问题描述
我正在使用CVXOPT进行二次编程，以使用均值方差优化来计算组合的最佳权重. http://abel.ee.ucla中有一个很好的例子.edu/cvxopt/userguide/coneprog.html#quadratic-programming .但是，参数采用正则化形式(根据作者).该示例是基本版本.我希望在其中做一些更复杂的问题:
I'm using CVXOPT to do quadratic programming to compute the optimal weights of a potfolio using mean-variance optimization. There is a great example at http://abel.ee.ucla.edu/cvxopt/userguide/coneprog.html#quadratic-programming. However, the arguments are in a regularized form (according to the author). The example is a basic version. I am looking to do a bit of a more complex problem where:
min:
x'Sx
s.t.:
x'a >= g
x'1 = 0
x >= -Wb
x <= c1 - Wb
where:
x: active weights of assets (active weight = portfolio weight - benchmark weight)
S: covariance matrix of asset returns
a: expected stock excess returns
g: target gain
Wb: weights of assets in the benchmark
c: upper limit (weight) of any asset in the portfolio
假定所有变量都是经过计算或已知的.
Assume all the variables are computed or known.
文档中提供的基本示例:
The basic example presented in the documentation:
min:
x'Sx
s.t.
p'x >= g
1'x = 1
其中p是资产收益.
我不知道的内容(请参考 http://abel.ee.ucla.edu/cvxopt/examples/book/portfolio.html 和上面的优化问题):
What I do not know (referring to the code at http://abel.ee.ucla.edu/cvxopt/examples/book/portfolio.html and optimization problem above):
1.我认为这些参数设置了约束，但我不确定:
1.I think these arguments setup the constraints but I'm not entirely sure:
G = matrix(0.0, (n,n))
G[::n+1] = -1.0
h = matrix(0.0, (n,1))
A = matrix(1.0, (1,n))
b = matrix(1.0)
2.我认为这是规范形式"的最小化问题的一部分，我不确定这意味着什么:
2.I believe this is part of the minimization problem in "regulated form", which I'm not sure what means:
mus = [ 10**(5.0*t/N-1.0) for t in xrange(N) ]
3.qp的参数是什么(solver.qp是二次优化器):
3.What the arguments to qp are (solver.qp is the quadratic optimizer):
xs = [ qp(mu*S, -pbar, G, h, A, b)['x'] for mu in mus ]
看看文档，我很确定mu * S(第一个参数)是要最小化的目标函数，而-pbar是返回值.但是，这看起来像一个最大化问题(最大化负收益).
Looking at the documentation, I'm pretty sure that mu*S (the first argument) is the objective function to be minimzed and -pbar are the returns. This looks like a maximization problem however (maximizing negative returns).
我不知道，但是其他参数如何使用.
I do not know, however how the other arguments are used.
鉴于上述最小化问题和约束，我正在寻找使用优化器的帮助.
I am looking for help using the optimizer given my minimization problem and constraints above.
推荐答案
我阅读了文档，并且我认为您必须将函数与以下参数一起使用.我假设x的大小为n:
I read the docs and I think you have to use the function with the following parameters. I assume that x has size n:
P = S
q = (0,....0)
A = (1, ...... 1)
b = (0)
G是从垂直堆叠的
-a
+I_n
-I_n
其中，I_n是大小为n x n的单位矩阵.相应的右侧h是
where I_n is the identity matrix of size n x n . And the corresponding right hand side h is
-g
Wb
...
Wb
C1-Wb
...
C1-Wb
即:一个-g，n乘Wb和n乘C1-Wb.
HTH.
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