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Subject: Re: Norm Constrained Portfolio Optimization
Date: Wed, 7 Jan 2009 03:56:02 +0000 (UTC)
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"Jeremiah Green" <jeremiah.green@gmail.com> wrote in message <gk13vh$7i8$1@fred.mathworks.com>...
> For portfolio optimization, my conceptual problem is to select weights (w) to minimize the following: min -1*(XRet*w)/(w'*C*w) subject to the constraint that abs(w)<=c, where c is some constant. I have been doing other constrained versions with the fmincon function. Can anyone tell me how I might solve this problem? Thanks
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  A few questions.  You entitled your thread "Norm Constrained Portfolio Optimization" but gave the constraint as "abs(w)<=c".  This latter is not the usual L2-norm (square root of sum of squares) but the L-infinity norm, which is the maximum absolute value.  Which do you actually wish to use?

  Also what are the sizes of the arrays, XRet, C, and w - how many rows and how many columns for each?

  Final disturbing point.  As it stands, it does not look like a well-defined problem, for the following reason.  For any particular w lying within the given constraint, as we change w along a line moving towards the zero point (w = 0), the function you defined must either approach plus infinity or minus infinity, because w is second order in the denominator and only first order in the numerator.  Passing through the zero vector along this line to the other side will reverse its sign.  Either way, the overall minimum would have to be minus infinity.  This means that 'fmincon' (as well as any other kind of optimizer function) would continue to crowd closer and closer to w = 0 without ever finding a true minimum (because there is no finite minimum.)

Roger Stafford