[GSoC-PortA] mapping function

[Ross Bennett]

Brian,

Thanks for sharing the paper and your ideas for the mapping function.

One of the main point I got from the methodology described in the paper is
that the sets of weights are omitted instead of transformed if they do not
meet the constraints.

   - generate random portfolios
   - test each set of portfolio weights
   - if a constraint is violated, omit that set of weights
   - compute the objective function on each remaining set of weights
   - select the set of optimal weights

Is this what you were getting at here?
"A slightly more rigorous treatment of the problem is given here:
http://papers.ssrn.com/sol3/**papers.cfm?abstract_id=1680224<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1680224>
It is possible that can use this method directly for random portfolios."

I could add a block of code at the end of random_portfolios that tests each
set of weights against the constraints and only keeps the weights that satisfy
the constraints. Thoughts?

If we have to transform the weights, here are my thoughts looking at the
specific cases for constraint types
Constraint Type

   - Leverage (min_sum and max_sum)
      - This is done in randomize_portfolio by randomly permuting and
      increasing or decreasing an individual element (asset weight)
until min_sum
      and max_sum constraints are satisfied while taking into account box
      constraints. These constraints are satisfied based on the way random
      portfolios are constructed.
      - This is done in constrained_objective by transforming the entire
      vector (
         - if(sum(w)>max_sum) { w<-(max_sum/sum(w))*w } # normalize to
         max_sum
         - if(sum(w)<min_sum) { w<-(min_sum/sum(w))*w } # normalize to
         min_sum
      - Implement by moving this into the mapping function... correct?
   - Box (min and max)
      - This is done in randomize_portfolio by construction
      - This is done in constrained_objective by penalizing weights outside
      box constraints.
      - Implement by using logic from randomize_portfolio to transform
      weights instead of penalizing them. Goal is to transform the
weights vector
      instead of penalize... correct?
   - Group (cLO and cUP)
      - One approach is to normalize the weights in each given group that
      violate cLO or cUP so that the group weights sum to cLO or cUP. This
      changes the sum of weights, so when the weights vector is normalized the
      group constraints will likely be violated, but it gets us close. See
      sandbox/testing_constrained_group.R
      - Another approach is to add this to randomize_portfolio so the group
      constraints as well as box and leverage are satisfied by
construction. Need
      to spend more time understanding code in randomize_portfolio to see how
      feasible this is.
   - turnover
      - Could we include this in constrained_objective as a penalty?
   - diversification
      - Could we include this in constrained_objective as a penalty?
   - volatility
      - Could we include this in constrained_objective as a penalty?
   - position_limit
      - This may be able to be implemented in randomize_portfolio by
      generating portfolios with the number of non-zero weights equal
to max.pos,
      then fill in weights of zero so the length of the weights vector is equal
      to the number of assets, then scramble the weights vector. The number of
      non-zero weights could also be random so that the number of non-zero
      weights is not always equal to max.pos. This could be implemented in the
      DEoptim solver with the mapping function. This might be do-able in Rglpk
      for max return and min ETL. Rglpk supports mixed integer types, but
      solve.QP does not. May be able to use branch-and-bound technique using
      solve.QP, but needs more research.


Regards,
Ross


On Sat, Jun 29, 2013 at 8:45 AM, Brian G. Peterson <brian at braverock.com>wrote:

> Based on side conversations with Ross and Peter, I thought I should talk a
> little bit about next steps related to the mapping function.
>
> Apologies for the long email, I want to be complete, and I hope that some
> of this can make its way to the documentation.
>
> The purpose of the mapping function is to transform a weights vector that
> does not meet all the constraints into a weights vector that does meet the
> constraints, if one exists, hopefully with a minimum of transformation.
>
> In the random portfolios code, we've used a couple of techniques pioneered
> by Pat Burns.  The philosophical idea is that your optimum portfolio is
> most likely to exist at the edges of the feasible space.
>
> At the first R/Finance conference, Pat used the analogy of a mountain
> lake, where the lake represents the feasible space.  With a combination of
> lots of different constraints, the shore of the lake will not be smooth or
> regular.  The lake (the feasible space) may not take up a large percentage
> of the terrain.
>
> If we randomly place a rock anywhere in the terrain, some of them will
> land in the lake, inside the feasible space, but most will land outside, on
> the slopes of the mountains that surround the lake.  The goal should be to
> nudge these towards the shores of the lake (our feasible space).
>
> Having exhausted the analogy, let's talk details.
>
> A slightly more rigorous treatment of the problem is given here:
> http://papers.ssrn.com/sol3/**papers.cfm?abstract_id=1680224<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1680224>
> It is possible that can use this method directly for random portfolios
> (and that we could add the ectra constraint types to DEoptim).  If so, much
> of the rest of what I'll write here is irrelevant.  I strongly suspect that
> there will be some constraint types that will still need to be 'adjusted'
> via a mapping method like the one laid out below, since a stochastic solver
> will hand us a vector that needs to be transformed at least in part to move
> into the feasible space.  It's alsom not entirely clear to me that the
> methods presented in the paper can satisfy all our constraint types.
>
>
> I think our first step should be to test each constraint type, in some
> sort of hierarchy, starting with box constraints (almost all solvers
> support box constraints, of course), since some of the other
> transformations will violate the box constraints, and we'll need to
> transform back again.
>
> Each constraint can be evaluated as a logical expression against the
> weights vector.  You can see code for doing something similar with time
> series data in the sigFormula function in quantstrat. It takes advantage of
> some base R functionality that can treat an R object (in this case the
> weights vector) as an environment or 'frame'. This allows the columns of
> the data to be addressed without any major manipulation, simply by column
> name (asset name in the weights vector, possibly after adding names back
> in).
>
> The code looks something like this:
> eval(parse(text=formula), data)
>
> So, 'data' is our weights vector, and 'formula' is an expression that can
> be evaluated as a formula by R.  Evaluating this formula will give us TRUE
> or FALSE to denote whether the weights vector is in compliance or in
> violation of that constraint.  Then, we'll need to transform the weight
> vector, if possible, to comply with that constraint.
>
> Specific Cases:
> I've implemented this transformation for box constraints in the random
> portfolios code.  We don't need the evaluation I'll describe next for box
> constraints, because each single weight is handled separately.
>
> min_sum and max_sum leverage constraints can be evaluated without using
> the formula, since the formula is simple, and can be expressed in simple R
> code.  The transformation can be accomplished by transforming the entire
> vector.  There's code to do this in both the random portfolios code and in
> constrained_objective.  It is probably preferable to do the transformation
> one weight at a time, as I do in the random portfolios code, to end closer
> to the edges of the feasible space, while continuing to take the box
> constraints into account.
>
> linear (in)equality constraints and group constraints can be evaluated
> generically via the formula method I've described above.  Then individual
> weights can be transformed taking the value of the constraint (<,>,=) into
> account (along with the box constraints and leverage constraints).
>
> and so on...
>
> Challenges:
> - recovering the transformed vector from a optimization solver that
> doesn't directly support a mapping function.  I've got some tricks for this
> using environments that we can revisit after we get the basic methodology
> working.
>
> -allowing for progressively relaxing constraints when the constraints are
> simply too restrictive.  Perhaps Doug has some documentation on this as
> he's done it in the past, or perhaps we can simply deal with it in the
> penalty part of constrained_objective()
>
> Hopefully this was helpful.
>
> Regards,
>
> Brian
>
> --
> Brian G. Peterson
> http://braverock.com/brian/
> Ph: 773-459-4973
> IM: bgpbraverock
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