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Nonnegative polynomials: from optimization to control and learning

Amir Ali Ahmadi, ORFE, Princeton University
Friday, May 17, 2019 - 2:30pm to 3:30pm
ECE 037

The problem of recognizing nonnegativity of a multivariate polynomial
has a celebrated history, tracing back to Hilbert’s 17th problem. In
recent years, there has been much renewed interest in the topic
because of a multitude of applications in applied and computational
mathematics and the observation that one can optimize over an
interesting subset of nonnegative polynomials using “sum of squares

In this talk, we give a brief overview of some of our recent
contributions to this area. In part (i), we propose more scalable
alternatives to sum of squares optimization and show how they impact
verification problems in control and robotics. Our algorithms do not
rely on semidefinite programming, but instead use linear programming,
or second-order cone programming, or are altogether free of
optimization. In particular, we present the first Positivstellensatz
that certifies infeasibility of a set of polynomial inequalities
simply by multiplying certain fixed polynomials together and checking
nonnegativity of the coefficients of the resulting product.

In part (ii), we study the problem of learning dynamical systems from
very limited data but in presence of “side information”, such as
physical laws or contextual knowledge. This is motivated by
safety-critical applications where an unknown dynamical system needs
to be controlled after a very short learning phase where a few of its
trajectories are observed. (Imagine, e.g., the task of autonomously
landing a passenger airplane that has gone through sudden wing
damage.) We show that sum of squares and semidefinite optimization are
particularly suited for exploiting side information in order to assist
the task of learning when data is limited. Joint work with A. Majumdar
and G. Hall (part (i)) and with B. El Khadir (part (ii)).