',x_current,x_next);
    
    x_current = x_next;
end

It give 5.543798e+01 as solution while the solution should either be 0.9 or 0.3 (local and global minimas, respectivily).

Whate am I missing here? can anyone help?

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Answer 1

Matt J 2019-9-17

1
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https://ww2.mathworks.cn/matlabcentral/answers/480830-steepest-descent-method-algorithm#answer_392335

alpha is too big. Try alpha=0.001.

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Luqman Saleem 2019-9-17

编辑：Luqman Saleem 2019-9-17

@Matt J It worked. I am so stupid.

With initial guess = 0, the solution converges to 0.3 (global minima) while with guess=1, the solution is 0.9 (local minima).

Do you know any way to bypass local minima and get to global minima always? I am trying to understand multiscaling, can you help me understanding this.

Thank you very much.

Saleh Msaddi 2020-3-9

In steepest descent, you would always get the local minima. You'd only get the global minima if you start with an initial point that would converge to the global minima; if you're lucky enough. If your stepping size is too small, your solution may converge too slow or might not converge to a local/global minima. On the contradictory, if you choose a big step size, your solution may miss the minimal point.

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