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I need to optimize a portfolio given side constraints on the past performance of the portfolio (in terms of Value-at-Risk). A simplified version of my problem is 

min t
s.t. t >= (w'H1w)^0.5 + (w'H2w)^0.5 = ||G1w||_2 + ||G2w||_2           (1)
          ...

where H1 and H2 are covariance matrices and w is a vector of portfolio weights. G1 and G2 are such that H = G'G. The dots indicate other constraints which I omit for brevity.

According to the paper, this is a second-order cone problem. I tried to do this in Mosek but I do not see how I could write (1) as a cone. If I had to minimize the sum of the variances, the task would be easy, but unfortunately, I have to minimize the sum of the standard deviations.

How can I write (1) in terms of a (rotated) quadratic cone?

LittleBobbyTables - Au Revoir
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BayerSe
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  • I'm voting to close this question as off-topic because it is a question of mathematics. Linear programming (of which SOCP is a variation) is not programming as generally accepted here on SO. – High Performance Mark Dec 09 '15 at 10:50

1 Answers1

2

The trick is to split the sum into two terms. You can write for example

min t1+t2 s.t. t1 >= (w'H1w)^0.5 and t2 >= (w'H2w)^0.5

and each constraint can be represented using a quadratic cone.

Fanfan
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