Multiple tracking error constraints - is this problem convex?

Question

Let's say I have a return forecast for each stock in the DAX index. I also have a covariance matrix for these 30 stocks.

I want to solve for the 30 weights by maximising the forecast portfolio return, subject to keeping the overall volatility at a certain target level, and subject to multiple tracking error constraints: the overall t.e. for the entire portfolio should be less than or equal to 1%, and tracking error for each sector to also be below or equal to 1% (relative to dax and each Dax sector, respectively).

So, the objective function is linear, but I have multiple quadratic constraints.

Is this a convex problem? Could there exist multiple local minima?

Answer 1

The problem being convex depends on the structure of the quadratic constraints in this case, particularly if the quadratic part is positive semi-definite. So you need to write out the constraints in matrix form and do the algebra to check.