In this chapter we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and the dual theory of moments. A crucial ingredient is a semidefinite characterization of the real radical ideal, consisting of all polynomials with the same real zero set as the system of poly- nomials to be solved. Combining this characterization with ideas from commutative algebra, (numerical) linear algebra and semidefinite optimization yields a new class of real algebraic algorithms. This chapter sheds some light on the underlying theory and the link to polynomial optimization.
Additional Metadata
Keywords polynomial equation, real root, moment, semidefinite programming, real algebraic geometry, sum of squares
THEME Logistics (theme 3)
Publisher Springer
Series International Series in Operations Research and Management Science
Project Semidefinite programming and combinatorial optimization
Citation
Laurent, M, & Rostalski, P. (2012). The approach of moments for polynomial equations. In Handbook on Semidefinite, Conic and Polynomial Optimization (pp. 25–60). Springer.