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# Gabor Pataki, Geometric Methods in Representation Theory Seminar

## March 31, 2017 @ 4:40 pm - 5:40 pm

**Title:** Combinatorial characterizations in semidefinite programming duality

**Abstract:** I will discuss optimization problems over affine slices of positive definite symmetric matrices. These problems, called semidefinite programs (SDPs), have numerous applications. Several basic properties of SDP’s are rooted in the fact that the linear image of the cone of symmetric positive semidefinite matrices may not be closed. Examples of these properties include the non-triviality of the infeasibility problem (when the set of positive semidefinite matrices has an empty intersection with an affine subspace) and pathological properties of dual SDP’s.

In this talk I survey recent, somewhat surprising combinatorial type characterizations for several fundamental problems in SDP duality. The main tool is very simple: we use elementary row operations – inherited from Gaussian elimination – to bring an SDP to a format in which properties – such as infeasibility – are trivial to recognize.

Part of this work is joint with Minghui Liu.