Active share constraint DCP problems [CVXPY]

Hi Everyone,

I am working on a portfolio optimisation case with an objective function like this, to minimise tracking error

"Minimize(quad_form(w, cov) - 2*w.T*cov_b)"


I am now trying to add a constraint on active share with a minimum threshold

"sum_entries(abs(w - df_in['BENCH_WEIGHT'].as_matrix())) >= 0.8 "


so I basically want to keep my active part of the portfolio above 80%. But I now get a DCP error, which I understand why, I can do

"sum_entries(abs(w - df_in['BENCH_WEIGHT'].as_matrix())) <= 0.8"


which will work fine. But I cant figure out how I can transform my problem into a DCP conform case, any guidance will be appreciated

2 responses

Hi Markus,

CVXPY (and, more generally, any convex optimization library) can only handle constraints where the set of all points satisfying that constraint forms a convex set. A set is convex if, for any two points a and b in the set, all points on the line between a and b are also in the set. Geometrically, that formulation corresponds to the idea that convex sets don't have any "holes" and that the boundary of a convex set has no "dents". For example, in two dimensions, circles and squares are convex, but donuts and crescents are not.

In the context of portfolio optimization, a "point" is tuple of portfolio weights, constraints correspond to sets of portfolios that satisfy the constraints we're interested in.

The constraint that you're interested in is:

sum_entries(abs(w - df_in['BENCH_WEIGHT'].as_matrix())) >= 0.8


The set of points satisfying this constraint corresponds geometrically to the set of points outside a hypercube centered at df_in['BENCH_WEIGHT']. Unfortunately, that set isn't convex. An easy to to see that is to notice that df_in['BENCH_WEIGHT'] doesn't satisfy the constraint, but df_in['BENCH_WEIGHT'] plus or minus 0.81 does satisfy it.

As a slightly handwavey rule, it's usually possible to place upper bounds on expressions involving absolute quantities, but not possible to place lower bounds on those expressions. This is why your constraint works when you replace the greater than constraint with a less than constraint.

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Hi Scott,
thanks for your reply, I think i set up the constraint wrong. According to literature it should be

1-sum_entries(min_elemwise(w, df_in['BENCH_WEIGHT'])) >= 0.8