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

Assume all the variables are computed or known.

The basic example presented in the documentation:

min:

x'Sx

s.t.

p'x >= g

1'x = 1

Where p are the asset returns.

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.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.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.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 ]

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).

I do not know, however how the other arguments are used.

I am looking for help using the optimizer given my minimization problem and constraints above.

回答1:

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 is vertically stacked from

-a

+I_n

-I_n

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

That is: one -g, n times Wb and n times C1-Wb.

HTH.

来源：https://stackoverflow.com/questions/7572698/python-using-cvxopt-for-quadratic-programming