# Sum of Squares Programming for Julia.

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This packages contains the Sum of Squares reformulation for polynomial optimization. When used in conjunction with MultivariatePolynomial.jl and PolyJuMP.jl, it provides a Sum of Squares Programming extension for JuMP. Enabling the creation of sum of squares variables and constraints.

The following example shows how to find lower bounds for the Goldstein-Price function using this package with MultivariatePolynomial.jl and PolyJuMP.jl.

```
using MultivariatePolynomials
using JuMP
using PolyJuMP
using SumOfSquares
# Create symbolic variables (not JuMP decision variables)
@polyvar x1 x2
# Create a JuMP model with the default SDP solver (you should have at least one installed)
m = Model()
# Create a JuMP decision variable for the lower bound
@variable m γ
# f(x) is the Goldstein-Price function
f1 = x1+x2+1
f2 = 19-14*x1+3*x1^2-14*x2+6*x1*x2+3*x2^2
f3 = 2*x1-3*x2
f4 = 18-32*x1+12*x1^2+48*x2-36*x1*x2+27*x2^2
f = (1+f1^2*f2)*(30+f3^2*f4)
# Constraints f(x) - γ to be sum of squares
@polyconstraint m f >= γ
@objective m Max γ
status = solve(m)
# The lower bound found is 3
println(getobjectivevalue(m))
```