Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of squares of polynomials) and functional analysis (moments of measures) with semidefinite optimization. Sums of squares are used to certify positive polynomials, combining an old idea of Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite optimization. The dual approach revisits the classical moment problem and leads to algorithmic methods for checking optimality of semidefinite relaxations and extracting global minimizers. We review some selected features of this general methodology, illustrate how it applies to some combinatorial graph problems, and discuss links with other relaxation methods.
Additional Metadata
Keywords Polynomial optimization, real algebraic geometry, semidefinite optimization, combinatorial optimization
THEME Logistics (theme 3)
Publisher Kyung Moon SA Co. Ltd.
Editor S.Y. Jang , Y.R. Kim , D.-W. Lee , I. Yie
Project Semidefinite programming and combinatorial optimization
Conference International Congress of Mathematicians
Citation
Laurent, M. (2014). Optimization over polynomials: Selected topics. In S.Y Jang, Y.R Kim, D.-W Lee, & I Yie (Eds.), Proceedings of International Congress of Mathematicians 2014 (0) (pp. 843–869). Kyung Moon SA Co. Ltd.