L1 Norm Constraint in Maximum-Sharpe-Ratio Portfolio
Dear Gurobi Community,
I successfully implemented a Minimum-Variance Portfolio with L1 Norm Constraint in order to control the short selling budget (e.g. to create a 130/30 long short strategy). Doing so, I chose to linearize the L1 Norm Constraint for subsequent representation purposes.
However, I am struggling now to transfer the linearized L1 Norm Constraint to a Maximum-Sharpe-Ratio Portfolio, because of the conversion which is necessary to optimize the portfolio in Gurobi, see:
I was able to model the Maximum-Sharpe-Ratio Portfolio with Short Selling Constraint due to Eli's awesome explanation, but does anyone know how to do the conversion in case of a linearized L1 Norm Constraint in Gurobi? All of my attempts failed so far.
The following displays the optimization problem and its corresponding conversion on paper:
The following displays a code snippet of my current approach to a Maximum-Sharpe-Ratio Portfolio with Short Selling Constraint in Python/Gurobi and serves as my starting point I'd like to augment for the linearized L1 Norm Constraint:
N = returns.shape # number of assets
S = returns.cov().to_numpy() # covariance matrix
mu = returns.mean().to_numpy() # expected returns
A = mu
b = np.array()
C = np.identity(N)
d = np.repeat(0,N)
model = gp.Model("MaxSharpeSSC")
y = model.addMVar((N,), lb=-gp.GRB.INFINITY)
model.setObjective(y @ S @ y, gp.GRB.MINIMIZE)
model.addConstr(A @ y == b)
model.addConstr(C @ y >= d)
Any ideas would be much appreciated.
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