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(Page Last updated 29th September 2021)
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, CSDP, SDPNAL, SDPNAL+, CDCS, SDPA and MOSEK. 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:
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.
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 or
CSCL at ASU: [http://control.asu.edu/sostools] or
Control Group at Oxford: http://www.eng.ox.ac.uk/control/sostools.
Older versions:
To install and run SOSTOOLS, you need:
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).
The software has been written and is maintained by:
For a detailed explanation of the theory and applications of sums of squares programming, as well as references to related work, please see:
For comments, bug reports, encouragement, suggestions, complaints, etc., please send email to: sostools@cds.caltech.edu. You can also do this on GitHub.
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, P. A. Parrilo, M. M. Peet and D. Jagt},
title = {{SOSTOOLS}: Sum of squares optimization toolbox for {MATLAB}},
note = {Available from \texttt{https://github.com/oxfordcontrol/SOSTOOLS}},
year = {2021},
address = {\texttt{http://arxiv.org/abs/1310.4716}},
}