Ncpol2sdpa is a tool to convert a polynomial optimization problem of either commutative or noncommutative variables to a sparse semidefinite programming (SDP) problem that can be processed by the SDPA family of solvers, MOSEK, or further processed by PICOS to solve the problem by CVXOPT . The optimization problem can be unconstrained or constrained by equalities and inequalities.

The objective is to be able to solve very large scale optimization problems. Example applications include:

• Ground-state energy problems: bosonic and fermionic systems, Pauli spin operators.
• Maximum quantum violation of Bell inequalities, also in multipartite scenarios.
• Nieto-Silleras hierarchy for quantifying randomness.
• Moroder hierarchy to enable PPT-style and other additional constraints.
• If using commutative variables, the hierarchy is identical to Lasserre's.

The implementation has an intuitive syntax for entering problems and it scales for a larger number of noncommutative variables using a sparse representation of the SDP problem.

The implementation requires SymPy and Numpy. The code is compatible with both Python 2 and 3, but using version 3 incurs a major decrease in performance.

While the default CPython interpreter is sufficient for small to medium-scale problems, execution time becomes excessive for larger problems. The code is compatible with Pypy. Using it yields a 10-20x speedup. If you use Pypy, you will need the Pypy fork of Numpy.

Optional dependencies include:

• SciPy allows faster execution with the default CPython interpreter, and enables removal of equations and chordal graph extensions.
• Chompack improves the sparsity of the chordal graph extension.
• PICOS is necessary for converting the problem to a PICOS problem.
• MOSEK Python module is necessary to work with the MOSEK converter.
• Cvxopt is required by both Chompack and PICOS.

Documentation is available online. The following code replicates the toy example from Pironio, S.; Navascues, M. & Acin, A. Convergent relaxations of polynomial optimization problems with noncommuting variables SIAM Journal on Optimization, SIAM, 2010, 20, 2157-2180.

from ncpol2sdpa import generate_variables, SdpRelaxation, write_to_sdpa

# Number of Hermitian variables
n_vars = 2
# Level of relaxation
level = 2

# Get Hermitian variables
X = generate_variables(n_vars, hermitian=True)

# Define the objective function
obj = X[0] * X[1] + X[1] * X[0]

# Inequality constraints
inequalities = [-X[1] ** 2 + X[1] + 0.5]

# Simple monomial substitutions
monomial_substitution = {}
monomial_substitution[X[0] ** 2] = X[0]

# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(level, objective=obj, inequalities=inequalities,
                             substitutions=monomial_substitution)
write_to_sdpa(sdpRelaxation, 'examplenc.dat-s')

Further instances are in the examples folder and also in the manual.

The code is available on PyPI, hence it can be installed by

$ sudo pip install ncpol2sdpa

If you want the latest git version, follow the standard procedure for installing Python modules:

$ sudo python setup.py install

This work is supported by the European Commission Seventh Framework Programme under Grant Agreement Number FP7-601138 PERICLES, by the Red Espanola de Supercomputacion grants number FI-2013-1-0008 and FI-2013-3-0004, and by the Swedish National Infrastructure for Computing project number SNIC 2014/2-7.

For more information refer to the following manuscript:

Wittek, P. (2014). Ncpol2sdpa -- Sparse Semidefinite Programming Relaxations for Polynomial Optimization Problems of Noncommuting Variables. To Appear in ACM Transactions on Mathematical Software.