## Invited Session Thu.1.H 2038

#### Thursday, 10:30 - 12:00 h, Room: H 2038

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

### New results in copositive and semidefinite optimization

**Chair: Mirjam Dür**

**Thursday, 10:30 - 10:55 h, Room: H 2038, Talk 1**

**Luuk Gijben**

Scaling relationship between the copositive cone and Parrilo's first level approximation

**Coauthors: Peter J.c. Dickinson, Mirjam Dür, Roland Hildebrand**

**Abstract:**

Several NP-complete problems can be turned into convex problems by formulating them as optimizion problems over the copositive cone. Unfortuntely checking membership in the copositive cone is a co-NP-complete problem in itself. To deal with this problem, several approximation schemes have been developed. One of them is the hierarchy of cones introduced by P. Parillo, membership of which can be checked via semidefinite programming. We know that for matrices of order *n ≤ 4* the zero order parillo cone equals the copositive cone. In this talk we will investigate the relation between the hierarchy and the copositive cone for order *n > 4*. In particular a surprising result is found for the case *n = 5*.

**Thursday, 11:00 - 11:25 h, Room: H 2038, Talk 2**

**Faizan Ahmed**

On connections between copositive programming and semi-infinite programming

**Coauthor: Georg Still**

**Abstract:**

In this presentation we will discuss about the connections between copositive programming(CP) and Linear Semi-infinite Programming(LSIP). We will view copositive programming as a special instance of linear semi-infinite programming. Discretization methods are well known for solving

LSIP (approximately). The connection between CP and LSIP will leads us to interpret certain approximation schemes for CP as a special instance of discretization methods for LSIP. We will provide an overview of error bound for these approximation schemes in terms of the mesh size. Examples will illustrate the structure of the programs.

**Thursday, 11:30 - 11:55 h, Room: H 2038, Talk 3**

**Bolor Jargalsaikhan**

Conic programming: Genericity results and order of minimizers

**Coauthors: Mirjam Dür, Georg Still**

**Abstract:**

We consider generic properties of conic programs like SDPs and copositive programs. In this context, a property is called generic, if it holds for "almost all'' problem instances. Genericity of properties like non-degeneracy and strict complementarity of solutions has been studied. In this talk, we discuss genericity of Slater's condition in conic programs, in particular for SDP and copositive programs. We also discuss the order of the minimizers in SDP and copositive problems, which has important consequences for the convergence rate in discretization methods.