# PolyJuMP.jl

PolyJuMP.jl is a JuMP extension for formulating and solving polynomial optimization problems. This extension includes the following:

• Polynomial functions on JuMP decisions variables. These can be solved with the PolyJuMP.QCQP.Optimizer or PolyJuMP.KKT.Optimizer.
• Constraints that a polynomial is nonnegative where the coefficients of the polynomials depend on JuMP decision variables. These nonnegativity constraints can be reformulated using sufficient conditions using PolyJuMP.RelativeEntropy submodule or SumOfSquares.jl.

PolyJuMP.jl is licensed under the MIT license.

## Installation

Install PolyJuMP using Pkg.add:

import Pkg
Pkg.add("PolyJuMP")

## Use with JuMP

### Polynomial nonnegativity constraints

PolyJuMP allows encoding a constraint that a polynomial should be nonnegative for all values of some symbolic variables defined with DynamicPolynomials.@polyvar or TypedPolynomials.@polyvar as follows. For instance, the following constrains the JuMP decision variable a to be such that a * x * y^2 + y^3 - a * x is nonnegative for all real values of x and y:

using DynamicPolynomials
@polyvar x y
using JuMP
model = Model()
@variable(model, a)
@constraint(model, a * x * y^2 + y^3 >= a * x)

Determining the nonnegativity of a multivariate polynomial is however NP-hard so sufficient conditions are used instead. You need to specify which sufficient condition is used explicitly. To use Sum-of-Arithmetic-Geometric-Exponentials (SAGE), use

import PolyJuMP
PolyJuMP.setpolymodule!(model, PolyJuMP.SAGE)

To use Sum-of-Squares (SOS), use

import SumOfSquares
PolyJuMP.setpolymodule!(model, SumOfSquares)

or replace model = Model() by model = SOSModel().

Alternatively, the nonnegativity constraint can be explicit:

@constraint(model, a * x * y^2 + y^3 - a * x in PolyJuMP.SAGE.Polynomials())
@constraint(model, a * x * y^2 + y^3 - a * x in SumOfSquares.SOSCone())

This allows mixing SAGE and SOS constraints in the same model.

### Polynomial optimization

PolyJuMP also allows solving polynomial optimization problems using the QCQP and KKT solvers. Polynomial optimization problems do not involve any symbolic variables from DynamicPolynomials or TypedPolynomials, instead all variables are JuMP decision variables.

The QCQP solver is parametrized by a nonconvex QCQP inner solver. It reformulates the polynomial optimization problem into a nonconvex QCQP and relies on the inner solver to solve it. For instance, to use the QCQP solver with JuMP with Gurobi.Optimizer as inner solver, use:

using JuMP, PolyJuMP, Gurobi
model = Model(() -> PolyJuMP.QCQP.Optimizer(Gurobi.Optimizer))

The KKT solver is parametrized by an inner solver of algebraic systems of equations implementing the SemialgebraicSets interface. It reformulates the polynomial optimization problem into a system of polynomial equations and relies on the inner solver to solve it. For instance, to use the QCQP solver with JuMP with HomotopyContinuation.SemialgebraicSetsHCSolver as inner solver, use:

using JuMP, PolyJuMP, HomotopyContinuation
model = Model(optimizer_with_attributes(
PolyJuMP.KKT.Optimizer,
"solver" => HomotopyContinuation.SemialgebraicSetsHCSolver(),
))

## Documentation

Documentation for PolyJuMP.jl is included in the documentation for SumOfSquares.jl.