Optimization over polynomials: Selected topics

Minimizing a polynomial function over a region defined by polynomial inequalities mod4 els broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic 5 approaches have emerged recently for computing the global minimum, by combining tools from real 6 algebra (sums of squares of polynomials) and functional analysis (moments of measures) with semidef7 inite optimization. Sums of squares are used to certify positive polynomials, combining an old idea of 8 Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite opti9 mization. The dual approach revisits the classical moment problem and leads to algorithmic methods 10 for checking optimality of semidefinite relaxations and extracting global minimizers. We review some 11 selected features of this general methodology, illustrate how it applies to some combinatorial graph 12 problems, and discuss links with other relaxation methods. 13 Mathematics Subject Classification (2010). Primary 44A60, 90C22, 90C27, 90C30; Secondary 14 14P10, 13J30, 15A99. 15