Symbolic linear matrix inequalities (LMI) and semi-definite programming (SDP) tools for Python
Project description
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 and NumPy; and optionally on CVXOPT and on SciPy (for sparse matrices). Single codebase supporting both Python 2.7 and Python 3.x. PyLMI-SDP is tested for various combinations of Python and sympy. See here.
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{matrix}x + 1 & y + 2\\y + 2 & x + z\end{matrix}\right] \succ 0
>>> print(latex(lmi.expanded(variables)))
\left[\begin{matrix}1.0 & 0.0\\0.0 & 1.0\end{matrix}\right] x + \left[\begin{matrix}0.0 & 1.0\\1.0 & 0.0\end{matrix}\right] y + \left[\begin{matrix}0.0 & 0.0\\0.0 & 1.0\end{matrix}\right] z + \left[\begin{matrix}1.0 & 2.0\\2.0 & 0.0\end{matrix}\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{matrix}- x & - y\\- y & - x - z\end{matrix}\right] \preceq \left[\begin{matrix}1 & 2\\2 & 0\end{matrix}\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
Install
From PyPi:
pip install PyLMI-SDP
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
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