# [R-SIG-Finance] solnp Problem Inverting Hessian

Paul Gilbert pgilbert902 at gmail.com
Tue Dec 15 19:09:54 CET 2015

```WRT scaling I should also add that, while it is almost always the right
thing to do numerically, it does not always make sense in a particular
application. One often does optimization in a situation where the
parameters have different units, or have no units, so the scales are
arbitrary to begin with and rescaling is numerically convenient. In
application like yours, you may be summing things that are all in the
same unit, dollars. Rescaling could mean some are in dollars and some in
millions or billions of dollars. Theoretically, at the maximum of an
unconstrained problem, scaling will not make any difference because the
gradient will be zero in all directions. Practically, there is a
difficulty that the numerical solution stops according to some rule when
it gets "close enough". By rescaling you will be emphasizing the
importance of parameters that really have no effect on the portfolio. It
might be better to simply eliminate the parameters that have no
contribution to the objective. (Take this all with a grain of salt, I
don't know anything about portfolio optimization.)

Paul

On 12/15/2015 10:45 AM, Michael Ashton wrote:
> Understood. I was just noting that Ret.vect spans zero, and thus some
> projected returns will always be pretty close to zero.
>
>> On Dec 15, 2015, at 10:43 AM, Paul Gilbert <pgilbert902 at gmail.com>
>> wrote:
>>
>> Just to be clear, when I said "difference between the largest and
>> smallest elements" I am think of the absolute value. It is the
>> scale not the sign that causes problems in optimization. And I
>> cannot commenting on whether you have the correct objective
>> function, just pointing out what would happen numerically with the
>> one you specified.
>>
>> Paul
>>
>>> On 12/15/2015 10:17 AM, Michael Ashton wrote: That is very
>>> interesting. Actually some of the projected returns are negative,
>>> some are very close to zero. None is exactly zero, which I guess
>>> is why it's not singular, but that's why it's ill-conditioned.
>>> Very interesting. And hard to solve...if I add 5% to all returns
>>> will I still get the same portfolio? Seems to me intuitively I
>>> shouldn't since the proportional risk/return tradeoff then is
>>> greater for the assets on the low end of the spectrum, but I
>>> might be wrong.
>>>
>>> -----Original Message----- From: Paul Gilbert
>>> [mailto:pgilbert902 at gmail.com] Sent: Tuesday, December 15, 2015
>>> 9:58 AM To: Michael Ashton Cc: Michael Weylandt;
>>> r-sig-finance at r-project.org Subject: Re: [R-SIG-Finance] solnp
>>> Problem Inverting Hessian
>>>
>>> I have not been paying close attention and I've missed something
>>> in this thread, but ...
>>>
>>> I think the hessian in the optimization will be the second
>>> derivative of this function
>>>
>>>>>>> opt.fun <- function(wgt.vect)
>>>>>>> {-crossprod(wgt.vect,t(Ret.vect))}
>>>
>>> WRT  wgt.vect. If there are zero elements in Ret.vect then that
>>> hessian will be singular. And if there are orders of magnitude
>>> difference between the largest and smallest elements in Ret.vect
>>> then the hessian will be ill-conditioned.
>>>
>>> The ill-conditioned case is a numerical problem and you might
>>> solve it with a different algorithm, or by better scaling. The
>>> zero case is a theoretical problem, there is not a unique optimal
>>> point but rather whole continuum of optimal points (your
>>> parameters corresponding to the zero elements will not make any
>>> difference in the function value).
>>>
>>> HTH, Paul
>>>
>>>> On 12/14/2015 11:39 PM, Michael Weylandt wrote: (+list -- I'm
>>>> not a numerical linear algebra expert or portfolio optimization
>>>> expert, but there are a number on this list who might chime
>>>> in)
>>>>
>>>> That is quite high, and almost certainly problematic.
>>>>
>>>> I'm not sure what an acceptable bound is (will depend on your
>>>> solver), but that's almost certainly above it for any
>>>> algorithm which will require inverting cov.mat.
>>>>
>>>> Two things you could try: - use a different
>>>> (non-Hessian-using) optimization algorithm; - use some form of
>>>> shrinkage/regularization (Ledoit-Wolf is a common choice) to
>>>>
>>>> Michael
>>>>
>>>> On Mon, Dec 14, 2015 at 10:10 PM, Michael Ashton
>>>> <m.ashton at enduringinvestments.com> wrote:
>>>>> Well, not sure whether this makes any sense but
>>>>> kappa(cov.mat) gives me 11148245007.
>>>>>
>>>>> That seems large. But I am not sure what it is supposed to
>>>>> be.
>>>>>
>>>>> -----Original Message----- From: Michael Weylandt
>>>>> [mailto:michael.weylandt at gmail.com] Sent: Monday, December
>>>>> 14, 2015 9:29 PM To: Michael Ashton Cc:
>>>>> r-sig-finance at r-project.org Subject: Re: [R-SIG-Finance]
>>>>> solnp Problem Inverting Hessian
>>>>>
>>>>> What's the condition number of your correlation/covariance
>>>>> matrix? (?kappa)
>>>>>
>>>>> I think I've used quadprog for portfolio optimization with
>>>>> success, but it's been a while.
>>>>>
>>>>> MW
>>>>>
>>>>> On Mon, Dec 14, 2015 at 8:06 PM, Michael Ashton
>>>>> <m.ashton at enduringinvestments.com> wrote:
>>>>>> Do you have a suggestion for such? I have in the past
>>>>>> tried fPortfolio, but it would not allow me to specify my
>>>>>> own projected return vectors rather than the historical
>>>>>> returns of the series (which is exactly backwards).
>>>>>>
>>>>>> As for whether the matrix is comfortably non-singular...I
>>>>>> suppose it depends in a Clintonian way on the meaning of
>>>>>> "comfortably," but I can create a Cholesky decomposition
>>>>>> without it blowing up, which is usually how I can tell if
>>>>>> I have done something stupid. Well, stupider than normal.
>>>>>>
>>>>>> -----Original Message----- From: Michael Weylandt
>>>>>> [mailto:michael.weylandt at gmail.com] Sent: Monday, December
>>>>>> 14, 2015 8:32 PM To: Michael Ashton Cc:
>>>>>> r-sig-finance at r-project.org Subject: Re: [R-SIG-Finance]
>>>>>> solnp Problem Inverting Hessian
>>>>>>
>>>>>> The Hessian is the matrix of second derivatives
>>>>>> (https://en.wikipedia.org/wiki/Hessian_matrix) -- in
>>>>>> scalar terms, you're finding a point where the second
>>>>>> derivative is zero and then trying to divide by the second
>>>>>> derivative to calculate the step size.
>>>>>>
>>>>>> I haven't gone through your code in any detail, but I'd
>>>>>> start by checking the covariance matrix since that's
>>>>>> proportional to the Hessian of your objective function. Is
>>>>>> it (comfortably) non-singular?
>>>>>>
>>>>>> Since you're just solving the standard Markowitz problem,
>>>>>> of a general non-linear solver. Should behave a bit
>>>>>> better.
>>>>>>
>>>>>> Michael
>>>>>>
>>>>>>
>>>>>> On Mon, Dec 14, 2015 at 6:13 PM, Michael Ashton
>>>>>> <m.ashton at enduringinvestments.com> wrote:
>>>>>>> I must admit to being flummoxed here, mainly because my
>>>>>>> linear algebra was 25 years ago and I can't remember what
>>>>>>> a Hessian is.
>>>>>>>
>>>>>>> I have a matrix of 60 securities' weekly returns, along
>>>>>>> with 60 projected returns. The returns are in a vector
>>>>>>> called Ret.vect and the covariance matrix of weekly
>>>>>>> returns in cov.mat . I have the minConstraints and
>>>>>>> maxConstraints that the parameters are permitted to take.
>>>>>>> I cycle through targeted risks and get the same error for
>>>>>>> each risk targeted...below I have removed the loop to
>>>>>>> focus on the risk=0.002.
>>>>>>>
>>>>>>> wgt.vect=c(rep(1/60, 60)) constr.fun <-
>>>>>>> function(wgt.vect) {; c1 = sqrt(crossprod(t(wgt.vect %*%
>>>>>>> cov.mat),wgt.vect)); c2 = sum(wgt.vect);
>>>>>>> return(c(c1,c2)); } ineqconstr.fun <- function(wgt.vect)
>>>>>>> { wgt.vect[1:60]; } opt.fun <- function(wgt.vect)
>>>>>>> {-crossprod(wgt.vect,t(Ret.vect))}
>>>>>>>
>>>>>>> OptimSolution <-
>>>>>>> solnp(wgt.vect,opt.fun,constr.fun,eqB=c(0.002,1),ineqconstr.fun,ine
>>
>>>>>>>
qL
>>>>>>> B =minConstraints,ineqUB=maxConstraints)
>>>>>>>
>>>>>>> I get the following error: solnp--> Solution not
>>>>>>> reliable....Problem Inverting Hessian.
>>>>>>>
>>>>>>> Well, that doesn't tell me very much! The parameters
>>>>>>> (weights) that are output for each run, as I cycle
>>>>>>> through the weights, are very scrambled...lots of little
>>>>>>> allocations, rather than clumping as you would expect to
>>>>>>> happen especially at the risky and riskless ends of the
>>>>>>> spectrum.
>>>>>>>
>>>>>>> Can anyone with more math than me give me a helping hand
>>>>>>> on the Hessian?
>>>>>>>
>>>>>>> Thanks,
>>>>>>>
>>>>>>>
>>>>>>> Mike
>>>>>>>
>>>>>>> Michael Ashton, CFA Managing Principal
>>>>>>>
>>>>>>> Enduring Investments LLC W: 973.457.4602 C: 551.655.8006
>>>>>>>
>>>>>>>
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