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?