## Invited Session Fri.1.H 2036

#### Friday, 10:30 - 12:00 h, Room: H 2036

**Cluster 4: Conic programming** [...]

### Algebraic geometry and conic programming I

**Chair: Lek-Heng Lim and Cordian Riener**

**Friday, 10:30 - 10:55 h, Room: H 2036, Talk 1**

**Tim Netzer**

Describing the feasible sets of semidefinite programming

**Coauthors: Daniel Plaumann, Andreas B. Thom**

**Abstract:**

The feasible sets of semidefinite programming, sometimes called

spectrahedra, are affine slices of the cone positive semidefinite matrices.

For a given convex set it might however be quite complicated to decide

whether it is such a slice or not. Alternative characterizations of

spectrahedra are thus highly desirable. This interesting problem turns out

to be related to algebra, algebraic geometry and non-commutative geometry. I

will explain some of the recent developments in the area.

**Friday, 11:00 - 11:25 h, Room: H 2036, Talk 2**

**Sabine Burgdorf**

Lasserre relaxation for trace-optimization of NC polynomials

**Coauthors: Kristijan Cafuta, Igor Klep, Janez Povh**

**Abstract:**

Given a polynomial *f* in noncommuting (NC) variables, what is the smallest trace *f(A)* can attain for a tuple *A * of symmetric matrices? This is a nontrivial extension of minimizing a polynomial in commuting variables or of eigenvalue optimization of an NC polynomial - two topics with various applications in several fields. We propose a sum of Hermitian squares relaxation for trace-minimization of an NC polynomial and its implementation as an SDP. We will discuss the current state of knowledge about this relaxation and compare it to the behavior of Lasserre relaxations for classical polynomial minimization and for eigenvalue optimization respectively.

**Friday, 11:30 - 11:55 h, Room: H 2036, Talk 3**

**Raman Sanyal**

Deciding polyhedrality of spectrahedra

**Coauthors: Avinash Bhardwaj, Philipp Rostalski**

**Abstract:**

Spectrahedra, the feasible regions of semidefinite programs, form a rich class of convex bodies that properly contains that of polyhedra. It is a theoretical interesting and practically relevant question to decide when a spectrahedron is a polyhedron. In this talk I will discuss how this can be done algorithmically by making use of the geometry as well as the algebraic structure of spectrahedra.