Checking Convexity in Sector-Constrained Portfolio Optimization

Summary

The document poses a portfolio optimization problem using forecast returns and a covariance matrix for DAX stocks. The objective is to maximize forecast portfolio return while limiting tracking error for each sector. Its central lesson is that convexity cannot be determined from the objective or the word “quadratic” alone: the quadratic parts of the constraints must be expressed in matrix form and checked for positive semidefiniteness.

The reply gives this as a diagnostic rather than working through the matrices or solving the optimization. It does not establish whether the specific sector constraints are convex, whether multiple local minima exist, or what assumptions apply to the covariance and sector tracking error calculations. A reader would need the precise constraint definitions to reach those conclusions.

Key ideas

• Convexity depends on the structure of the quadratic constraints.
• Write each tracking error constraint in matrix form before assessing it.
• Check whether each relevant quadratic matrix is positive semidefinite.
• The document does not determine convexity or local minima for the specific problem.

Tags

# Dax 30 Portfolio optimisation


# Dax 30 Portfolio optimisation












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 tracking error constraints for each sector being less than 1%.

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

## Answer by river_rat (score 1)

https://quant.stackexchange.com/a/54117

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.