# Sums of squares

Saturday, January 20, 2007 - 11:00am - 11:50am

Salma Kuhlmann (University of Saskatchewan)

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

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

Saturday, January 20, 2007 - 9:30am - 10:20am

Masakazu Kojima (Tokyo Institute of Technology)

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

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

Thursday, October 26, 2006 - 1:40pm - 2:30pm

Pablo Parrilo (Massachusetts Institute of Technology)

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

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