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Symmetry reduction (#175)
* Symmetry reduction * Switch to Github CI
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‎.github/workflows/CompatHelper.yml

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name: CompatHelper
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on:
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schedule:
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- cron: '00 00 * * *'
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workflow_dispatch:
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jobs:
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CompatHelper:
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runs-on: ${{ matrix.os }}
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strategy:
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matrix:
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julia-version: [1.2.0]
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julia-arch: [x86]
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os: [ubuntu-latest]
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steps:
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- name: Pkg.add("CompatHelper")
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run: julia -e 'using Pkg; Pkg.add("CompatHelper")'
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- name: CompatHelper.main()
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env:
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GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
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run: julia -e 'using CompatHelper; CompatHelper.main()'

‎.github/workflows/TagBot.yml

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name: TagBot
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on:
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schedule:
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- cron: 0 * * * *
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issue_comment:
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types:
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- created
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workflow_dispatch:
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jobs:
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TagBot:
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if: github.event_name == 'workflow_dispatch' || github.actor == 'JuliaTagBot'
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runs-on: ubuntu-latest
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steps:
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- uses: JuliaRegistries/TagBot@v1
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with:
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token: ${{ secrets.GITHUB_TOKEN }}
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ssh: ${{ secrets.DOCUMENTER_KEY }}

‎.github/workflows/ci.yml

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name: CI
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on:
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push:
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branches: [master]
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pull_request:
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types: [opened, synchronize, reopened]
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jobs:
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test:
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name: Julia ${{ matrix.version }} - ${{ matrix.os }} - ${{ matrix.arch }} - ${{ github.event_name }}
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runs-on: ${{ matrix.os }}
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strategy:
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fail-fast: false
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matrix:
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include:
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- version: '1.0'
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os: ubuntu-latest
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arch: x64
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- version: '1'
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os: ubuntu-latest
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arch: x64
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- version: '1'
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os: ubuntu-latest
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arch: x86
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- version: 'nightly'
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os: ubuntu-latest
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arch: x64
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steps:
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- uses: actions/checkout@v2
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- uses: julia-actions/setup-julia@v1
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with:
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version: ${{ matrix.version }}
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arch: ${{ matrix.arch }}
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- uses: actions/cache@v1
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env:
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cache-name: cache-artifacts
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with:
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path: ~/.julia/artifacts
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key: ${{ runner.os }}-test-${{ env.cache-name }}-${{ hashFiles('**/Project.toml') }}
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restore-keys: |
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${{ runner.os }}-test-${{ env.cache-name }}-
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${{ runner.os }}-test-
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${{ runner.os }}-
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- uses: julia-actions/julia-buildpkg@v1
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- uses: julia-actions/julia-runtest@v1
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with:
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depwarn: error
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- uses: julia-actions/julia-processcoverage@v1
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- uses: codecov/codecov-action@v1
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with:
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file: lcov.info

‎.github/workflows/documentation.yml

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name: Documentation
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on:
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push:
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branches: [master]
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tags: '*'
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pull_request:
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types: [opened, synchronize, reopened]
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jobs:
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build:
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runs-on: ubuntu-latest
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steps:
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- uses: actions/checkout@v2
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- uses: julia-actions/setup-julia@latest
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with:
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# Build documentation on the latest Julia 1.x
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version: '1'
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- name: Install dependencies
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run: julia --project=docs/ -e 'using Pkg; Pkg.instantiate();
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Pkg.add("DynamicPolynomials");
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Pkg.add(PackageSpec(url="https://github.com/blegat/Documenter.jl", rev="latexout"));
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Pkg.develop(PackageSpec(path=pwd()))'
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- name: Build and deploy
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env:
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GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} # For authentication with GitHub Actions token
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DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} # For authentication with SSH deploy key
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run: julia --project=docs/ docs/make.jl

‎.github/workflows/examples.yml

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name: Examples
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on:
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push:
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branches: [master]
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pull_request:
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types: [opened, synchronize, reopened]
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jobs:
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build:
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runs-on: ubuntu-latest
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steps:
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- uses: actions/checkout@v2
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- uses: julia-actions/setup-julia@latest
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with:
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version: '1'
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- name: Install dependencies
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run: julia --project=examples -e 'using Pkg; Pkg.instantiate(); Pkg.develop(PackageSpec(path=pwd()))'
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- name: Run examples
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run: julia --project=examples --color=yes examples/run_examples.jl

‎.travis.yml

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‎Project.toml

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[extras]
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CSDP = "0a46da34-8e4b-519e-b418-48813639ff34"
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Cyclotomics = "da8f5974-afbb-4dc8-91d8-516d5257c83b"
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DynamicPolynomials = "7c1d4256-1411-5781-91ec-d7bc3513ac07"
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LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
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PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
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Pkg = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
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PolyJuMP = "ddf597a6-d67e-5340-b84c-e37d84115374"
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SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"

‎appveyor.yml

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This file was deleted.

‎docs/Project.toml

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[deps]
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CSDP = "0a46da34-8e4b-519e-b418-48813639ff34"
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Cyclotomics = "da8f5974-afbb-4dc8-91d8-516d5257c83b"
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DynamicPolynomials = "7c1d4256-1411-5781-91ec-d7bc3513ac07"
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Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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Literate = "98b081ad-f1c9-55d3-8b20-4c87d4299306"
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MultivariateBases = "be282fd4-ad43-11e9-1d11-8bd9d7e43378"
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MultivariatePolynomials = "102ac46a-7ee4-5c85-9060-abc95bfdeaa3"
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MutableArithmetics = "d8a4904e-b15c-11e9-3269-09a3773c0cb0"
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PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
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SumOfSquares = "4b9e565b-77fc-50a5-a571-1244f986bda1"

‎docs/make.jl

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"Sum-of-Squares Matrices.jl",
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"Noncommutative variables.jl",
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"Sums of Hermitian squares.jl",
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"Symmetry reduction.jl",
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]
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for example in EXAMPLES
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"Sum-of-Squares Matrices" => "generated/Sum-of-Squares Matrices.md",
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"Noncommutative variables" => "generated/Noncommutative variables.md",
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"Sums of Hermitian squares" => "generated/Sums of Hermitian squares.md",
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"Symmetry reduction" => "generated/Symmetry reduction.md",
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]
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],
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# The following ensures that we only include the docstrings from

‎examples/Project.toml

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[deps]
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CSDP = "0a46da34-8e4b-519e-b418-48813639ff34"
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Cyclotomics = "da8f5974-afbb-4dc8-91d8-516d5257c83b"
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DynamicPolynomials = "7c1d4256-1411-5781-91ec-d7bc3513ac07"
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IJulia = "7073ff75-c697-5162-941a-fcdaad2a7d2a"
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JuMP = "4076af6c-e467-56ae-b986-b466b2749572"
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MultivariateBases = "be282fd4-ad43-11e9-1d11-8bd9d7e43378"
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MultivariatePolynomials = "102ac46a-7ee4-5c85-9060-abc95bfdeaa3"
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MutableArithmetics = "d8a4904e-b15c-11e9-3269-09a3773c0cb0"
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PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
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SumOfSquares = "4b9e565b-77fc-50a5-a571-1244f986bda1"

‎examples/Symmetry reduction.jl

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# # Lyapunov Function Search
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#md # [![](https://mybinder.org/badge_logo.svg)](@__BINDER_ROOT_URL__/generated/Lyapunov Function Search.ipynb)
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#md # [![](https://img.shields.io/badge/show-nbviewer-579ACA.svg)](@__NBVIEWER_ROOT_URL__/generated/Lyapunov Function Search.ipynb)
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# **Adapted from**: SOSTOOLS' SOSDEMO2 (See Section 4.2 of [SOSTOOLS User's Manual](http://sysos.eng.ox.ac.uk/sostools/sostools.pdf))
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using Pkg
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pkg"add https://github.com/kalmarek/SymbolicWedderburn.jl#tw/ex_sos"
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import MutableArithmetics
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const MA = MutableArithmetics
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using MultivariatePolynomials
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const MP = MultivariatePolynomials
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using MultivariateBases
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const MB = MultivariateBases
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using Test #src
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using DynamicPolynomials
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@polyvar x[1:4]
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# We would like to find the minimum value of the polynomial
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poly = sum(x) + sum(x.^2)
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# As we can decouple the problem for each `x[i]` for which `x[i] + x[i]^2` has
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# minimum value 0.25, we would expect to get `-1` as answer.
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# Can this decoupling be exploited by SumOfSquares as well ?
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# For this, we need to use a certificate that can exploit the permutation symmetry of the polynomial.
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# This is still a work in progress in SumOfSquares, so we need to define things here:
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using SymbolicWedderburn
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using PermutationGroups
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using Cyclotomics
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using SumOfSquares
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function SymbolicWedderburn.OnPoints(basis::Union{MonomialVector, AbstractVector{<:Monomial}})
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basis_exps = Vector{Vector{Int}}(undef, length(basis))
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basis_dict = Dict{Vector{Int}, Int}()
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sizehint!(basis_dict, length(basis))
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for (i, b) in enumerate(basis)
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e = MP.exponents(b) # so that we allocate exponents only once
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basis_exps[i] = e
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basis_dict[e] = i
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end
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return SymbolicWedderburn.OnPoints(basis_exps, basis_dict)
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end
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function symmetry_adapted_basis(G, mvec)
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chars_vars = SymbolicWedderburn.characters_dixon(G)
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chars = chars_vars
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basis = mvec
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@assert all(χ.inv_of == first(chars).inv_of for χ in chars)
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induced_action = SymbolicWedderburn.OnPoints(basis)
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ccG_large = induced_action.(conjugacy_classes(first(chars)))
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let ccls = ccG_large, large_gens = induced_action.(gens(G)) # double check:
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G_large = PermGroup(large_gens)
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ccG_large = conjugacy_classes(G_large)
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@assert all(Set.(collect.(ccG_large)) .== Set.(collect.(ccls)))
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end
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chars_mvec = [SymbolicWedderburn.Character(values(χ), χ.inv_of, ccG_large) for χ in chars]
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vr_chars = SymbolicWedderburn.real_vchars(chars_mvec)
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U = filter!(x->all(!iszero, x), [SymbolicWedderburn.matrix_projection(χ) for χ in vr_chars])
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R = map(U) do c_u
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u = last(c_u)
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if all(isreal, u)
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image_coeffs, pivots = SymbolicWedderburn.row_echelon_form(float.(u))
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dim = length(pivots)
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image_coeffs[1:dim, :]
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else
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throw("Not Implemented")
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end
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end
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return map(R) do Ri
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FixedPolynomialBasis(Ri * mvec)
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end
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end
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function MP.polynomialtype(::Type{<:MB.AbstractPolynomialVectorBasis{PT}}, T::Type) where PT
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C = MP.coefficienttype(PT)
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U = MA.promote_operation(*, C, T)
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V = MA.promote_operation(+, U, U)
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return MP.polynomialtype(PT, V)
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end
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struct SymmetricIdeal{CT, GT} <: Certificate.AbstractIdealCertificate
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cone::CT
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group::GT
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end
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SumOfSquares.matrix_cone_type(::Type{<:SymmetricIdeal{CT}}) where {CT} = SumOfSquares.matrix_cone_type(CT)
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Certificate.get(::Type{<:SymmetricIdeal}, ::SumOfSquares.Certificate.GramBasisType) = Vector{MB.FixedPolynomialBasis}
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Certificate.zero_basis_type(::Type{<:SymmetricIdeal}) = MB.MonomialBasis
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Certificate.zero_basis(::SymmetricIdeal) = MB.MonomialBasis
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Certificate.get(::SymmetricIdeal, ::Certificate.ReducedPolynomial, poly, domain) = poly
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function Certificate.get(cert::SymmetricIdeal, ::Certificate.GramBasis, poly)
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basis = Certificate.maxdegree_gram_basis(MB.MonomialBasis, MP.variables(poly), MP.maxdegree(poly))
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return symmetry_adapted_basis(cert.group, basis.monomials)
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end
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G = PermGroup([perm"(1,2,3,4)"])
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# We can use this certificate as follows:
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import CSDP
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solver = CSDP.Optimizer
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model = Model(solver)
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@variable(model, t)
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@objective(model, Max, t)
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con_ref = @constraint(model, sum(x) + sum(x.^2) - t in SOSCone(), ideal_certificate = SymmetricIdeal(SOSCone(), G))
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optimize!(model)
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@test value(t) ≈ -1 #src
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value(t)
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# We indeed find `-1`, let's verify that symmetry was exploited:
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@test length(gram_matrix(con_ref).sub_gram_matrices) == 3 #src
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gram_matrix(con_ref)

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