具有 CVXPY 约束的投资组合优化具有 DCP 错误
Portfolio Optimization with CVXPY constraint has DCP error
我正在优化一个投资组合，其中我最大化活跃 return。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，主动风险必须等于小于或等于 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 returns）是一个形状为 (99,1) 的矩阵。 W_bench 是每只股票在基准中的权重，它的形状与 alpha 相同。 V 是形状为 (99,99) 的协方差矩阵。从我下面的代码中可以看出，Active_Risk 就是有些人所说的跟踪误差。下面的代码是我如何设置优化：
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)
我可以使用excel的解算器成功解决这个问题，虽然这需要相当长的时间。但是，我似乎无法在 Python 上使用 CVXPY 让它工作。我很乐意上传我的数据，以便其他人可以帮助我解决问题，但我不知道该怎么做。我也考虑过用随机数据制作这个问题的较小版本，但我担心最佳解决方案可能不可行。
当我 运行 我的代码 result = prob.solve() 我得到以下错误：
---------------------------------------------------------------------------
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]
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[3.91275909e-04]
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[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]
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[3.65845535e-02]
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[1.16324940e-04]
[9.39765398e-04]
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[1.53355187e-04]
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[3.65823985e-04]
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[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]
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[5.22032722e-02]
[2.96349926e-03]
[6.83703630e-04]
[3.92175651e-04]
[1.55757896e-03]
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[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，不完全了解 DCP 规则。任何帮助，将不胜感激。谢谢
这个问题的情况很糟糕，因为每个试图解决它的人 运行 都需要做一些工作，而您本可以通过研究一些独立的示例来避免这些工作（是的：即使您对某些最初的失败尝试发表了评论，我认为自己做应该比让我们做更容易。
还有因素上下文。 DCP是一个基于规则的框架，能够表达很多凸问题，但不是所有的凸问题。有一些迹象表明这个问题是 DCP 兼容的，但是一个明确的答案将再次导致我们这边做更多的工作。
这一切导致我方对以下方面的保证有限：
您的问题在：
Active_Risk = cp.sqrt((weights - W_bench).T @ V @ (weights - W_bench))
- 这是变量的乘积，通常是非凸的，因此，在没有进一步假设的情况下处理一般情况不能在 DCP 中表达
- 你得到了额外的假设，比如 V 是 PSD（因为它是一个协方差矩阵），但是 cvxpy 不能推断出这个给定的形式
- 您需要更清楚地向 cvxpy 表达这个额外的结构/假设
而不是使用：
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 感觉不好，但你可以试试。
重要的是要看到，使用这个非 sqrt quad_form 可以通过更改其他模型组件（缩放参数）来纠正。
所以代替：
0.07 >= Active_Risk
你将拥有：
0.07^2 >= Active_Risk
您不需要触摸 objective（是否平方；objective 会改变，但解向量不会）。
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：在未指定求解器或使用 ECOS 时，我遇到了几个错误。 SCS 工作完美，但需要大量迭代才能找到最佳解决方案。无论哪种方式，代码 运行.
只需要几秒钟
我正在优化一个投资组合，其中我最大化活跃 return。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，主动风险必须等于小于或等于 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 returns）是一个形状为 (99,1) 的矩阵。 W_bench 是每只股票在基准中的权重，它的形状与 alpha 相同。 V 是形状为 (99,99) 的协方差矩阵。从我下面的代码中可以看出，Active_Risk 就是有些人所说的跟踪误差。下面的代码是我如何设置优化：
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)
我可以使用excel的解算器成功解决这个问题，虽然这需要相当长的时间。但是，我似乎无法在 Python 上使用 CVXPY 让它工作。我很乐意上传我的数据，以便其他人可以帮助我解决问题，但我不知道该怎么做。我也考虑过用随机数据制作这个问题的较小版本，但我担心最佳解决方案可能不可行。
当我 运行 我的代码 result = prob.solve() 我得到以下错误：
---------------------------------------------------------------------------
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，不完全了解 DCP 规则。任何帮助，将不胜感激。谢谢
这个问题的情况很糟糕，因为每个试图解决它的人 运行 都需要做一些工作，而您本可以通过研究一些独立的示例来避免这些工作（是的：即使您对某些最初的失败尝试发表了评论，我认为自己做应该比让我们做更容易。
还有因素上下文。 DCP是一个基于规则的框架，能够表达很多凸问题，但不是所有的凸问题。有一些迹象表明这个问题是 DCP 兼容的，但是一个明确的答案将再次导致我们这边做更多的工作。
这一切导致我方对以下方面的保证有限：
您的问题在：
Active_Risk = cp.sqrt((weights - W_bench).T @ V @ (weights - W_bench))
- 这是变量的乘积，通常是非凸的，因此，在没有进一步假设的情况下处理一般情况不能在 DCP 中表达
- 你得到了额外的假设，比如 V 是 PSD（因为它是一个协方差矩阵），但是 cvxpy 不能推断出这个给定的形式
- 您需要更清楚地向 cvxpy 表达这个额外的结构/假设
而不是使用：
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 感觉不好，但你可以试试。
重要的是要看到，使用这个非 sqrt quad_form 可以通过更改其他模型组件（缩放参数）来纠正。
所以代替：
0.07 >= Active_Risk
你将拥有：
0.07^2 >= Active_Risk
您不需要触摸 objective（是否平方；objective 会改变，但解向量不会）。
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：在未指定求解器或使用 ECOS 时，我遇到了几个错误。 SCS 工作完美，但需要大量迭代才能找到最佳解决方案。无论哪种方式，代码 运行.
只需要几秒钟