Tutorial

The implementation follows an object-oriented design. The core object is SdpRelaxation. There are three steps to generate the relaxation:

  • Instantiate the SdpRelaxation object.
  • Get the relaxation.
  • Write the relaxation to a file or solve the problem.

The second step is the most time consuming, often running for hours as the number of variables increases. Once the solution is obtained, it can be studied further with some helper functions.

To instantiate the SdpRelaxation object, you need to specify the variables. You can use any SymPy symbolic variable, as long as the adjoint operator is well-defined. The library also has helper functions to generate commutative or noncommutative variables or operators.

Getting the relaxation requires at least the level of relaxation, and the matching method, SdpRelaxation.get_relaxation, will generate the moment matrix. Additional elements of the problem, such as the objective function, inequalities, equalities, and constraints on the moments.

The last step in is to either solve or export the relaxation. The function solve_sdp or the class method SdpRelaxation.solve autodetects the possible solvers: SDPA, MOSEK, and CVXOPT. Alternatively, the method write_to_file exports the file to sparse SDPA format, which can be solved externally on a supercomputer, in MATLAB, or by any other means that accepts this input format.

Defining a Polynomial Optimization Problem of Commuting Variables

Consider the following polynomial optimization problem:

\[\min_{x\in \mathbb{R}^2}2x_1x_2\]

such that

\[-x_2^2+x_2+0.5\geq 0\]
\[x_1^2-x_1=0.\]

The equality constraint is a simple projection. We either substitute it with two inequalities or treat the equality as a monomial substitution. The second option leads to a sparser SDP relaxation. The code samples below take this approach. In this case, the monomial basis is \(\{1, x_1, x_2, x_1x_2, x_2^2\}\). The corresponding level-2 relaxation is written as

\[\min_{y}2y_{12}\]

such that

\[\begin{split}\left[ \begin{array}{c|cc|cc}1 & y_{1} & y_{2} & y_{12} & y_{22}\\ \hline{}y_{1} & y_{1} & y_{12} & y_{12} & y_{122}\\ y_{2} & y_{12} & y_{22} & y_{122} & y_{222}\\ \hline{}y_{12} & y_{12} & y_{122} & y_{122} & y_{1222}\\ y_{22} & y_{122} & y_{222} & y_{1222} & y_{2222}\end{array} \right] \succeq{}0\end{split}\]
\[\begin{split}\left[ \begin{array}{c|cc}-y_{22}+y_{2}+0.5 & -y_{122}+y_{12}+0.5y_{1} & -y_{222}+y_{22}+0.5y_{2}\\ \hline{}-y_{122}+y_{12}+0.5y_{1} & -y_{122}+y_{12}+0.5y_{1} & -y_{1222}+y_{122}+0.5y_{12}\\ -y_{222}+y_{22}+0.5y_{2} & -y_{1222}+y_{122}+0.5y_{12} & -y_{2222}+y_{222}+0.5y_{22} \end{array}\right]\succeq{}0.\end{split}\]

Apart from the matrices being symmetric, notice other regular patterns between the elements – these are recognized in the relaxation and the same SDP variables are used for matching moments. To generate the relaxation, first we set up a few helper variables, including the symbolic variables used to define the polynomial objective function and constraint. The symbolic manipulations are based on SymPy.

from ncpol2sdpa import *

n_vars = 2 # Number of variables
level = 2  # Requested level of relaxation
x = generate_variables('x', n_vars)

By default, the generated variables are commutative. Alternatively, you can use standard SymPy symbols, but it is worth declaring them as real. With these variables, we can define the objective and the inequality constraint.

obj = x[0]*x[1] + x[1]*x[0]
inequalities = [-x[1]**2 + x[1] + 0.5>=0]

We can also write all inequality-type constraints assuming to be in the form \(\ge 0\) as

inequalities = [-x[1]**2 + x[1] + 0.5]

This is more convenient when we have a large number of constraints.

The equality, as discussed, is entered as a substitution rule:

substitutions = {x[0]**2 : x[0]}

Generating and Solving the Relaxation

After we defined the problem, we need to initialize the SDP relaxation object with the variables, and request generating the relaxation given the constraints:

sdp = SdpRelaxation(x)
sdp.get_relaxation(level, objective=obj, inequalities=inequalities,
                   substitutions=substitutions)

For large problems, getting the relaxation can take a long time. Once we have the relaxation, we can try to solve it solve it. Currently three solvers are supported fully: SDPA, MOSEK, and CVXOPT. If any of them are available, we obtain the solution by calling the solve method:

sdp.solve()
print(sdp.primal, sdp.dual, sdp.status)

This gives a solution close to the optimum around -0.7321. The solution and some status information and the time it takes to solve it become part of the relaxation object.

If no solver is detected, or you want more control over the parameters of the solver, or you want to solve the problem in MATLAB, you export the relaxation to SDPA format:

sdp.write_to_file('example.dat-s')

You can also specify a solver if you wish. For instance, if you want to use the arbitrary-precision solver that you have available in the path, along with a matching parameter file, you can call

sdp.solve(solver='sdpa', solverparameters={"executable":"sdpa_gmp",
                                           "paramsfile"="params.gmp.sdpa"})

If you have multiple solvers available, you might want to specify which exactly you want to use. For CVXOPT, call

sdp.solve(solver='cvxopt')
print(sdp.primal, sdp.dual)

This solution also requires PICOS on top of CXOPT. Alternatively, if you have MOSEK installed and it is callable from your Python distribution, you can request to use it:

sdp.solve(solver=’mosek’) print(sdp.primal, sdp.dual)

Analyzing the Solution

We can study individual entries of the solution matrix by providing the monomial we are interested in. For example:

sdp[X[0]*X[1]]

The sums-of-square (SOS) decomposition is extracted from the dual solution:

sigma = sdp.get_sos_decomposition()

If we solve the SDP with the arbitrary-precision solver sdpa_gmp, we can find a rank loop at level two, indicating that convergence has been achieved.

sdp.solve(solver='sdpa', solverparameters={"executable":"sdpa_gmp",
                                           "paramsfile"="params.gmp.sdpa"})
sdp.find_solution_ranks()

The output for this problem is [2, 3], not showing a rank loop at this level of relaxation.

Debugging the SDP Relaxation

It often happens that solving a relaxation does not yield the expected results. To help understand what goes wrong, Ncpol2sdpa provides a function to write the relaxation in a comma separated file, in which the individual cells contain the respective monomials. The first line of the file is the objective function.

sdp.write_to_file("examples.csv")

Furthermore, the library can write out which SDP variable corresponds to which monomial by calling

sdp.save_monomial_index("monomials.txt")

Defining and Solving an Optimization Problem of Noncommuting Variables

Consider a slight variation of the problem discussed in the previous sections: change the algebra of the variables from commutative to Hermitian noncommutative, and use the following objective function:

\[\min_{x\in \mathbb{R}^2}x_1x_2+x_2x_1\]

The constraints remain identical:

\[-x_2^2+x_2+0.5\geq 0\]
\[x_1^2-x_1=0.\]

Defining the problem, generating the relaxation, and solving it follow a similar pattern, but we request operators instead of variables.

X = generate_operators('X', n_vars, hermitian=True)
obj_nc = X[0] * X[1] + X[1] * X[0]
inequalities_nc = [-X[1] ** 2 + X[1] + 0.5]
substitutions_nc = {X[0]**2 : X[0]}
sdp_nc = SdpRelaxation(X)
sdp_nc.get_relaxation(level, objective=obj_nc, inequalities=inequalities_nc,
                      substitutions=substitutions_nc)
sdp_nc.solve()

This gives a solution very close to the analytical -3/4. Let us export the problem again:

sdp.write_to_file("examplenc.dat-s")

Solving this with the arbitrary-precision solver, we discover a rank loop:

sdp.solve(solver='sdpa', solverparameters={"executable":"sdpa_gmp",
                                           "paramsfile"="params.gmp.sdpa"})
sdp.find_solution_ranks()

The output is [2, 2], indicating a rank loop and showing that the noncommutative case of the relaxation converges faster.