Python凸优化工具包——cvxopt

二次规划

二次规划标准型

\[\begin{array}{l}
{\rm{minimize \quad}}\left( {1/2} \right){x^T}Px + {q^T}x\\
{\rm{subject\ to \quad}}Gx \le h\\
{\rm{ \qquad \qquad }}Ax = b
\end{array}\]

例如

\[\begin{array}{l}
{\rm{minimize \quad }}2x_1^2 + x_2^2 + {x_1}{x_2} + {x_1} + {x_2}\\
{\rm{subject{\ }to \quad }}{x_1} \ge 0\\
{\rm{ \qquad \qquad }}{x_2} \ge 0\\
{\rm{ \qquad \qquad }}{x_1} + {x_2} = 1
\end{array}\]

其中

\[\begin{array}{l}
P = \left[ {\begin{array}{*{20}{c}}
{4,}&1\\
{1,}&2
\end{array}} \right] \qquad q = \left[ {\begin{array}{*{20}{c}}
1\\
1
\end{array}} \right]\\
G = \left[ {\begin{array}{*{20}{c}}
{ - 1,}&0\\
{0,}&{ - 1}
\end{array}} \right] \qquad h = \left[ {\begin{array}{*{20}{c}}
0\\
0
\end{array}} \right]\\
A = \left[ {\begin{array}{*{20}{c}}
{1,}&1
\end{array}} \right] \qquad b = 1
\end{array}\]

代码如下：

from cvxopt import matrix, solvers
Q = 2*matrix([ [2, .5], [.5, 1] ])
p = matrix([1.0, 1.0])
G = matrix([[-1.0,0.0],[0.0,-1.0]])
h = matrix([0.0,0.0])
A = matrix([1.0, 1.0], (1,2))
b = matrix(1.0)
sol=solvers.qp(Q, p, G, h, A, b)
print(sol['x'])

注：

matrix元素的类型必须是double类型，可以通过如下语句设置。

A = matrix([1.0, 1.0], (1,2), 'd')

参考链接：

https://cvxopt.org/examples/index.html

https://cvxopt.org/userguide/index.html