Arithmetic to ensure positives

This is not possible to do without at least one comparison, and comparisons are not pure arithmetic. Your inspiration (a-b)^2 + (c-d)^2 + (e-f)^2 = 0 involves a comparison and so is not pure arithmetic.

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Walter Roberson 2015-12-27

When you say that your optimization scheme is very fragile, are you talking about attempting to code constraints in a manner that is differentiable?

Amit 2016-1-3

Yes actually Walter. In deed, hoping for that. Thanks in deed for bringing me closer to asking proper question here.

In the last few days, I tried in built GRG Non Linear Scheme with 'min' function, as a work around. Hoping to do something better.

Thanks for your attention.

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Answer 2

Walter Roberson 2016-1-4

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https://ww2.mathworks.cn/matlabcentral/answers/261656-arithmetic-to-ensure-positives#answer_204990

(a-b)^2 + (c-d)^2 + (e-f)^2 = 0 is differentiable only because it is smoothly invertible, that it can be translated into a series of variable reductions. Inequalities cannot be inverted that way. You cannot even code a > 0 invertibly -- if you could then c>=d could be coded as (c-d)^2 - delta_c = 0 together with however you coded delta_c > 0.

Unless, that is, you are okay with coding Heaviside functions, in which delta_c > 0 translates to Heaviside(delta_c) - 1 = 0 after having defined Heaviside(0) as 0 (Heaviside(0) does not have a fixed value, not really; one of the common conventions says Heaviside(0) = 1/2).

But diff(Heaviside(delta_c),delta_c) is Dirac(delta_c) and that is considered a distribution rather than a particular value, definitely not continuously differentiable. I would not consider it suitable for the use in this situation, but perhaps the theory of GRG is more flexible than I am.

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Amit 2016-1-4

Dear Walter:

Thanks much. I learnt a few things from you answer. I understood what you are saying. Though this is pushing my problem in 'high mathematics' rather than come out of it.

Perhaps what I asked is not available, but let me seek a little bit more by keeping it still as an open question.

Meanwhile, the GRG scheme in deed appears to be accommodating.

Look forward to more thoughts, if any.

Regards, Amit

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