Category Archives: Spring 2019

Amir Ali Ahmadi; Nonnegative polynomials: from optimization to control and learning

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

Poster PDF

*Title:
Nonnegative polynomials: from optimization to control and learning

*Speaker:
Amir Ali Ahmadi
Princeton, ORFE

*Abstract:
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
optimization”.

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)).

*Bio:
Amir Ali Ahmadi ( http://aaa.princeton.edu/ ) is a Professor at the
Department of Operations Research and Financial Engineering at
Princeton University and an Associated Faculty member of the Program
in Applied and Computational Mathematics, the Department of Computer
Science, the Department of Mechanical and Aerospace Engineering, and
the Center for Statistics and Machine Learning. Amir Ali received his
PhD in EECS from MIT and was a Goldstine Fellow at the IBM Watson
Research Center prior to joining Princeton. His research interests are
in optimization theory, computational aspects of dynamics and control,
and algorithms and complexity. Amir Ali’s distinctions include the
Sloan Fellowship in Computer Science, a MURI award from the AFOSR, the
NSF CAREER Award, the AFOSR Young Investigator Award, the DARPA
Faculty Award, the Google Faculty Award, the Howard B. Wentz Junior
Faculty Award as well as the Innovation Award of Princeton University,
the Goldstine Fellowship of IBM Research, and the Oberwolfach
Fellowship of the NSF. His undergraduate course at Princeton (ORF 363,
“Computing and Optimization’’) has received the 2017 Excellence in
Teaching of Operations Research Award of the Institute for Industrial
and Systems Engineers and the 2017 Phi Beta Kappa

Rainer Sinn and Daniel Plaumann; From conic programming to real algebraic geometry and back

Rainer Sinn, Fachbereich Mathematik und Informatik, Freie Universität Berlin
Daniel Plaumann, Fakultät für Mathematik, Technische Universität Dortmund 

Tuesday May 14, 2019, More Hall 234, 4pm-5:30pm

Poster PDF

Title: From conic programming to real algebraic geometry and back

Abstract: In this talk, we will present interactions between conic programming, a branch of optimization, and real algebraic geometry, the algebraic and geometric study of real polynomial systems. This has led to a number of beautiful mathematical theorems, both improving our understanding of the geometric picture as well as sharpening our tools for applications.

On the optimization side, we will take a look at hyperbolic programs, an instance of conic programs first studied in the 1990s that comprises linear programs as well as semidefinite programs. On the algebraic side, it is based on the theory of hyperbolic respectively real stable polynomials, which show up in many parts of mathematics, ranging from control theory to combinatorics. In many ways, hyperbolic polynomials behave like determinants of families of symmetric matrices. We will compare these two paradigms, one based in matrix calculus, the other purely in algebraic geometry, through a series of examples, pictures, theorems, and conjectures.

 

Bios:

Daniel Plaumann has been associate professor of algebra and its applications at TU Dortmund, Germany, since 2016. His research interests are real and classical algebraic geometry, positive polynomials, matrix inequalities, and applications to optimization and functional analysis. He received his PhD in 2008 from the University of Konstanz under the supervision of Claus Scheiderer. In 2010, he was Feodor Lynen Fellow of the Alexander von Humboldt Foundation at UC Berkeley and in 2014 Research Fellow at the Nanyang Technological University in Singapore. From 2013-2016 he was Research Felllow at the Zukunftskolleg of the University of Konstanz. He is currently a long-term visitor at the Simons Institute for the Theory of Computing in Berkeley.
 
Rainer Sinn has been assistant professor for discrete geometry at Freie Universität Berlin, Germany, since 2017. His research interests are discrete, convex, and real algebraic geometry. He graduated from the University of Konstanz in 2014 under the supervision of Claus Scheiderer with a fellowship of the German National Academic Foundation. In 2014, before moving to the Georgia Institute of Technology for a postdoc position with Greg Blekherman, he was a visiting researcher at the National Institute of Mathematical Sciences in Daejeon, Korea, during a Thematic Program on Applied Algebraic Geometry. In 2017, he was at the Max Planck Institute for Mathematics in the Sciences in Leipzig as a member of the Nonlinear Algebra group led by Bernd Sturmfels. He is currently a research fellow at the Simons Institute for the Theory of Computing in Berkeley as part of the program on Geometry of Polynomials.