(Page Last updated 1st July 2016)

## Introduction

SOSTOOLS is a free MATLAB toolbox for formulating and solving sums of squares (SOS) optimization programs. SOSTOOLS can be used to specify and solve sum of squares polynomial problems using a very simple, flexible, and intuitive high-level notation. The SOS programs can be solved using SeDuMi, SDPT3 and more recently with CSDP, SDPNAL, SDPNAL+ and SDPA. All these are well-known semidefinite programming solvers, with SOSTOOLS handling internally all the necessary reformulations and data conversion.

What is a "sum of squares optimization program"? Why would I want such a thing?

A sum of squares (SOS) program, in the simplest case, has the form:

minimize: c_1 * u_1 + ... + c_n * u_n

subject to constraints:

P_i(x) := A_i0(x) + A_i1(x) * u_1 + ... + A_in(x) * u_n

are sums of squares of polynomials (for i=1..n).

Here, the A_ij(x) are multivariate polynomials, and the decision variables u_i are scalars. This is a convex optimization problem, since the objective function is linear and the set of feasible u_i is convex.

While this looks quite nice, perhaps you are actually interested in more concrete problems such as:

• Constrained or unconstrained optimization of polynomial functions.
• Mixed continuous-discrete optimization.
• Finding Lyapunov or Bendixson-Dulac functions for nonlinear dynamical systems (with polynomial vector fields).
• Deciding copositivity of a matrix.
• Inequalities in probability theory.
• Distinguishing separable from entangled states in quantum systems.
• Or, more generally, problems that deal with basic semialgebraic sets (sets defined by polynomial equalities and inequalities)..

Although most of these problems are NP-hard, it turns out that useful bounds (or even exact solutions) for all these problems can be found by formulating them in a sum of squares optimization framework.

Hopefully, by now you'll be intrigued, and a bit more inclined to think that this sum of squares stuff may actually be useful to you. If interested, you'll find a much more detailed explanation of the toolbox, some of the applications, and the concepts behind it in the SOSTOOLS user's guide, and the references below.

## Distribution and release information

SOSTOOLS is freely available under the GNU license from either site:

CDS at Caltech: http://www.cds.caltech.edu/sostools or

LIDS at MIT: http://www.mit.edu/~parrilo/sostools.

Control Group at Oxford: http://www.eng.ox.ac.uk/control/sostools.

Older versions:

## System requirements

To install and run SOSTOOLS, you need:

• MATLAB version 6.0 or later.
• MATLAB Symbolic Math Toolbox version 2.1.2 (optional) for SOSTOOLS versions 2.05 and earlier, or the current version of the MATLAB Symbolic Math Toolbox for SOSTOOLS version 3.00 and later.
• An SDP solver, either SeDuMi, SDPT3, CSDP, SDPNAL, SDPNAL+ or SDPA. These solvers and their documentation can be downloaded for free. For information on how to install them, you are referred to their installation instructions.

SOSTOOLS can easily be run on Windows or MAC OSX machines. It utilizes MATLAB sparse matrix representation for good performance and to reduce the amount of memory needed.

Detailed installation instructions are available in the SOSTOOLS user's guide (also included with the standard distribution).

## Authors

The software has been written and is maintained by:

## References

For a detailed explanation of the theory and applications of sums of squares programming, as well as references to related work, please see:

• Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization
California Institute of Technology, Pasadena, CA, May 2000.
Abstract, postscript, gzipped postscript, pdf.
• Semidefinite programming relaxations for semialgebraic problems.
P. A. Parrilo,
Abstract, postscript, gzipped postscript.
• Minimizing polynomial functions
P. A. Parrilo, B. Sturmfels,
http://www.arxiv.org/abs/math.OC/0103170
• For more references please see http://hot.caltech.edu/math.html but also the authors' websites.

## Feedback

For comments, bug reports, encouragement, suggestions, complaints, etc., please send email to: sostools@cds.caltech.edu.

If you use SOSTOOLS for research purposes, we'd be happy to hear about it and mention it in the reference guide. Please drop us a line, to sostools@cds.caltech.edu

Here's the bibtex entry, for citation:

```@manual{sostools,
author = {A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Prajna, P. Seiler and P. A. Parrilo},
title = {{SOSTOOLS}: Sum of squares optimization toolbox for {MATLAB}},
note = {Available from \texttt{http://www.eng.ox.ac.uk/control/sostools}, \texttt{http://www.cds.caltech.edu/sostools} and \texttt{http://www.mit.edu/\~{}parrilo/sostools}},
year = {2013},