find_minimum_on_interval() uses the wrong scipy function

[aghitza]

This was reported by Dean Moore on sage-support. Consider:

sage: f(x) = -x*sin(x^2)
sage: f.find_minimum_on_interval(-2.5, 2)
(-1.3076194129914434, 1.35521114057)
sage: f.find_minimum_on_interval(-2.5, -1)
(-2.1827697846777219, -2.19450274985)

So find_minimum_on_interval() returns a local minimum as opposed to the global one. (The same issue applies to find_maximum_on_interval.) This is due to the fact that the function wraps scipy.optimize.fminbound, which is only a local optimizer (Carl Witty pointed this out). We should instead use one of the global optimizers, i.e. scipy.optimize.anneal or scipy.optimize.brute.

Apply:

• attachment: trac2607.patch
• attachment: trac2607-whitespace.2.patch
• attachment: trac_2607_renaming.3.patch
• attachment: trac2607-doctest-and-spacing.patch
• attachment: trac_2607_update_deprecation.patch

Depends on #13109

CC: @robert-marik @kcrisman

Component: calculus

Keywords: sd31, sd40.5

Author: Dan Drake, Andrey Novoseltsev, Andrzej Giniewicz, Volker Braun

Reviewer: Karl-Dieter Crisman, Mike Hansen, Andrey Novoseltsev

Merged: sage-5.3.beta0

Issue created by migration from https://trac.sagemath.org/ticket/2607

[robert-marik]

comment:3

It seems that there are two global optimizers in scipy: scipy.optimize.anneal and scipy.optimize.brute. Does anybody more experienced in numerics know, which one is better for including into Sage?

[jasongrout]

comment:5

Robert, that sounds like a great question for either sage-devel or the scipy list.

[kcrisman]

comment:7

This Scipy tutorial page should be relevant. I will try to resolve this soon.

[kcrisman]

Changed keywords from none to sd31

[kcrisman]

comment:8

#5960 was a dup. Here are the examples from there.

sage: h(x) =  -sin(x) - 2*sin(2*x)
sage: h.find_minimum_on_interval(0, 2*pi)
(-1.3271810224585345, 3.8298351449342838)
But there is another local minimum at h(0.8666760871050464) = -2.73581510406


sage: find_minimum_on_interval(x*(x-1)*(x+1), -2, 2)
(-0.38490017945975047, 0.57735026913115706)
The minimum on this interval is the endpoint h(-2) = 6.


sage: find_minimum_on_interval((x-2)*(x-1)*x*(x+1) - x, -2, 2)
(-0.43749999999999994, -0.49999999973911674)

but
sage: find_minimum_on_interval((x-2)*(x-1)*x*(x+1) - x, 0, 2)
(-2.6642135623730949, 1.7071067879138031)

[kcrisman]

comment:9

The brent algorithm will also not work.

Triple (a,b,c) where (a<b<c) and func(b) < func(a),func(c). If bracket consists of two numbers (a,c) then they are assumed to be a starting interval for a downhill bracket search (see bracket); it doesn’t always mean that the obtained solution will satisfy a<=x<=c.

Which is not the same as constrained minimization, for us.