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

**Cristian Dobre**

Infinite dimensional semidefinite programming

**Coauthors: Mirjam Dür, Frank Vallentin**

**Abstract:**

In this talk we investigate the infinite dimensional analogue of the primal and dual semidefinite matrix cones. Whereas in the finite case the cone of positive semidefinite matrices is self-dual this is no longer true in infinite dimensions. We introduce the suitable infinite dimensional objects, formulate the pair of primal-dual semidefinite programs and characterize the extremal rays of the dual infinite semidefinite cone. The technique we use employs the theory of reproducing kernels. Applying the same technique to the finite case gives a new proof and interesting new insights on the extremal semidefinite matrices.

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

**Juan C. Vera**

Exploiting symmetry in copositive programs

**Coauthor: Cristian Dobre**

**Abstract:**

We study the solution of copositive programs using a sequence of improving relaxations, as the ones used by Gaddar-Vera-Anjos for polynomial programs. This method consists of using interactively a master-subproblem scheme; the master solves a conic-relaxation of the original problem, while the subproblem improves the cone used in the relaxation using dual information from the master.

We show how symmetry of the original copositive formulation can be used to reduce both the master and subproblem. To reduce the master, techniques to exploit symmetry in semidefinite programming - which are becoming standard nowadays - are used; reducing the subproblem requires exploding the symmetry of Polya-like representations for copositive polynomials in a novel manner.