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tools to export LMIs to [CVXOPT](http://abel.ee.ucla.edu/cvxopt/userguide/coneprog.html#semidefinite-programming) SDP input and to the [SDPA](http://sdpa.sourceforge.net/) format.\n\nDepends on [SymPy](http://sympy.org) and [NumPy](http://www.numpy.org/); and optionally on [CVXOPT](http://cvxopt.org) and on [SciPy](http://www.scipy.org) (for sparse matrices).\nSingle codebase supporting both Python 2.7 and Python 3.x.\nPyLMI-SDP is tested for various combinations of Python and sympy. See [here](https://github.com/cdsousa/PyLMI-SDP/actions/workflows/ci.yaml).\n\nPyLMI-SDP is at [GitHub](https://github.com/cdsousa/PyLMI-SDP).\n\n[![Build Status](https://github.com/cdsousa/PyLMI-SDP/actions/workflows/ci.yaml/badge.svg?branch=master)](https://github.com/cdsousa/PyLMI-SDP/actions/workflows/ci.yaml)\n[![Coverage Status](https://coveralls.io/repos/cdsousa/PyLMI-SDP/badge.png?branch=master)](https://coveralls.io/r/cdsousa/PyLMI-SDP?branch=master)\n\nLMI Definition\n--------------\n\n### Examples\n\n```Python\n\u003e\u003e\u003e from sympy import symbols, Matrix\n\u003e\u003e\u003e from lmi_sdp import LMI_PD, LMI_NSD\n\u003e\u003e\u003e variables = symbols('x y z')\n\u003e\u003e\u003e x, y, z = variables\n\u003e\u003e\u003e lmi = LMI_PD(Matrix([[x+1, y+2], [y+2, z+x]]))\n\u003e\u003e\u003e lmi\nMatrix([\n[x + 1, y + 2],\n[y + 2, x + z]]) \u003e 0\n\n```\n\n```Python\n\u003e\u003e\u003e from lmi_sdp import init_lmi_latex_printing\n\u003e\u003e\u003e from sympy import latex\n\u003e\u003e\u003e init_lmi_latex_printing()\n\u003e\u003e\u003e print(latex(lmi))\n\\left[\\begin{matrix}x + 1 \u0026 y + 2\\\\y + 2 \u0026 x + z\\end{matrix}\\right] \\succ 0\n\n```\n![lmi](http://latex.codecogs.com/gif.latex?%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7Dx%2B1%26y%2B2%5C%5Cy%2B2%26x%2Bz%5Cend%7Bmatrix%7D%5Cright%5D%5Csucc0)\n\n```Python\n\u003e\u003e\u003e print(latex(lmi.expanded(variables)))\n\\left[\\begin{matrix}1.0 \u0026 0.0\\\\0.0 \u0026 1.0\\end{matrix}\\right] x + \\left[\\begin{matrix}0.0 \u0026 1.0\\\\1.0 \u0026 0.0\\end{matrix}\\right] y + \\left[\\begin{matrix}0.0 \u0026 0.0\\\\0.0 \u0026 1.0\\end{matrix}\\right] z + \\left[\\begin{matrix}1.0 \u0026 2.0\\\\2.0 \u0026 0.0\\end{matrix}\\right] \\succ 0\n\n```\n![lmi.expanded(variables)](http://latex.codecogs.com/gif.latex?%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D1.0%260.0%5C%5C0.0%261.0%5Cend%7Bmatrix%7D%5Cright%5Dx%2B%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D0.0%261.0%5C%5C1.0%260.0%5Cend%7Bmatrix%7D%5Cright%5Dy%2B%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D0.0%260.0%5C%5C0.0%261.0%5Cend%7Bmatrix%7D%5Cright%5Dz%2B%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D1.0%262.0%5C%5C2.0%260.0%5Cend%7Bmatrix%7D%5Cright%5D%5Csucc0)\n\n```Python\n\u003e\u003e\u003e lmi_2 = LMI_NSD( Matrix([[-x, -y], [-y, -z-x]]), Matrix([[1, 2], [2, 0]]))\n\u003e\u003e\u003e lmi_2\nMatrix([\n[-x,     -y],\n[-y, -x - z]]) \u003c= Matrix([\n[1, 2],\n[2, 0]])\n\u003e\u003e\u003e lmi_2.canonical()\nMatrix([\n[x + 1, y + 2],\n[y + 2, x + z]]) \u003e= 0\n\n```\n\n```Python\n\u003e\u003e\u003e print(latex(lmi_2))\n\\left[\\begin{matrix}- x \u0026 - y\\\\- y \u0026 - x - z\\end{matrix}\\right] \\preceq \\left[\\begin{matrix}1 \u0026 2\\\\2 \u0026 0\\end{matrix}\\right]\n\n```\n![lmi_2](http://latex.codecogs.com/gif.latex?%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D-x%26-y%5C%5C-y%26-x-z%5Cend%7Bmatrix%7D%5Cright%5D%5Cpreceq%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D1%262%5C%5C2%260%5Cend%7Bmatrix%7D%5Cright%5D)\n\nConvertion to CVXOPT SDP\n------------------------\n\n### Example\n\n(from CVXOPT [SDP example](http://abel.ee.ucla.edu/cvxopt/userguide/coneprog.html#semidefinite-programming))\n\n```Python\n\u003e\u003e\u003e from sympy import symbols, Matrix\n\u003e\u003e\u003e from lmi_sdp import LMI_NSD, init_lmi_latex_printing\n\u003e\u003e\u003e\n\u003e\u003e\u003e init_lmi_latex_printing()\n\u003e\u003e\u003e\n\u003e\u003e\u003e variables = symbols('x1 x2 x3')\n\u003e\u003e\u003e x1, x2, x3 = variables\n\u003e\u003e\u003e\n\u003e\u003e\u003e min_obj = x1 - x2 + x3\n\u003e\u003e\u003e\n\u003e\u003e\u003e LMI_1 = LMI_NSD(\n...     x1*Matrix([[-7, -11], [-11, 3]]) +\n...     x2*Matrix([[7, -18], [-18, 8]]) +\n...     x3*Matrix([[-2, -8], [-8, 1]]),\n...     Matrix([[33, -9], [-9, 26]]))\n\u003e\u003e\u003e\n\u003e\u003e\u003e LMI_2 = LMI_NSD(\n...     x1*Matrix([[-21, -11, 0], [-11, 10, 8], [0, 8, 5]]) +\n...     x2*Matrix([[0, 10, 16], [10, -10, -10], [16, -10, 3]]) +\n...     x3*Matrix([[-5, 2, -17], [2, -6, 8], [-17, 8, 6]]),\n...     Matrix([[14, 9, 40], [9, 91, 10], [40, 10, 15]]))\n\u003e\u003e\u003e\n\u003e\u003e\u003e min_obj\nx1 - x2 + x3\n\n```\n![min_obj](http://latex.codecogs.com/gif.latex?x_%7B1%7D-x_%7B2%7D%2Bx_%7B3%7D)\n\n```Python\n\u003e\u003e\u003e LMI_1.expanded(variables)\nMatrix([\n[ -7.0, -11.0],\n[-11.0,   3.0]])*x1 + Matrix([\n[  7.0, -18.0],\n[-18.0,   8.0]])*x2 + Matrix([\n[-2.0, -8.0],\n[-8.0,  1.0]])*x3 \u003c= Matrix([\n[33, -9],\n[-9, 26]])\n\n```\n![LMI_1.expanded(variables)](http://latex.codecogs.com/gif.latex?%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D-7.0%26-11.0%5C%5C-11.0%263.0%5Cend%7Bmatrix%7D%5Cright%5Dx_%7B1%7D%2B%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D7.0%26-18.0%5C%5C-18.0%268.0%5Cend%7Bmatrix%7D%5Cright%5Dx_%7B2%7D%2B%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D-2.0%26-8.0%5C%5C-8.0%261.0%5Cend%7Bmatrix%7D%5Cright%5Dx_%7B3%7D%5Cpreceq%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D33%26-9%5C%5C-9%2626%5Cend%7Bmatrix%7D%5Cright%5D)\n\n```Python\n\u003e\u003e\u003e LMI_2.expanded(variables)\nMatrix([\n[-21.0, -11.0, 0.0],\n[-11.0,  10.0, 8.0],\n[  0.0,   8.0, 5.0]])*x1 + Matrix([\n[ 0.0,  10.0,  16.0],\n[10.0, -10.0, -10.0],\n[16.0, -10.0,   3.0]])*x2 + Matrix([\n[ -5.0,  2.0, -17.0],\n[  2.0, -6.0,   8.0],\n[-17.0,  8.0,   6.0]])*x3 \u003c= Matrix([\n[14,  9, 40],\n[ 9, 91, 10],\n[40, 10, 15]])\n\n```\n![LMI_2.expanded(variables)](http://latex.codecogs.com/gif.latex?%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D-21.0%26-11.0%260.0%5C%5C-11.0%2610.0%268.0%5C%5C0.0%268.0%265.0%5Cend%7Bmatrix%7D%5Cright%5Dx_%7B1%7D%2B%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D0.0%2610.0%2616.0%5C%5C10.0%26-10.0%26-10.0%5C%5C16.0%26-10.0%263.0%5Cend%7Bmatrix%7D%5Cright%5Dx_%7B2%7D%2B%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D-5.0%262.0%26-17.0%5C%5C2.0%26-6.0%268.0%5C%5C-17.0%268.0%266.0%5Cend%7Bmatrix%7D%5Cright%5Dx_%7B3%7D%5Cpreceq%5Cleft%5B%5Cbegin%7Bmatrix%7D%7B%7D14%269%2640%5C%5C9%2691%2610%5C%5C40%2610%2615%5Cend%7Bmatrix%7D%5Cright%5D)\n\n```Python\n\u003e\u003e\u003e from cvxopt import solvers\n\u003e\u003e\u003e from lmi_sdp import to_cvxopt\n\u003e\u003e\u003e\n\u003e\u003e\u003e solvers.options['show_progress'] = False\n\u003e\u003e\u003e\n\u003e\u003e\u003e c, Gs, hs = to_cvxopt(min_obj, [LMI_1, LMI_2], variables)\n\u003e\u003e\u003e\n\u003e\u003e\u003e sol = solvers.sdp(c, Gs=Gs, hs=hs)\n\u003e\u003e\u003e print(sol['x'])\n[-3.68e-01]\n[ 1.90e+00]\n[-8.88e-01]\n\u003cBLANKLINE\u003e\n\n```\n\nExport to SDPA Format\n---------------------\n\n### Example\n\n```Python\n\u003e\u003e\u003e from sympy import symbols, Matrix\n\u003e\u003e\u003e from lmi_sdp import LMI_PSD, to_sdpa_sparse\n\u003e\u003e\u003e\n\u003e\u003e\u003e variables = x1, x2 = symbols('x1 x2')\n\u003e\u003e\u003e\n\u003e\u003e\u003e min_obj = 10*x1 + 20*x2\n\u003e\u003e\u003e lmi_1 = LMI_PSD(\n...     -Matrix([[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 3, 0], [0, 0, 0, 4]]) +\n...     Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])*x1 +\n...     Matrix([[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 5, 2], [0, 0, 2, 6]])*x2)\n\u003e\u003e\u003e lmi_1\nMatrix([\n[x1 - 1,           0,        0,        0],\n[     0, x1 + x2 - 2,        0,        0],\n[     0,           0, 5*x2 - 3,     2*x2],\n[     0,           0,     2*x2, 6*x2 - 4]]) \u003e= 0\n\u003e\u003e\u003e\n\u003e\u003e\u003e dat = to_sdpa_sparse(min_obj, lmi_1, variables, comment='test sparse')\n\u003e\u003e\u003e print(dat)\n\"test sparse\"\n2 = ndim\n3 = nblocks\n1 1 2 = blockstruct\n10.0, 20.0 = objcoeffs\n0 1 1 1 1.0\n0 2 1 1 2.0\n0 3 1 1 3.0\n0 3 2 2 4.0\n1 1 1 1 1.0\n1 2 1 1 1.0\n2 2 1 1 1.0\n2 3 1 1 5.0\n2 3 1 2 2.0\n2 3 2 2 6.0\n\u003cBLANKLINE\u003e\n\n```\n\n\nAuthor\n------\n\n[Cristóvão Duarte Sousa](https://github.com/cdsousa)\n\nInstall\n-------\n\nFrom PyPi:\n\n    pip install PyLMI-SDP\n\nFrom git source:\n\n    git clone https://github.com/cdsousa/PyLMI-SDP.git\n    cd PyLMI-SDP\n    python setup.py install\n\nLicense\n-------\n\nSimplified BSD License. See [License File](LICENSE.txt)\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fcdsousa%2Fpylmi-sdp","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fcdsousa%2Fpylmi-sdp","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fcdsousa%2Fpylmi-sdp/lists"}