Amir Ali Ahmadi

Laboratory for Information and Decision Systems

Massachusetts Institute of Technology

**The Interplay of Convexity and Algorithmic Algebra in Optimization and Systems Analysis**

Exciting recent developments at the interface of computational algebra and convex optimization have led to
major
algorithmic advances in a broad range of problems in optimization and systems theory. I will start this talk
by giving an overview of these techniques and presenting applications in continuous and combinatorial
optimization, statistics and control theory. I will then focus on two recent results on computational and
algebraic aspects of *convexity* in optimization: (i) I will show that deciding convexity of polynomials
of degree as low as four is strongly NP-hard. This solves a problem that appeared as one of seven open
problems in complexity theory for numerical optimization in 1992. (ii) I will introduce an algebraic,
semidefinite programming (SDP) based sufficient condition for convexity known as
*sum-of-squares-convexity* and present a complete charactization of the cases where it is equivalent to
convexity. This characterization draws an interesting parallel to a seminal 1888 result of Hilbert in real
algebraic geometry.

In the final part of the talk, I will move on to a problem with numerous applications in engineering and
economics: understanding the asymptotic behavior of linear dynamical systems under uncertainty. I will tackle
this problem with a novel class of SDP-based algorithms (with provable guarantees) that are based on new
connections between ideas from control theory and the theory of finite automata.

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