Networks
Training
Training week 17
On 8 - 11 April 2024 NETWORKS organizes the 17th Training Week in De Schildkamp in Asperen. Monique Laurent (CWI & UvT) and Rajat Hazra (UL) are the lecturers of the minicourses.
Next to the minicourses various Networks members will give talks about their recent results during a research presentation or will present an open problem. On Wednesday afternoon there will be a social event.
Sums of squares approximations for polynomial optimization
Minicourse by Monique Laurent (CWI)
In these lectures we present polynomial optimization, which is the problem of
minimizing a polynomial over a feasible region defined by polynomial inequalities. This models a broad range of problems in discrete or continuous optimization and applications, including operations research, energy optimization, control theory, quantum information.
Finding this minimum value is a computationally hard problem in general. Hence, one is interested in getting tractable bounds for this minimum.
We will present the moment/sum-of-squares (sos) approach, which leads to hierarchies of semidefinite lower bounds for the minimum. Our objective is to introduce the main ideas and mathematical tools underlying this approach:
- recognizing sums of squares of polynomials via semidefinite programming
- real algebraic results about sums of squares that underlie the construction of the bounds
- duality of the moment/sos bounds
- asymptotic convergence (in the compact case)
- finite convergence (via the so-called flatness criterion) and extraction of global minimizers
We may also touch upon the following topics:
- quantitative analysis of the bounds
- extension to the noncommutative setting, with application to quantum information
Spectrum of random graphs
Mincourse by Rajat Hazra (UL)
n this lecture series, I will provide a brief overview of the methods used to study the spectrum of random graphs, with a special focus on the bulk and largest eigenvalues of the adjacency matrices. I will describe the method of moments and the Stieltjes transform to derive the scaling limit of the empirical spectral distribution for both sparse and dense Erdos-Renyi random graphs. The behavior of the largest eigenvalues and their fluctuations differs between the sparse and dense cases. I will explain how the theory of local weak convergence is useful in the sparse case, while the theory of graphons is useful in the dense case.
Location
Conferentiecentrum De Schildkamp, Leerdamseweg 44, 4147 BM Asperen.
