Tolerances are relative or not in interior-point for fmincon?

Question

lkscan 2017-3-16

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https://ww2.mathworks.cn/matlabcentral/answers/330191-tolerances-are-relative-or-not-in-interior-point-for-fmincon

编辑： Nick Van Oosterwyck 2023-3-27

Hi, my colleges,

The document for Tolerances and Stopping Criteria in Matlab says, StepTolerance and FunctionTolerance could be relative or absolute. https://fr.mathworks.com/help/optim/ug/tolerances-and-stopping-criteria.html

But in the introducetion for interior-point for fmincon, there is not any information about this method whether the tolerances are relative or absolute.

https://fr.mathworks.com/help/optim/ug/choosing-the-algorithm.html#btr9d6u

http://fr.mathworks.com/help/optim/ug/fmincon.html

Is there anyone know about it?

Best regards.

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Answer 1

Alan Weiss 2017-3-16

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https://ww2.mathworks.cn/matlabcentral/answers/330191-tolerances-are-relative-or-not-in-interior-point-for-fmincon#answer_258989

在 MATLAB Online 中打开

You can find out the answer using the exit message link stopping criteria details. Here is what I just saw after running a small problem:

Optimization completed: The relative first-order optimality measure, 4.268868e-08,
is less than options.OptimalityTolerance = 1.000000e-06, and the relative maximum constraint
violation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.
Optimization Metric                                            Options
relative first-order optimality =   4.27e-08       OptimalityTolerance =   1e-06 (default)
relative max(constraint violation) =   0.00e+00    ConstraintTolerance =   1e-06 (default)

Clearly, relative tolerances were used.

Alan Weiss

MATLAB mathematical toolbox documentation

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Mohammad 2020-4-21

Hi

How can find an exact definition of relative first-order optimality?

I search a lot in MATLAB document and net but I can not find any.

Tnx

Huayu Tian 2020-10-16

Hi,

Yes that's also my question. I cannot find the exact definition for the relative first-order optimality.

In the link below it mentions the scaling factor to define a relative first order optimality can be either (1) the infinity norm of the gradient at the starting point, or (2) the infinity norm of inputs to the solver, but it does not state specifically what scaling factor fmincon is using. I guess it's using (1) but I am not sure.

https://www.mathworks.com/help/optim/ug/first-order-optimality-measure.html

I also checked exit message but it only contains a link to first-order optimality measure, not a definition of relative one.

Does anybody know the definition of relative first-order optimality for each solver (especially fmincon)?

Many thanks.

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