10:00am - 12:00pmPolynomial optimization and its applications
Chair(s): Timo de Wolff (Technische Universität Berlin, Germany), Simone Naldi (Université de Limoges, France), João Gouveia (Universidade de Coimbra, Portugal)
The importance of polynomial (aka semi-algebraic) optimization is highlighted by the large number of its interactions with different research domains of mathematical sciences. These include, but are not limited to, automatic control, combinatorics, and quantum information. The mini-symposium will focus on the development of methods and algorithms dedicated to the general polynomial optimization problem. Both the theoretical and more applicative viewpoints will be covered.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Limitations on the expressive power of convex cones without long chains of faces
James Saunderson
Monash University, Melbourne, Australia
Recently Averkov showed that various convex cones related to nonnegative polynomials do not have K-lifts (representations as projections of linear sections of another convex cone K) where K is a Cartesian product of positive semidefinite cones of "small" size. In this talk I'll explain how to extend Averkov's approach to show that cones with certain neighborliness properties do not have K-lifts whenever K is a Cartesian product of cones, each of which does not have any long chains of faces (such as smooth cones, low-dimensional cones, and cones defined by hyperbolic polynomials of low degree).
On the exactness of Lasserre relaxations and pure states over real closed fields
Markus Schweighofer1, Tom-Lukas Kriel2
1Universität Konstanz, Germany, 2TNG Technology Consulting GmbH
Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $SsubseteqR^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation.
High-dimensional estimation via sum-of-squares proofs
David Steurer1, Prasad Raghavendra2, Tselil Schramm3
1ETH Zürich, Switzerland, 2University of California, Berkeley, CA, USA, 3MIT, Cambridge, MA, USA
Estimation is the computational task of approximately recovering a hidden parameter x associated with a distribution D_x given a draw y from the distribution D_x. Numerous interesting questions in statistics, machine learning, and signal processing are captured in this way, for example, sparse linear regression, Gaussian mixture models, topic models, and stochastic block models.
In many cases, there is currently a large gap between the statistical guarantees of computationally efficient algorithms and the guarantees of computationally inefficient methods; it is an open question if this gap is inherent in these cases or if better computationally efficient estimation algorithms exist.
In this talk, I will present a meta-algorithm for estimation problems based on the sum-of-squares method of Shor, Parrilo, and Lasserre. For some problems, e.g., learning mixtures of spherical Gaussians, this meta-algorithm is able to close previous long-standing gaps and achieve nearly optimal statistical guarantees. Furthermore, it is plausible that, for a wide range of estimation problems, the statistical guarantees that this meta-algorithm achieves are best possible among all efficient algorithms.
This talk is based on an ICM proceedings article with Prasad Raghavendra and Tselil Schramm.
Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM ExponentialsLog-concave polynomials, entropy, and approximate counting
Henning Seidler1, Victor Magron2, Timo de Wolff1
1Technische Universität Berlin, Germany, 2CNRS-LAAS, Toulouse, France
We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm for polynomials lying in the interior of the SAGE cone. Each framework, inspired by previous contributions of Parrilo and Peyrl, is a rounding-projection procedure. For a polynomial lying in the interior of the SAGE cone, we prove that the decision algorithm terminates within a number of arithmetic operations, which is polynomial in the degree and number of terms of the input, and singly exponential in the number of variables. We also provide experimental comparisons regarding the implementation of the two optimization algorithms.