## Invited Session Thu.3.H 0110

#### Thursday, 15:15 - 16:45 h, Room: H 0110

**Cluster 16: Nonlinear programming** [...]

### Polynomial optimization and semidefinite programming

**Chair: Jiawang Nie**

**Thursday, 15:15 - 15:40 h, Room: H 0110, Talk 1**

**Lihong Zhi**

Computing real solutions of polynomial systems via low-rank moment matrix completion

**Coauthor: Yue Ma**

**Abstract:**

We propose a new algorithm for computing real

roots of polynomial equations or a subset of real roots in a

given semi-algebraic set described by additional polynomial

inequalities. The algorithm is based on using modified fixed

point continuation method for solving Lasserre's hierarchy

of moment relaxations. We establish convergence properties

for our algorithm. For a large-scale polynomial system with

only few real solutions in a given area, we can extract them

quickly. Moreover, for a polynomial system with an infinite

number of real solutions, our algorithm can also be used

to find some isolated real solutions or real solutions on the

manifolds.

**Thursday, 15:45 - 16:10 h, Room: H 0110, Talk 2**

**Cordian Riener**

Symmetry in polynomial optimization

**Abstract:**

Solving polynomial optimization problems is known to be

a hard task in general. In order to turn the recently emerged

relaxation paradigms into efficient tools for these optimization questions it is necessary to

exploit further structure whenever presented in the problem structure.

In this talk we will focus on the situation of optimization problems

that are given by symmetric polynomials in order to highlight several

approaches to take advantage of symmetry.

The techniques presented in the talk will also give a better

understanding of the cones of symmetric sums of squares and symmetric

non negative forms and the symmetric mean inequalities associated to

these. In particular, we will show that in degree four, symmetric mean

inequalities are characterized by sum of squares decomposition.

**Thursday, 16:15 - 16:40 h, Room: H 0110, Talk 3**

**Markus Schweighofer**

The sums of squares dual of a semidefinite program

**Coauthor: Igor Klep**

**Abstract:**

It is now commonly known that many polynomial optimization problems can be modeled or at least approximated by semidefinite programs (SDPs). In this talk we go the other way around: We consider an SDP from the perspective of polynomial optimization and real algebraic geometry. We will see that, from this perspective, the standard duality of an SDP can be seen as a theorem on representation of positive polynomials. We will prove that natural variants of this theorem hold true, and these can be refined with some effort to yield what we call the sums of squares dual of a semidefinite program. This dual has polynomial size in the primal and, unlike the standard dual, ensures always strong duality. Based on completely different ideas, another dual with this property has already been given by Matt Ramana in 1995. However, we think that our dual is interesting because it shows that ideas from real algebraic geometry might lead very naturally to seriously exploitable concepts in optimization.