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Welcome to the CaΣoS wiki! Here you'll find information about polynomial optimization, sum-of-squares and related cones, and general usage of the software.

About CaΣoS

CaΣoS aims to bring nonlinear optimization into sum-of-squares programming as well as facilitate the integration of pre-computed polynomials into optimal control solutions. CaΣoS is developed by researchers at the Institute of Flight Mechanics and Control of the University of Stuttgart and distributed open-source under the GPL-3.0 license.

Getting Started

The following requirements need to be met in order to use all functionalities of CaΣoS:

1. Download CasADi v3.6.x and add it to your Matlab path.

2. Download and install at least one solver for semidefinite (conic) programs, and add the solver(s) to the Matlab Path.

   • Currently supported are Mosek (v10.1), SCS (v3.2.4), and SeDuMi (v1.3).
   • Note: Mosek requires a separate license. Academic Licenses are available.

3. Add the CaΣoS root folder (the one that contains the directory +casos) to your Matlab path.

Important

CaΣoS requires CasADi version 3.6.0 or newer.

You can check if CaΣoS is ready to go by running the install script in utils/install.m.

Examples

Nonnegativity of scalar polynomial

With sossol, CaΣoS can verify nonnegativity of a polynomial. Consider for example

$$ f(x) = x^4 + 10x. $$

For which $\gamma>0$ is $f(x)+\gamma$ nonnegative? This can be checked using polynomial optimization:

% Polynomial optimization.

% indeterminate variable
x = casos.Indeterminates('x');
% some polynomial
f = x^4 + 10*x;
% scalar decision variable
g = casos.PS.sym('g');

% define SOS problem:
%   min g s.t. (f + g) is SOS
sos = struct('x',g,'f',g,'g',f+g);
% constraint is scalar SOS cone
opts = struct('Kc',struct('sos',1));

% solve by relaxation to SDP
S = casos.sossol('S','sedumi',sos,opts);
% evaluate
sol = S();

fprintf('Minimum is %g.\n', full(sol.f))

Quasiconvex sum-of-squares programming

TODO: add example (ROA)