The approach of moments for polynomial equations
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.
|Keywords||polynomial equation, real root, moment, semidefinite programming, real algebraic geometry, sum of squares|
|THEME||Logistics (theme 3)|
|Series||International Series in Operations Research and Management Science|
|Project||Semidefinite programming and combinatorial optimization|
Laurent, M, & Rostalski, P. (2012). The approach of moments for polynomial equations. In Handbook on Semidefinite, Conic and Polynomial Optimization (pp. 25–60). Springer.