Can I calculate a standard deviation as a pyomo variable using ipopt?

[Jaime de la Mota Sanchis]

Hello everyone. I am now trying to make my model stochastic and put some restrictions using the mean and standard deviation assuming a normal random variable.

I have constructed a pyomo code in which I calculate the weighted mean of 3 vectors as

def meanX_definition(model, i):
      return m.meanX[i] ==(m.x1[i]+4*m.x2[i]+m.x3[i])/6

m.meanX_const = Constraint(m.N, rule = meanX_definition)

And then, I have calculated the variance as

 def varX_definition(model, i):
       return m.varX[i] ==((m.x1[i]-m.meanX[i])**2+ 4*(m.x2[i]-m.meanX[i])**2+(m.x3[i]-m.meanX[i])**2)/6

m.varX_const = Constraint(m.N, rule = varX_definition)

Since my model is a stochastic one, I want to put restrictions as x1, x2 and x3 have to be localidez between the mean and +-2 standard deviations.

For that, I have attempted to calculate the standard deviation as

def stdX_definition(model, i):
      return m.desvestX[i] == sqrt(m.varX[i])
m.stdX_const = Constraint(m.N, rule = stdX_definition)

and

def stdX_definition(model, i):
      return m.desvestX[i] == m.varX[i]**(0.5)
m.stdX_const = Constraint(m.N_notinitial_notfinal, rule = stdX_definition)

I have also tried to define the standard deviation at all points but the first and the last in case the error was the standard deviation of 0 at [0] and [n], since the start and endpoints are the same for all 3 vectors.

Unfortunately, an error occurs

ApplicationError: Solver (ipopt) did not exit normally

Can someone please tell me if this issue can be fixed so I can get the standard deviation?

I will attach my code so that if someone is interested can take a look.

Thanks.

Jaime.

[Erling D. Andersen]

Since the standard dev. includes a 2 norm which is non smooth at 0 (not differentiable), then traditional optimizers like ipopt might have problems with the std. dev.

Whether that this is the issue you are fighting I cannot say.

Alternatively standard deviation can be modeled using a conic quadratic constraint i.e.

  https://pyomo.readthedocs.io/en/stable/library_reference/kernel/conic.html

but unless you can state the complete model on conic form then that most likely is not useful for you.

[Siirola, John D]

One additional note (because this comes up frequently):  Erling is correct that the 2-norm as generally written can cause problems at 0.  There is a “standard” reformulation that can have significantly better behavior, for example, instead of:

 

                m.desvestX[i] == sqrt(m.varX[i])

 

try expressing the constraint as

 

                m.desvestX[i]**2 == m.varX[i]

 

John

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[Jaime de la Mota Sanchis]

Thank you very much to both of you for your answers. I have been able to fix the issue.

Regards.

Jaime.

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