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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Index · SumOfSquares</title><meta name="title" content="Index · SumOfSquares"/><meta property="og:title" content="Index · SumOfSquares"/><meta property="twitter:title" content="Index · SumOfSquares"/><meta name="description" content="Documentation for SumOfSquares."/><meta property="og:description" content="Documentation for SumOfSquares."/><meta property="twitter:description" content="Documentation for SumOfSquares."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link 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src="assets/themeswap.js"></script><link href="assets/citations.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>SumOfSquares</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Index</a><ul class="internal"><li><a class="tocitem" href="#Contents"><span>Contents</span></a></li></ul></li><li><a class="tocitem" href="sumofsquares/">Sum-of-Squares Programming</a></li><li><a class="tocitem" href="variables/">Variables</a></li><li><a class="tocitem" href="constraints/">Constraints</a></li><li><span class="tocitem">Tutorials</span><ul><li><input class="collapse-toggle" id="menuitem-5-1" type="checkbox"/><label class="tocitem" for="menuitem-5-1"><span class="docs-label">Getting started</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="generated/Getting started/circle/">Nonnegative over a variety</a></li><li><a class="tocitem" href="generated/Getting started/dualization/">On the importance of Dualization</a></li><li><a class="tocitem" href="generated/Getting started/getting_started/">Getting started</a></li><li><a class="tocitem" href="generated/Getting started/motzkin/">Motzkin</a></li><li><a class="tocitem" href="generated/Getting started/sos_decomposition/">A trivial SOS decomposition example</a></li><li><a class="tocitem" href="generated/Getting started/sum-of-squares_matrices/">Sum-of-Squares matrices</a></li><li><a class="tocitem" href="generated/Getting started/univariate/">Minimization of a univariate polynomial</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5-2" type="checkbox"/><label class="tocitem" for="menuitem-5-2"><span class="docs-label">Polynomial Optimization</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="generated/Polynomial Optimization/bilinear/">Bilinear terms</a></li><li><a class="tocitem" href="generated/Polynomial Optimization/bound_on_global_extremum/">Bound on Global Extremum</a></li><li><a class="tocitem" href="generated/Polynomial Optimization/ellipsoid/">Exterior of ellipsoid</a></li><li><a class="tocitem" href="generated/Polynomial Optimization/goldstein_price/">Goldstein-price function</a></li><li><a class="tocitem" href="generated/Polynomial Optimization/min_univariate/">Maximizing as minimum</a></li><li><a class="tocitem" href="generated/Polynomial Optimization/polynomial_optimization/">Polynomial Optimization</a></li><li><a class="tocitem" href="generated/Polynomial Optimization/qcqp/">Nonconvex quadratically constrained quadratic programs</a></li><li><a class="tocitem" href="generated/Polynomial Optimization/qp/">Nonconvex QP</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5-3" type="checkbox"/><label class="tocitem" for="menuitem-5-3"><span class="docs-label">Systems and Control</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="generated/Systems and Control/barrier_certificate/">Barrier certificates for collision avoidance</a></li><li><a class="tocitem" href="generated/Systems and Control/julia_set/">Outer approximation of Julia set</a></li><li><a class="tocitem" href="generated/Systems and Control/lyapunov_function_search/">Lyapunov Function Search</a></li><li><a class="tocitem" href="generated/Systems and Control/stabilization_of_nonlinear_systems/">Stabilization of nonlinear systems</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5-4" type="checkbox"/><label class="tocitem" for="menuitem-5-4"><span class="docs-label">Other Applications</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="generated/Other Applications/bounds_in_probability/">Bounds in Probability</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5-5" type="checkbox"/><label class="tocitem" for="menuitem-5-5"><span class="docs-label">Noncommutative and Hermitian</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="generated/Noncommutative and Hermitian/noncommutative_variables/">Noncommutative variables</a></li><li><a class="tocitem" href="generated/Noncommutative and Hermitian/sums_of_hermitian_squares/">Sums of Hermitian squares</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5-6" type="checkbox"/><label class="tocitem" for="menuitem-5-6"><span class="docs-label">Sparsity</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="generated/Sparsity/sign_symmetry/">Sign symmetry</a></li><li><a class="tocitem" href="generated/Sparsity/term_sparsity/">Term sparsity</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5-7" type="checkbox"/><label class="tocitem" for="menuitem-5-7"><span class="docs-label">Symmetry</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="generated/Symmetry/cyclic/">Cyclic symmetry for Sums of Hermitian Squares</a></li><li><a class="tocitem" href="generated/Symmetry/dihedral/">Dihedral symmetry of the Robinson form</a></li><li><a class="tocitem" href="generated/Symmetry/even_reduction/">Even reduction</a></li><li><a class="tocitem" href="generated/Symmetry/permutation_symmetry/">Symmetry reduction</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5-8" type="checkbox"/><label class="tocitem" for="menuitem-5-8"><span class="docs-label">Extension</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="generated/Extension/certificate/">Certificate</a></li><li><a class="tocitem" href="generated/Extension/hypercube/">Hypercube</a></li><li><a class="tocitem" href="generated/Extension/typed/">Multivariate polynomials implementations</a></li><li><a class="tocitem" href="generated/Extension/univariate_solver/">Univariate Solver</a></li></ul></li></ul></li><li><a class="tocitem" href="bibliography/">Bibliography</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Index</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Index</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/SumOfSquares.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/SumOfSquares.jl/blob/master/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SumOfSquares-–-Sum-of-Squares-Programming-for-Julia"><a class="docs-heading-anchor" href="#SumOfSquares-–-Sum-of-Squares-Programming-for-Julia">SumOfSquares –- Sum of Squares Programming for Julia</a><a id="SumOfSquares-–-Sum-of-Squares-Programming-for-Julia-1"></a><a class="docs-heading-anchor-permalink" href="#SumOfSquares-–-Sum-of-Squares-Programming-for-Julia" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/SumOfSquares.jl">SumOfSquares</a> implements Sum of Squares reformulation for <a href="https://github.com/jump-dev/PolyJuMP.jl">PolyJuMP</a>, enabling the creation of sum of squares variables and constraints in <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>.</p><p>The polynomial can be represented by any library implementing the <a href="https://github.com/JuliaAlgebra/MultivariatePolynomials.jl">MultivariatePolynomial.jl</a> interface. That is, you can currently choose between <a href="https://github.com/JuliaAlgebra/MultivariatePolynomials.jl">DynamicPolynomials</a> and <a href="https://github.com/JuliaAlgebra/MultivariatePolynomials.jl">TypedPolynomials</a>. As a rule of thumb, if you know at compile time (or at the time you are writing your code), the number of variable and that this number is small, use TypedPolynomials, otherwise, use DynamicPolynomials.</p><h2 id="Contents"><a class="docs-heading-anchor" href="#Contents">Contents</a><a id="Contents-1"></a><a class="docs-heading-anchor-permalink" href="#Contents" title="Permalink"></a></h2><ul><li><a href="sumofsquares/#Sum-of-Squares-Programming">Sum-of-Squares Programming</a></li><li class="no-marker"><ul><li><a href="sumofsquares/#Quadratic-forms-and-Semidefinite-programming">Quadratic forms and Semidefinite programming</a></li><li><a href="sumofsquares/#Polynomial-nonnegativity-and-Semidefinite-programming">Polynomial nonnegativity and Semidefinite programming</a></li><li><a href="sumofsquares/#When-is-nonnegativity-equivalent-to-sum-of-squares-?">When is nonnegativity equivalent to sum of squares ?</a></li></ul></li><li><a href="variables/#Polynomial-and-Sum-of-Squares-variables">Polynomial and Sum-of-Squares variables</a></li><li class="no-marker"><ul><li><a href="variables/#Polynomial-variables">Polynomial variables</a></li><li><a href="variables/#Nonnegative-polynomial-variables">Nonnegative polynomial variables</a></li><li><a href="variables/#Choosing-a-polynomial-basis">Choosing a polynomial basis</a></li><li><a href="variables/#Reference">Reference</a></li></ul></li><li><a href="constraints/#Constraints">Constraints</a></li><li class="no-marker"><ul><li><a href="constraints/#Equality-constraints-between-polynomials">Equality constraints between polynomials</a></li><li><a href="constraints/#Inequality-constraints-between-polynomials">Inequality constraints between polynomials</a></li><li><a href="constraints/#Changing-the-polynomial-basis">Changing the polynomial basis</a></li><li><a href="constraints/#Polynomial-nonnegativity-on-a-subset-of-the-space">Polynomial nonnegativity on a subset of the space</a></li><li><a href="constraints/#Dual-of-polynomial-constraints">Dual of polynomial constraints</a></li><li><a href="constraints/#API-Reference">API Reference</a></li></ul></li></ul></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="sumofsquares/">Sum-of-Squares Programming »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Thursday 21 December 2023 13:46">Thursday 21 December 2023</span>. 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