portfolio-optimization | python application | Portfolio library

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As you note, the tricky constraint is that sum(pmin(w_bench, w_pf)) > 0.7 (actually, it turns out to be very tough to have strict inequality, so I will be doing >= instead of >; you could of course re-solve with >= 0.7+epsilon for some small epsilon). To approach this, we will create a new variable y_i for each element i in our portfolio, and we will add constraints y_i <= wpf_i (aka wpf_i - y_i >= 0) and y_i <= wbench_i (aka -y_i >= -wbench_i), where wpf_i is the proportion of i in our selected portfolio (a decision variable) and wbench_i is the proportion of i in the benchmark portfolio (input data). This constrains y_i to be no larger than the minimum of these two values. Finally, we will add the constraint \sum_i y_i >= 0.7, requiring that these minimum values sum to at least 0.7.

All that remains is to implement this in the quadprog package. Setting up with your problem data:

I am trying to understand your data: