Square root in objective function must appear as nonlinear constraints for optimization?
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The title already described my question. Let me give a specific example:
If I have 
in my objective function for fmincon to optimize. During the iterations, 
may be smaller than 0.
in my objective function for fmincon to optimize. During the iterations, may be smaller than 0. Therefore, do I need to write 
as nonlinear constraints or does Matlab already take account of the problem?
as nonlinear constraints or does Matlab already take account of the problem?Thank you very much!
回答（4 个）
You should omit square roots, because they introduce both unneccesary additional computation and non-differentiability into the cost function, which breaks the theoretical assumptions of fmincon. In your example, the problem could be equivalently posed without square roots as
min x^2
s.t. x^2>=100
6 个评论
Frank
2019-3-6
Sandeep's advice won't work reliably, unfortunately, because fmincon does not obey the nonlinear constraints at all iterations. Even if you specify constraints, it can still attempt to evaluate your objective in regions where the square root argument is negative.
Also, even if the nonlinear constraints were obeyed at all iterations, the non-differentiability issue still remains.
You have to avoid the square roots... As for how to do that, I recommend you repost your question with an example more representative of your actual problem.
John D'Errico
2019-3-6
@Frank - you have decided to take Sandeep's advice here, because you think it makes your solution easy. In the end, Matt is giving you the advice you need, whether or not you like the consequence. The easy solution is not always the best one.
Lastly, Matt, could you please kindly tell me what is the non-differentiability issue? I can differentiate my objective function and nonlinear constraints easily.
The function sqrt(z) has no derivative at z=0. In your case, the function f and hence also t(f) is non-differentiable wherever -x^3+2*x+99=0. So, if the optimal solution lies near such a point, fmincon will not be able to use gradients to find its way there.
What is |t(f)-data|? Is it a least squares objective or is it the L1 norm error between t(f) and |data|? The L1 norm also has differentiability issues.
One general way to get rid of sqrt expressions in the objective is to replace them with an additional nonlinearly constrained variable, e.g., instead of
f=a+b^2+sqrt(-x^3+2*x+99)
have instead,
f=a+b^2+c
s.t. c^2=-x^3+2*x+99
c>=0;
All of the above expressions, both in the objective and the constraints, are now differentiable everywhere.
1 个评论
Pedro Luis Camuñas
2020-8-5
Thanks, this canswer helped me a lot
SandeepKumar R
2019-3-6
0 个投票
Yes, you need to specify that constraint as ignoring this will lead infeasibilty. This is true for any optimizer
Walter Roberson
2019-3-6
0 个投票
if you have fractional power of a polynomial (not a multinomial) then manually solve for the bounds and express them as bounds constraints . Bounds constraints are respected in most situations .
You might end up with discontinuous ranges. If so it can often be more efficient to run the ranges as separate problems and take the best solution afterwards . Nonlinear constraints are much less efficient to deal with .
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2020-8-5
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