I am optimizing a portfolio where I maximize active return. I also have four constraints: the sum of weights must be equal to 1, all weights must be less than or equal to 0.05, all weights must be greater than or equal to 0.0005, and active risk must be equal less than or equal 7%.
I have 99 stocks. This means that I have 3 matrices being used in my calculations. alphas (expected returns) is a matrix with a shape of (99,1). W_bench is the weight each stock has in the benchmark and it has the same shape as alphas. V is a covariance matrix with a shape of (99,99). As can be seen in my code below, Active_Risk is what some people call tracking error. The code below is how I am setting up the optimization:
weights = cp.Variable((99,1))
Active_Return = weights.T @ alphas
Active_Risk = cp.sqrt((weights - W_bench).T @ V @ (weights - W_bench))
constraints = [cp.sum(weights) == 1, 0.07 >= Active_Risk, 0.0005 <= weights, weights <= 0.05]
prob = cp.Problem(cp.Maximize(Active_Return), constraints)
I can successfully solve this problem using excel's solver, though it takes quite a long time. However, I can't seem to get it to work using CVXPY on Python. I would happily upload my data so that others can help me fix the problem but I do not know how to do that. Also I considered making a smaller version of this problem with random data but I am afraid an optimal solution might not be feasible.
When I run my code result = prob.solve() I get the following error:
---------------------------------------------------------------------------
DCPError Traceback (most recent call last)
<ipython-input-15-e01c2b878783> in <module>
1 # The optimal objective value is returned by `prob.solve()`.
----> 2 result = prob.solve()
/opt/anaconda3/lib/python3.7/site-packages/cvxpy/problems/problem.py in solve(self, *args, **kwargs)
287 else:
288 solve_func = Problem._solve
--> 289 return solve_func(self, *args, **kwargs)
290
291 @classmethod
/opt/anaconda3/lib/python3.7/site-packages/cvxpy/problems/problem.py in _solve(self, solver, warm_start, verbose, parallel, gp, qcp, **kwargs)
565 solver, warm_start, verbose, **kwargs)
566
--> 567 self._construct_chains(solver=solver, gp=gp)
568 data, solving_inverse_data = self._solving_chain.apply(
569 self._intermediate_problem)
/opt/anaconda3/lib/python3.7/site-packages/cvxpy/problems/problem.py in _construct_chains(self, solver, gp)
508
509 except Exception as e:
--> 510 raise e
511
512 def _solve(self,
/opt/anaconda3/lib/python3.7/site-packages/cvxpy/problems/problem.py in _construct_chains(self, solver, gp)
497
498 self._intermediate_chain = \
--> 499 construct_intermediate_chain(self, candidate_solvers, gp=gp)
500 self._intermediate_problem, self._intermediate_inverse_data = \
501 self._intermediate_chain.apply(self)
/opt/anaconda3/lib/python3.7/site-packages/cvxpy/reductions/solvers/intermediate_chain.py in construct_intermediate_chain(problem, candidates, gp)
68 append += ("\nHowever, the problem does follow DQCP rules. "
69 "Consider calling solve() with `qcp=True`.")
---> 70 raise DCPError("Problem does not follow DCP rules. Specifically:\n" + append)
71
72 elif gp and not problem.is_dgp():
DCPError: Problem does not follow DCP rules. Specifically:
The following constraints are not DCP:
power(var58 + -[[5.39791975e-03]
[1.26209297e-03]
[8.00351893e-05]
[5.26548876e-02]
[7.59655721e-03]
[2.57535232e-03]
[5.03372951e-04]
[1.47480336e-03]
[1.38599909e-04]
[5.74108704e-03]
[2.17434475e-03]
[2.60441630e-04]
[2.15412708e-04]
[7.40692609e-03]
[1.87030133e-04]
[1.99318918e-04]
[4.49890561e-04]
[5.54053889e-04]
[2.40066410e-04]
[3.25109445e-05]
[6.29530293e-02]
[9.26141788e-03]
[9.06847951e-02]
[3.82569606e-04]
[3.28101977e-04]
[9.46531949e-04]
[3.65845535e-02]
[6.64869333e-04]
[1.82867338e-03]
[1.16324940e-04]
[9.39765398e-04]
[3.12166211e-03]
[5.94047522e-04]
[2.93656014e-04]
[8.15666860e-04]
[1.53355187e-04]
[5.43259663e-04]
[2.11729826e-04]
[1.25169822e-04]
[7.45171899e-04]
[3.65823985e-04]
[5.55365318e-04]
[7.91907415e-05]
[5.30872851e-03]
[8.73601572e-04]
[9.36948807e-04]
[1.03941556e-02]
[2.95556711e-04]
[7.67666485e-03]
[2.14996147e-04]
[3.91275909e-04]
[2.27743262e-04]
[2.31689803e-04]
[6.84002188e-03]
[7.09758365e-02]
[2.27532699e-04]
[8.05783401e-04]
[4.63372193e-04]
[1.91341960e-03]
[3.45573641e-04]
[2.30427458e-02]
[8.18612558e-04]
[1.43341614e-03]
[1.53359342e-04]
[1.72991720e-04]
[3.30942207e-04]
[2.12224011e-02]
[2.93271371e-04]
[5.22032722e-02]
[2.96349926e-03]
[6.83703630e-04]
[3.92175651e-04]
[1.55757896e-03]
[2.73614114e-04]
[7.23199807e-03]
[1.06194086e-02]
[1.89837834e-04]
[3.06087369e-03]
[2.11597860e-04]
[2.87620087e-04]
[3.61744658e-04]
[4.09530072e-04]
[3.72525152e-04]
[2.96987842e-02]
[3.82775868e-01]
[1.09826720e-03]
[1.49363231e-03]
[2.34448326e-04]
[6.51708448e-03]
[3.43523667e-03]
[2.72356446e-03]
[1.01445242e-03]
[3.21249033e-04]
[1.73042944e-02]
[2.56326349e-03]
[1.53653802e-03]
[2.31159852e-04]
[1.11857284e-03]
[1.00845275e-02]].T * [[ 1.88468032 0.08627036 -0.81308165 ... -2.0625901 -0.64064643
-1.12708076]
[ 0.08627036 0.33892061 0.07628398 ... 0.19302222 0.14115192
-0.02078213]
[-0.81308165 0.07628398 0.58311836 ... 1.07460175 0.38954524
0.53023314]
...
[-2.0625901 0.19302222 1.07460175 ... 2.86645017 0.95771264
1.34641816]
[-0.64064643 0.14115192 0.38954524 ... 0.95771264 0.47074258
0.43406168]
[-1.12708076 -0.02078213 0.53023314 ... 1.34641816 0.43406168
0.7897458 ]] * (var58 + -[[5.39791975e-03]
[1.26209297e-03]
[8.00351893e-05]
[5.26548876e-02]
[7.59655721e-03]
[2.57535232e-03]
[5.03372951e-04]
[1.47480336e-03]
[1.38599909e-04]
[5.74108704e-03]
[2.17434475e-03]
[2.60441630e-04]
[2.15412708e-04]
[7.40692609e-03]
[1.87030133e-04]
[1.99318918e-04]
[4.49890561e-04]
[5.54053889e-04]
[2.40066410e-04]
[3.25109445e-05]
[6.29530293e-02]
[9.26141788e-03]
[9.06847951e-02]
[3.82569606e-04]
[3.28101977e-04]
[9.46531949e-04]
[3.65845535e-02]
[6.64869333e-04]
[1.82867338e-03]
[1.16324940e-04]
[9.39765398e-04]
[3.12166211e-03]
[5.94047522e-04]
[2.93656014e-04]
[8.15666860e-04]
[1.53355187e-04]
[5.43259663e-04]
[2.11729826e-04]
[1.25169822e-04]
[7.45171899e-04]
[3.65823985e-04]
[5.55365318e-04]
[7.91907415e-05]
[5.30872851e-03]
[8.73601572e-04]
[9.36948807e-04]
[1.03941556e-02]
[2.95556711e-04]
[7.67666485e-03]
[2.14996147e-04]
[3.91275909e-04]
[2.27743262e-04]
[2.31689803e-04]
[6.84002188e-03]
[7.09758365e-02]
[2.27532699e-04]
[8.05783401e-04]
[4.63372193e-04]
[1.91341960e-03]
[3.45573641e-04]
[2.30427458e-02]
[8.18612558e-04]
[1.43341614e-03]
[1.53359342e-04]
[1.72991720e-04]
[3.30942207e-04]
[2.12224011e-02]
[2.93271371e-04]
[5.22032722e-02]
[2.96349926e-03]
[6.83703630e-04]
[3.92175651e-04]
[1.55757896e-03]
[2.73614114e-04]
[7.23199807e-03]
[1.06194086e-02]
[1.89837834e-04]
[3.06087369e-03]
[2.11597860e-04]
[2.87620087e-04]
[3.61744658e-04]
[4.09530072e-04]
[3.72525152e-04]
[2.96987842e-02]
[3.82775868e-01]
[1.09826720e-03]
[1.49363231e-03]
[2.34448326e-04]
[6.51708448e-03]
[3.43523667e-03]
[2.72356446e-03]
[1.01445242e-03]
[3.21249033e-04]
[1.73042944e-02]
[2.56326349e-03]
[1.53653802e-03]
[2.31159852e-04]
[1.11857284e-03]
[1.00845275e-02]]), 1/2) <= 0.07 , because the following subexpressions are not:
|-- var58 + -[[5.39791975e-03]
[1.26209297e-03]
[8.00351893e-05]
[5.26548876e-02]
[7.59655721e-03]
[2.57535232e-03]
[5.03372951e-04]
[1.47480336e-03]
[1.38599909e-04]
[5.74108704e-03]
[2.17434475e-03]
[2.60441630e-04]
[2.15412708e-04]
[7.40692609e-03]
[1.87030133e-04]
[1.99318918e-04]
[4.49890561e-04]
[5.54053889e-04]
[2.40066410e-04]
[3.25109445e-05]
[6.29530293e-02]
[9.26141788e-03]
[9.06847951e-02]
[3.82569606e-04]
[3.28101977e-04]
[9.46531949e-04]
[3.65845535e-02]
[6.64869333e-04]
[1.82867338e-03]
[1.16324940e-04]
[9.39765398e-04]
[3.12166211e-03]
[5.94047522e-04]
[2.93656014e-04]
[8.15666860e-04]
[1.53355187e-04]
[5.43259663e-04]
[2.11729826e-04]
[1.25169822e-04]
[7.45171899e-04]
[3.65823985e-04]
[5.55365318e-04]
[7.91907415e-05]
[5.30872851e-03]
[8.73601572e-04]
[9.36948807e-04]
[1.03941556e-02]
[2.95556711e-04]
[7.67666485e-03]
[2.14996147e-04]
[3.91275909e-04]
[2.27743262e-04]
[2.31689803e-04]
[6.84002188e-03]
[7.09758365e-02]
[2.27532699e-04]
[8.05783401e-04]
[4.63372193e-04]
[1.91341960e-03]
[3.45573641e-04]
[2.30427458e-02]
[8.18612558e-04]
[1.43341614e-03]
[1.53359342e-04]
[1.72991720e-04]
[3.30942207e-04]
[2.12224011e-02]
[2.93271371e-04]
[5.22032722e-02]
[2.96349926e-03]
[6.83703630e-04]
[3.92175651e-04]
[1.55757896e-03]
[2.73614114e-04]
[7.23199807e-03]
[1.06194086e-02]
[1.89837834e-04]
[3.06087369e-03]
[2.11597860e-04]
[2.87620087e-04]
[3.61744658e-04]
[4.09530072e-04]
[3.72525152e-04]
[2.96987842e-02]
[3.82775868e-01]
[1.09826720e-03]
[1.49363231e-03]
[2.34448326e-04]
[6.51708448e-03]
[3.43523667e-03]
[2.72356446e-03]
[1.01445242e-03]
[3.21249033e-04]
[1.73042944e-02]
[2.56326349e-03]
[1.53653802e-03]
[2.31159852e-04]
[1.11857284e-03]
[1.00845275e-02]].T * [[ 1.88468032 0.08627036 -0.81308165 ... -2.0625901 -0.64064643
-1.12708076]
[ 0.08627036 0.33892061 0.07628398 ... 0.19302222 0.14115192
-0.02078213]
[-0.81308165 0.07628398 0.58311836 ... 1.07460175 0.38954524
0.53023314]
...
[-2.0625901 0.19302222 1.07460175 ... 2.86645017 0.95771264
1.34641816]
[-0.64064643 0.14115192 0.38954524 ... 0.95771264 0.47074258
0.43406168]
[-1.12708076 -0.02078213 0.53023314 ... 1.34641816 0.43406168
0.7897458 ]] * (var58 + -[[5.39791975e-03]
[1.26209297e-03]
[8.00351893e-05]
[5.26548876e-02]
[7.59655721e-03]
[2.57535232e-03]
[5.03372951e-04]
[1.47480336e-03]
[1.38599909e-04]
[5.74108704e-03]
[2.17434475e-03]
[2.60441630e-04]
[2.15412708e-04]
[7.40692609e-03]
[1.87030133e-04]
[1.99318918e-04]
[4.49890561e-04]
[5.54053889e-04]
[2.40066410e-04]
[3.25109445e-05]
[6.29530293e-02]
[9.26141788e-03]
[9.06847951e-02]
[3.82569606e-04]
[3.28101977e-04]
[9.46531949e-04]
[3.65845535e-02]
[6.64869333e-04]
[1.82867338e-03]
[1.16324940e-04]
[9.39765398e-04]
[3.12166211e-03]
[5.94047522e-04]
[2.93656014e-04]
[8.15666860e-04]
[1.53355187e-04]
[5.43259663e-04]
[2.11729826e-04]
[1.25169822e-04]
[7.45171899e-04]
[3.65823985e-04]
[5.55365318e-04]
[7.91907415e-05]
[5.30872851e-03]
[8.73601572e-04]
[9.36948807e-04]
[1.03941556e-02]
[2.95556711e-04]
[7.67666485e-03]
[2.14996147e-04]
[3.91275909e-04]
[2.27743262e-04]
[2.31689803e-04]
[6.84002188e-03]
[7.09758365e-02]
[2.27532699e-04]
[8.05783401e-04]
[4.63372193e-04]
[1.91341960e-03]
[3.45573641e-04]
[2.30427458e-02]
[8.18612558e-04]
[1.43341614e-03]
[1.53359342e-04]
[1.72991720e-04]
[3.30942207e-04]
[2.12224011e-02]
[2.93271371e-04]
[5.22032722e-02]
[2.96349926e-03]
[6.83703630e-04]
[3.92175651e-04]
[1.55757896e-03]
[2.73614114e-04]
[7.23199807e-03]
[1.06194086e-02]
[1.89837834e-04]
[3.06087369e-03]
[2.11597860e-04]
[2.87620087e-04]
[3.61744658e-04]
[4.09530072e-04]
[3.72525152e-04]
[2.96987842e-02]
[3.82775868e-01]
[1.09826720e-03]
[1.49363231e-03]
[2.34448326e-04]
[6.51708448e-03]
[3.43523667e-03]
[2.72356446e-03]
[1.01445242e-03]
[3.21249033e-04]
[1.73042944e-02]
[2.56326349e-03]
[1.53653802e-03]
[2.31159852e-04]
[1.11857284e-03]
[1.00845275e-02]])
I am new to CVXPY and don't fully understand DCP rules. Any help would be appreciated. Thanks.
I was able to get the optimization to work with the following code:
weights = cp.Variable((99,1))
Active_Return = weights.T @ alphas
Active_Risk = cp.quad_form((weights - W_bench), V)
constraints = [sum(weights) == 1, 0.07**2 >= Active_Risk, weights >= 0.0005, weights <= 0.05]
prob = cp.Problem(cp.Maximize(Active_Return), constraints)
result = prob.solve(qcp=True, verbose= False, solver= 'SCS', eps= 1e-10, max_iters = 100000, warm_start= True)
Note: I had several errors when either not specifying the solver or when using ECOS. SCS works perfectly but needs a large number of iterations to find the optimal solution. Either way it only takes a few seconds for the code to run.