我正在优化投资组合，以最大程度地提高积极回报。我还有四个约束条件：权重之和必须等于1，所有权重必须小于或等于0.05，所有权重必须大于或等于0.0005，活动风险必须小于或等于7％ 。
我有99只股票。这意味着我在计算中使用了3个矩阵。 alphas（预期收益）是形状为（99,1）的矩阵。 W_bench是基准中每种股票的权重，并且具有与alpha相同的形状。 V是形状为（99,99）的协方差矩阵。从下面的代码中可以看出，Active_Risk是某些人称为跟踪错误的东西。下面的代码是我设置优化的方式：
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)
尽管需要很长时间，但我可以使用excel的求解器成功解决此问题。但是，我似乎无法在Python上使用CVXPY使其工作。我很乐意上传数据，以便其他人可以帮助我解决问题，但我不知道该怎么做。我也考虑使用随机数据对这个问题进行较小的设计，但是我担心最佳解决方案可能不可行。
运行代码result = prob.solve()时，出现以下错误：
---------------------------------------------------------------------------
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]])
我是CVXPY的新手，并不完全了解DCP规则。任何帮助，将不胜感激。谢谢。
答案 0 :(得分：0)
这个问题的形式很糟糕，因为每个试图使其运行的人都需要通过编写一些独立的示例来避免的工作（是的：即使您对某些最初的失败尝试发表评论，我认为它也应该更容易自己去做，而不是问我们。）
还有因素 context 。 DCP是基于规则的框架，能够表达很多凸问题，但不是全部凸问题。有迹象表明，这个问题与DCP兼容，但是对此的确定答案将再次导致我们这方面的工作。
这一切都导致我方对以下方面的有限保证：
您的问题位于：
Active_Risk = cp.sqrt((weights - W_bench).T @ V @ (weights - W_bench))
代替使用：
cp.sqrt((weights - W_bench).T @ V @ (weights - W_bench))
您将需要使用：
cp.quad_form((weights - W_bench), V)
我对上面一行中丢失的外部cp.sqrt感到不好，但是您可以尝试一下。
但是，很重要的一点是，可以通过更改其他模型组件（缩放参数）来纠正使用此非sqd Quad_form的问题。
所以代替：
0.07 >= Active_Risk
您将拥有：
0.07^2 >= Active_Risk
您无需触摸物镜（无论是否平方；物镜都会变化，但解矢量不会）。
答案 1 :(得分：0)
我能够使用以下代码进行优化：
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)
注意：未指定求解器或使用ECOS时出现多个错误。 SCS可以完美运行，但是需要大量迭代才能找到最佳解决方案。无论哪种方式，代码都只需要几秒钟即可运行。