NCTSSOS

NCTSSOS is a non-commutative polynomial optimization tool based on the sparsity adapted moment-SOHS hierarchies. To use NCTSSOS in Julia, run

pkg> add https://github.com/wangjie212/NCTSSOS

Documentation

Dependencies

• Julia
• JuMP
• Mosek or COSMO
• ChordalGraph

NCTSSOS has been tested on WINDOW 10, Julia 1.6, JuMP 1.11.1 and MOSEK 10.0.

Usage

Unconstrained non-commutative polynomial optimization

Taking $f=1+x_1^4+x_2^4+x_3^4+x_1x_2+x_2x_1+x_2x_3+x_3x_2$ as an example, to execute the first level of the NCTSSOS hierarchy, run

using NCTSSOS
using DynamicPolynomials
@ncpolyvar x[1:3]
obj = 1+x[1]^4+x[2]^4+x[3]^4+x[1]*x[2]+x[2]*x[1]+x[2]*x[3]+x[3]*x[2]
opt,data = nctssos_first(obj, x, TS="MD", obj="eigen")

Two vectors will be output. The first vector includes the sizes of PSD blocks and the second vector includes the numbers of PSD blocks with sizes corresponding to the first vector.

To execute higher levels of the NCTSSOS hierarchy, repeatedly run

opt,data = nctssos_higher!(data, TS="MD")

Options:
obj: "eigen" (implements eigenvalue minimization), "trace" (implements trace minimization)
TS (term sparsity): "block" (using the maxmial chordal extension), "MD" (using approximately smallest chordal extention), false (without term sparsity)

Constrained non-commutative polynomial optimization

Taking the objective $f=2-x_1^2+x_1x_2^2x_1-x_2^2$ and constraints $g_1=4-x_1^2-x_2^2\ge0$, $g_2=x_1x_2+x_2x_1-2=0$ as an example, to execute the first level of the NCTSSOS hierarchy, run

@ncpolyvar x[1:2]
obj = 2-x[1]^2+x[1]*x[2]^2*x[1]-x[2]^2
ineq = [4-x[1]^2-x[2]^2]
eq = [x[1]*x[2]+x[2]*x[1]-2]
pop = [obj; ineq; eq]
d = 2 # the relaxation order
opt,data = nctssos_first(pop, x, d, numeq=1, TS="MD", obj="eigen")

To execute higher levels of the NCTSSOS hierarchy, repeatedly run

opt,data = nctssos_higher!(data, TS="MD")

Options:
obj: "eigen" (implements eigenvalue minimization), "trace" (implements trace minimization)
TS: "block" (using the maxmial chordal extension), "MD" (using approximately smallest chordal extention), false (without term sparsity)

To use correlative-term sparsity, run

opt,data = cs_nctssos_first(pop, x, d, TS="block", obj="eigen")

and

opt,data = cs_nctssos_higher!(data, TS="MD")

Trace polynomial optimization

Check out /examples/traceopt.jl.

State polynomial optimization

Check out /examples/stateopt.jl for state polynomial optimization over real numbers and /examples/state_complex.jl for state polynomial optimization over complex numbers.

References

[1] Exploiting Term Sparsity in Noncommutative Polynomial Optimization, 2021.
[2] Sparse polynomial optimization: theory and practice, 2023.
[3] State polynomials: positivity, optimization and nonlinear Bell inequalities, 2023.

Contact

Jie Wang: wangjie212@amss.ac.cn