I have an initial set of N weights, W. They sum up to 1.
Now, these weights need to be adjusted such that they fullfill the following constraint:
- The aggregated weight of the weights exceeding $\gamma$ shall not exceed $\Gamma$ (e.g. the sum of all weights exceeding 2% shall not exceed 20%)
Therefore, each weight as an adjustment factor, $\lambda$, which is defined as the ratio of the final weight to its initial weight.
The final weights are selected such that the following objectives are minimized:
- $\sum_{i=1}^{N}|\lambda_{i} - 1|$ and
- $\frac{1}{N}\sum_{i=1}^{N}(\lambda_{i}-\bar{\lambda})^{2}$
Of course, the final weights shall sum up to 1.
This process is conducted under the following principals:
- In general, the ranked order of the final weights is consistent with the ranked order of the initial weights
- All uncapped weights share an adjustment factor, as this minimizes objective 2) (the variance of the adjustment factors).
How would you set up this optimization problem and how would you solve it? Unfortunately, I am not very familiar with non-linear optimization so I would be super thankful for any hints.