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tennican

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02-13-2012
05:43 PM

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Reformulating a constrained minimization problem into a constrained nonlinear least square problem

thanks, Scott

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mecej4

Black Belt

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02-13-2012
05:59 PM

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If the objective function has the form

F = f^{2} + g^{2}+ h^{2}+ ...

the answer is yes.

F = f

the answer is yes.

tennican

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02-13-2012
06:43 PM

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I'd like to find the smoothing parameters which minimize the sum of the squared errors of a triple exponential smoothing fit. So, F is just the sum of the squared difference between the true values and the fitted values. But, the fitted values are computed iteratively over the time series from updating values of the level, trend and season components.

How would I formulate this problem to use the solver?

thanks, Scott

mecej4

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02-14-2012
02:34 AM

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N = x

(where there are N observations x, and their mean appears in the smoothing expression) cannot be handled directly. You can try to recast your problem in this form; if you post details of your smoothing function, I'll try to help.

Should it prove impossible to recast the problem as a bound-constrained least-squares problem, you will need to use another solver than the one in MKL. See the most useful Decision Tree for Optimization Software by Professor H.J. Mittelmann.

tennican

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02-14-2012
03:21 PM

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Since the nonlinear least square problems is:

min(sum(Fi(x)^2)) where i=1..m and x=(x1,..,xn) and m >= n

where the constraints are on Fi(x)

My problem could be:

min(sum(Ft(alpha,beta,gamma)^2)) where t=1..T and x=(alpha,beta,gamma) and T >= 3

where Ft returns the residual for period t

But, my constraints are: 0 <= alpha, beta, gamma <= 1 when I need constaints on F instead.

And, I don't see any way to reformulate the constraints in terms of F.

Plus, the number of periods T=m could be large and I don't think this solver is intended to handle large m.

Thanks for the pointer to Decision Tree site.

I'll find something outside of MKL.

mecej4

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02-14-2012
04:19 PM

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The MKL routine should handle expressions with a few hundred terms without running into problems.

tennican

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02-17-2012
01:28 PM

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There is no way to derive constraints on F from the contraints on X because F isn't that simple.

Anyway, I've implemented constrained multivariate steepest descent which seems to work fine for now.

What I really want is for MKL to implement some general optimization.

In particular, I would like: L-BFGS-C.

mecej4

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02-19-2012
02:26 AM

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Before you made the claim in #4, there was nothing stated to indicate that you had constraints on F

For each equality constraint on a

L-BFGS is an algorithm for

Steepest descent is a very inefficient algorithm for unconstrained minimization.

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