Sums of squares

Approximation of positive polynomials by sums of squares has important
applications to polynomial optimisation. In this talk, I will survey
the main recent results achieved on that topic:
I will consider positive (respectively, non-negative) polynomials on
compact (respectively, unbounded) semi-algebraic sets. I will discuss
representations in the associated preorderings (respectively, linear
representations in the associated quadratic module). The
representation often depends on

A polynomial optimization problem (POP) is a problem of minimizing a polynomial
objective function subject to polynomial equalities and inequalities. It is getting
popular to apply the sum of squares (SOS) relaxation to compute global minimum
solutions of a POP. The SOS relaxation problem is reduced to a semidefinite
programming problem (SDP), which we can solve by applying the primal-dual interior-point
method. In this process, exploiting sparsity is essential in solving a large-scale POP. We present

Sum of squares (SOS) programs are a particular class of convex optimization
problems, that combine in a very appealing way notions from algebraic and numeric computation (in particular, semidefinite programming). They are based on the sum of squares decomposition for multivariate polynomials, and have found many interesting applications, mainly through semidefinite relaxations of polynomial optimization problems.

In this talk we will quickly review the basic SOS framework, focusing on the