Lower bound for global minimum with monotonic behavior?
- From: hmobahi@xxxxxxxxx
- Date: 20 Feb 2007 12:44:43 -0800
Hello,
I am working on a least squares problem where the objective function
F
is multivariate and in form of sum of squared terms and each squared
term is nonlinear. Although finding the value of the objective F* at
the global minimum is time-consuming, there are many efficient lower
bound (LB) estimators when we focus on a certain box within the
domain
of F.
My goal is to identify the term in the sum that has the least effect
on the value of F*=Min(F). Now if there exists any LB that is
monotonic in the actual minimum, I will be able to such a term by the
following procedure:
If in F, I eliminate one of the terms of the sum, say K'th term, I
will name the new function FK. FK is the same as F but lacks the K'th
term of the sum. Now if the LB possesses the mentioned property, i.e.
LB(FJ)<LB(FK) implies MIN(FJ)<MIN(FK), then I can compute LB(FK) for
all K's and then check which one is the closest to LB(F).
Consequently, this reveals which term in has the least contribution
to
the value of F*. Note that all LB's are computed over the same box
and
that box is guaranteed to enclose the global minimum.
For example, if F=(x^2)^2+(3+X^2)^2 then MIN(F1)=MIN((x^2)^2)=0 and
MIN(F2)=MIN((3+X^2)^2)=9. Now If there is a LB that in the interval
of
[-123,536] behaves like LB((x^2)^2)=-1000 and LB((0.1+X^3)^2)=-1173 I
will be happy with that LB. The need here is only monotonicity of the
LB and not its sharpness as the example shows.
Is there such a LB at all? If there is such a LB even for very
specific objective functions, for example polynomials or rational
functions or anything, I will still be interested in learning about
that.
I would really appreciate your advice.
Regards
H.M.
.
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