PyLMI-SDP

Symbolic linear matrix inequalities (LMI) and semi-definite programming (SDP) tools for Python

This package includes a set of classes to represent and manipulate LMIs symbolically using SymPy. It also includes tools to export LMIs to CVXOPT SDP input and to the SDPA format.

Depends on SymPy 0.7.3 and on NumPy 1.7.1; and optionally on CVXOPT and on SciPy (for sparse matrices). Single codebase supporting both Python 2.7 and Python 3.3. PyLMI-SDP is tested in these versions but it may work in others.

PyLMI-SDP is at GitHub

LMI Definition

Examples

>>> from sympy import symbols, Matrix
>>> from lmi_sdp import LMI_PD, LMI_NSD
>>> variables = symbols('x y z')
>>> x, y, z = variables
>>> lmi = LMI_PD(Matrix([[x+1, y+2], [y+2, z+x]]))
>>> lmi
Matrix([
[x + 1, y + 2],
[y + 2, x + z]]) > 0

>>> from lmi_sdp import init_lmi_latex_printing
>>> from sympy import latex
>>> init_lmi_latex_printing()
>>> print(latex(lmi))
\left[\begin{smallmatrix}{}x + 1 & y + 2\\y + 2 & x + z\end{smallmatrix}\right] \succ 0

>>> print(latex(lmi.expanded(variables)))
\left[\begin{smallmatrix}{}1.0 & 0.0\\0.0 & 1.0\end{smallmatrix}\right] x + \left[\begin{smallmatrix}{}0.0 & 1.0\\1.0 & 0.0\end{smallmatrix}\right] y + \left[\begin{smallmatrix}{}0.0 & 0.0\\0.0 & 1.0\end{smallmatrix}\right] z + \left[\begin{smallmatrix}{}1.0 & 2.0\\2.0 & 0.0\end{smallmatrix}\right] \succ 0

>>> lmi_2 = LMI_NSD( Matrix([[-x, -y], [-y, -z-x]]), Matrix([[1, 2], [2, 0]]))
>>> lmi_2
Matrix([
[-x,     -y],
[-y, -x - z]]) <= Matrix([
[1, 2],
[2, 0]])
>>> lmi_2.canonical()
Matrix([
[x + 1, y + 2],
[y + 2, x + z]]) >= 0

>>> print(latex(lmi_2))
\left[\begin{smallmatrix}{}- x & - y\\- y & - x - z\end{smallmatrix}\right] \preceq \left[\begin{smallmatrix}{}1 & 2\\2 & 0\end{smallmatrix}\right]

Convertion to CVXOPT SDP

Example

(from CVXOPT SDP example)

>>> from sympy import symbols, Matrix
>>> from lmi_sdp import LMI_NSD, init_lmi_latex_printing
>>>
>>> init_lmi_latex_printing()
>>>
>>> variables = symbols('x1 x2 x3')
>>> x1, x2, x3 = variables
>>>
>>> min_obj = x1 - x2 + x3
>>>
>>> LMI_1 = LMI_NSD(
...     x1*Matrix([[-7, -11], [-11, 3]]) +
...     x2*Matrix([[7, -18], [-18, 8]]) +
...     x3*Matrix([[-2, -8], [-8, 1]]),
...     Matrix([[33, -9], [-9, 26]]))
>>>
>>> LMI_2 = LMI_NSD(
...     x1*Matrix([[-21, -11, 0], [-11, 10, 8], [0, 8, 5]]) +
...     x2*Matrix([[0, 10, 16], [10, -10, -10], [16, -10, 3]]) +
...     x3*Matrix([[-5, 2, -17], [2, -6, 8], [-17, 8, 6]]),
...     Matrix([[14, 9, 40], [9, 91, 10], [40, 10, 15]]))
>>>
>>> min_obj
x1 - x2 + x3

>>> LMI_1.expanded(variables)
Matrix([
[ -7.0, -11.0],
[-11.0,   3.0]])*x1 + Matrix([
[  7.0, -18.0],
[-18.0,   8.0]])*x2 + Matrix([
[-2.0, -8.0],
[-8.0,  1.0]])*x3 <= Matrix([
[33, -9],
[-9, 26]])

>>> LMI_2.expanded(variables)
Matrix([
[-21.0, -11.0, 0.0],
[-11.0,  10.0, 8.0],
[  0.0,   8.0, 5.0]])*x1 + Matrix([
[ 0.0,  10.0,  16.0],
[10.0, -10.0, -10.0],
[16.0, -10.0,   3.0]])*x2 + Matrix([
[ -5.0,  2.0, -17.0],
[  2.0, -6.0,   8.0],
[-17.0,  8.0,   6.0]])*x3 <= Matrix([
[14,  9, 40],
[ 9, 91, 10],
[40, 10, 15]])

>>> from cvxopt import solvers
>>> from lmi_sdp import to_cvxopt
>>>
>>> solvers.options['show_progress'] = False
>>>
>>> c, Gs, hs = to_cvxopt(min_obj, [LMI_1, LMI_2], variables)
>>>
>>> sol = solvers.sdp(c, Gs=Gs, hs=hs)
>>> print(sol['x'])
[-3.68e-01]
[ 1.90e+00]
[-8.88e-01]
<BLANKLINE>

Export to SDPA Format

Example

>>> from sympy import symbols, Matrix
>>> from lmi_sdp import LMI_PSD, to_sdpa_sparse
>>>
>>> variables = x1, x2 = symbols('x1 x2')
>>>
>>> min_obj = 10*x1 + 20*x2
>>> lmi_1 = LMI_PSD(
...     -Matrix([[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 3, 0], [0, 0, 0, 4]]) +
...     Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])*x1 +
...     Matrix([[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 5, 2], [0, 0, 2, 6]])*x2)
>>> lmi_1
Matrix([
[x1 - 1,           0,        0,        0],
[     0, x1 + x2 - 2,        0,        0],
[     0,           0, 5*x2 - 3,     2*x2],
[     0,           0,     2*x2, 6*x2 - 4]]) >= 0
>>>
>>> dat = to_sdpa_sparse(min_obj, lmi_1, variables, comment='test sparse')
>>> print(dat)
"test sparse"
2 = ndim
3 = nblocks
1 1 2 = blockstruct
10.0, 20.0 = objcoeffs
0 1 1 1 1.0
0 2 1 1 2.0
0 3 1 1 3.0
0 3 2 2 4.0
1 1 1 1 1.0
1 2 1 1 1.0
2 2 1 1 1.0
2 3 1 1 5.0
2 3 1 2 2.0
2 3 2 2 6.0
<BLANKLINE>

Author

Cristóvão Duarte Sousa

Install

From git source:

git clone https://github.com/cdsousa/PyLMI-SDP.git
cd PyLMI-SDP
python setup.py install

License

Simplified BSD License. See License File