Ca{\Sigma}oS: A nonlinear sum-of-squares optimization suite

TL;DR

CaΣoS is a MATLAB toolbox for nonlinear sum-of-squares optimization that leverages symbolic algebra and sparsity to enable fast, repeated problem evaluations and solve various classes of optimization problems efficiently.

Contribution

It introduces a novel MATLAB software suite that integrates symbolic polynomial algebra with sum-of-squares optimization, improving computational efficiency and flexibility.

Findings

Significant reduction in computation time for benchmark problems.

Successful application to region-of-attraction and reachable set estimation.

Open-source availability facilitates adoption and further development.

Abstract

We present Ca$\Sigma$oS, the first MATLAB software specifically designed for nonlinear sum-of-squares optimization. A symbolic polynomial algebra system allows to formulate parametrized sum-of-squares optimization problems and facilitates their fast, repeated evaluations. To that extent, we make use of CasADi's symbolic framework and realize concepts of monomial sparsity, linear operators (including duals), and functions between polynomials. Ca$\Sigma$oS currently provides interfaces to the conic solvers SeDuMi, Mosek, and SCS as well as methods to solve quasiconvex optimization problems (via bisection) and nonconvex optimization problems (via sequential convexification). Numerical examples for benchmark problems including region-of-attraction and reachable set estimation for nonlinear dynamic systems demonstrate significant improvements in computation time compared to existing toolboxes. Ca$\Sigma$oS is available open-source at https://github.com/ifr-acso/casos.