different minimum result of convex function by using fminsearch

A function which I used is already proved as a convex function.

1. initial point = [0,0,0,0,0,0]

myfun = @(x)(......)
[x, fval] = fminsearch(myfun, [0,0,0,0,0,0])

2. initial point = [0.2,0.2,0.2,0.2,0.2,0.2]

myfun = @(x)(......)
[x, fval] = fminsearch(myfun, [0.2,0.2,0.2,0.2,0.2,0.2])

• _ | | The "fval" results are quite different.||_*

I expect that if "myfun" is a convex function, the result should be similar. Although the fminsearch find out the local minimum, since the myfun is convex, so the result should be one whatever the initial point is.

What I guess:

# Lack of Number of iteration?

Help me!

[Walter Roberson]

Please post a sample myfun for us to test with.

[Hyeyun Yang]

myfun1 = @(x)(s1p12(x)+p12p13(x)+t1p13(x)+s2p12(x)+p12p23(x)+t2p23(x)+s3p13(x)+p13p23(x)+t3p23(x));
[x1,fval(1)] = fminsearch(myfun1,[0.2,0.2,0.2,0.2,0.2,0.2])

[Walter Roberson]

s1p12 does not appear to be a Mathworks function or to exist in anything in the File Exchange, and that combination of functions cannot be located in a Google search. I must suspect that those are your own functions, which makes it difficult for us to test the behavior.

[Hyeyun Yang]

sx = [0.4994 0.4994 0.2967];
sy = [0.8971 0.6985 0.3978];
tx = [0.1008 0.9003 0.6976];
ty = [0.2051 0.2051 0.4036];

s1p12 = @(x)((sx(1)-x(1))*(sx(1)-x(1))+(sy(1)-x(2))*(sy(1)-x(2))).^.5;
s1p13 = @(x)((sx(1)-x(3))*(sx(1)-x(3))+(sy(1)-x(4))*(sy(1)-x(4))).^.5;
p12p13 = @(x)((x(1)-x(3))*(x(1)-x(3))+(x(2)-x(4))*(x(2)-x(4))).^.5;
t1p12 = @(x)((tx(1)-x(1))*(tx(1)-x(1))+(ty(1)-x(2))*(ty(1)-x(2))).^.5;
t1p13 = @(x)((tx(1)-x(3))*(tx(1)-x(3))+(ty(1)-x(4))*(ty(1)-x(4))).^.5;

s2p12 = @(x)((sx(2)-x(1))*(sx(2)-x(1))+(sy(2)-x(2))*(sy(2)-x(2))).^.5;
s2p23 = @(x)((sx(2)-x(5))*(sx(2)-x(5))+(sy(2)-x(6))*(sy(2)-x(6))).^.5;
p12p23 = @(x)((x(1)-x(5))*(x(1)-x(5))+(x(2)-x(6))*(x(2)-x(6))).^.5;
t2p12 = @(x)((tx(2)-x(1))*(tx(2)-x(1))+(ty(2)-x(2))*(ty(2)-x(2))).^.5;
t2p23 = @(x)((tx(2)-x(5))*(tx(2)-x(5))+(ty(2)-x(6))*(ty(2)-x(6))).^.5;

s3p13 = @(x)((sx(3)-x(3))*(sx(3)-x(3))+(sy(3)-x(4))*(sy(3)-x(4))).^.5;
s3p23 = @(x)((sx(3)-x(5))*(sx(3)-x(5))+(sy(3)-x(6))*(sy(3)-x(6))).^.5;
p13p23 = @(x)((x(3)-x(5))*(x(3)-x(5))+(x(4)-x(6))*(x(4)-x(6))).^.5;
t3p13 = @(x)((tx(3)-x(3))*(tx(3)-x(3))+(ty(3)-x(4))*(ty(3)-x(4))).^.5;
t3p23 = @(x)((tx(3)-x(5))*(tx(3)-x(5))+(ty(3)-x(6))*(ty(3)-x(6))).^.5;

[Hyeyun Yang]

Here you are:)

Answer 1

[John D'Errico]

fminsearch is a very simple optimizer, so I am not at all surprised to see this happen.

1. A 6 dimensional problem is near the upper limit of where I would recommend using fminsearch.

2. ANY optimizer will converge to slightly different solutions, stopping when it decides it is close enough. Close enough is determined by when the objective function is not changing by a significant amount. If the solver hits the iteration or function evaluation limit, then it will also stop looking.

3. If your objective function is flat enough, then an optimizer can also easily terminate far away from the true minimum.

Solution? Start by using a better optimizer. Even then, don't be surprised to see if some other numerical optimizer does exactly the same thing.

[Hyeyun Yang]

Thank you. Then I will search for the better optimizer.

One more Q:

"function is flat enough"

What's "flat" like?

[Walter Roberson]

A function with a slope of (for example) 1E-15 might have all of the probe values within the current tolerance, with it not being detectable that the slope is real and important to chase for long ways.

You can construct functions of the form

-1/(delta + (x-c)^p)

for large even integer p and positive delta. These functions have a sharp valley close to x == c, but the delta and the fact that (x-c)^even is non-negative prevents the function from being a singularity; instead it has a minimum at -1/delta and the "effective" catch basin of the minimum can be made arbitrarily tight by increasing p. Calculus has no problem dealing with the situation, but numeric algorithms have finite tolerance and give up.