TSSOS

TSSOS is a sparse polynomial optimization tool based on (quasi) block Moment-SOS hierarchies. To use TSSOS in Julia, run

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

Dependencies

Julia
MOSEK 8.1

The Julia version of TSSOS has been tested on WINDOW 10, Julia 1.2.0, and MOSEK 8.1.

Usage

Unconstrained polynomial optimization problems

The unconstrained polynomial optimization problem formulizes as

Inf{f(x): x\in R^n}

where f is a polynomial with variables x1,...,xn and of degree d.

Taking f=x1^4+x2^4-x1*x2 as an example, to execute the first block hierarchy, run

julia> using TSSOS
julia> using DynamicPolynomials
julia> @polyvar x[1:2]
f=x[1]^4+x[2]^4-x[1]*x[2]
julia> opt,sol,data=blockupop_first(f,x)

By default, a monomial basis computed by the Newton polytope method will be used. If we set the key newton=0 in the input,

julia> opt,sol,data=blockupop_first(f,x,newton=0)

then the standard monomial basis will be used.

Two vectors will be outputed. The first vector is the size of blocks and the second vector is the number of blocks with size corresponding to the first vector.

You can use the option method="chordal" which is usually more efficient.

In many cases, the first block hierarchy already obtains the same optimum as the dense Moment-SOS relaxation.

To execute higher block hierarchies, repeatedly run

julia> opt,sol,data=blockupop_higher!(data)

Constrained polynomial optimization problems

The constrained polynomial optimization problem formulizes as

Inf{f(x): x\in K}

where f is a polynomial and K is the basic semi-algebraic set

K={x\in R^n: g_j(x)>=0, j=1,...,m, g_k(x)=0, k=m+1,...,m+l},

for some polynomials g_j, j=1,...,m+l.

Taking f=x1^4+x2^4-x1*x2 and g_1=2-x1^2-2*x2^2, g_2=x1^2-x2^2-1 as an example, to execute the first block hierarchy, run

julia> @polyvar x[1:2]
f=x[1]^4+x[2]^4-x[1]*x[2]
g_1=2-x[1]^2-2*x[2]^2
g_2=x[1]^2-x[2]^2-1
pop=[f,g_1,g_2]
d=2 # the order of Lasserre's hierarchy
julia> opt,sol,data=blockcpop_first(pop,x,d,numeq=1)

You can also use the option method="chordal" which is usually more efficient.

In many cases, the first block hierarchy already obtains the same optimum as the dense Moment-SOS relaxation.

To execute higher block hierarchies, repeatedly run

julia> opt,sol,data=blockcpop_higher!(data,numeq=1)

Reference

For more details about TSSOS, please refer to TSSOS: A Moment-SOS hierarchy that exploits term sparsity and Chordal-TSSOS: a moment-SOS hierarchy that exploits term sparsity with chordal extension. If there are any problems, you can contact Jie Wang: jwang@laas.fr.

Jarmill / TSSOS