This Week at ICERM
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March 19, 2023
There are no events currently scheduled for March 19th.
March 20, 2023

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space
March 21, 2023

9:00  10:00 am EDTProfessional Development: Hiring ProcessProfessional Development  11th Floor Lecture Hall

10:30  11:15 am EDTMemoryefficient approximation algorithm for SDPs and its application to MaxCut, MaxkCut, and Correlation ClusteringSeminar  11th Floor Lecture Hall
 Nimita Shinde, IITBMonash Research Academy
Abstract
Semidefinite programs (SDPs) are a special class of convex optimization problems that have a wide range of applications in areas such as control theory, statistical modeling, correlation clustering, community detection, angular synchronization, and combinatorial optimization. We discuss three fundamental graph partitioning problems MaxCut, MaxkCut, and the MaxAgree variant of correlation clustering. Given a graph G = (V, E), MaxkCut aims to divide the nodes into k partitions such that the weight of the edges crossing the partitions is maximized. MaxCut is a special case of MaxkCut with k = 2. While MaxAgree requires additional information about the ‘similarity’ or ‘dissimilarity’ of each pair of nodes (i, j) in E, and seeks to cluster the nodes of the graph such that the sum of the ‘similar’ edges within the cluster and ‘dissimilar’ edges across the clusters is maximized. For largescale instances of problems, the memory required to solve SDPs becomes a key computational bottleneck. We review a Gaussian samplingbased technique that replaces the decision variable of SDP with a zeromean Gaussian random vector whose covariance is equal to the value of the decision variable. This leads to a FrankWolfe based algorithmic framework that tracks the random vectors instead of the matrix iterates. We then apply this technique to the three combinatorial optimization problems, MaxCut, MaxkCut, and MaxAgree. We show that the proposed algorithmic framework uses O(V  + E) memory and generates an approximate solution to the input problem in polynomial time that nearly achieves the best existing approximation guarantees.

11:15 am  12:00 pm EDTBurerMonteiro lowrank optimization for sumofsquare certification11th Floor Lecture Hall
 Shixuan Zhang, Brown University
Abstract
To certify a sum of $k$ squares on a real projective variety, one can minimize the distance of the sum of squares of $k$ linear forms from it in the space of quadrics. When $k$ is smaller than the dimension of linear forms, the certification problem can be applied in lowrank semidefinite relaxation of polynomial optimization, dual to the BurerMonteiro method. In this talk, we give some preliminary results on the existence of spurious stationary points in this certification problem by explicit examples and some further study on their structure when the varieties have minimal degrees. For general varieties, we propose algorithmic remedies that guarantee the convergence to global minima.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space
March 22, 2023

3:30  4:00 pm EDTMentoring Coffee / TeaCoffee Break  11th Floor Collaborative Space
March 23, 2023

2:00  2:30 pm EDTFourier on convex compact bodies: Siegel meets MinkowskiPost Doc/Graduate Student Seminar  11th Floor Lecture Hall
 Sammy Khalife, Johns Hopkins University
Abstract
In 1889, Minkowski provided a sufficient and necessary condition for a symmetric convex body K in R^d to contain an integral point in its interior, given by Vol(K) > 2^{d}. After introducing the relevant facts in Fourier Analysis, I will discuss how one can use them to get a stronger statement expressing the gap between Vol(K) and 2^{d} in general. These results can be easily extended to intersection with Lattices and to K being compact. An application will be to deduce an original characterization of the extremal bodies, i.e. convex symmetric bodies such that Vol(K)=2^d. They correspond to the bodies which, when scaled by half, tile R^d after translations by the integral points.

2:30  3:00 pm EDTPPAD hardness – the complexity of Nash equilibriumPost Doc/Graduate Student Seminar  11th Floor Lecture Hall
 Chengyue He, Columbia University
Abstract
PPAD is a class of search problems where the solution is shown to exist by a parity argument. Whether a PPADcomplete problem admits a polytime algorithm raises significant attention in the algorithmic game theory community. We will present a brief introduction to this class by both discrete (Sperner’s lemma) and continuous (Brouwer fixed point theorem) perspectives. In addition, we will see how this relates to some famous results like Nash equilibrium, Scarf’s lemma and hypergraphic stable matching.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space
March 24, 2023

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space
March 25, 2023
There are no events currently scheduled for March 25th.
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