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Exploiting sparsity in polynomial optimization - lecture 2

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Authors : Magron, Victor (Author of the conference)
CIRM (Publisher )

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Abstract : Polynomial optimization methods often encompass many major scalability issues on the practical side. Fortunately, for many real-world problems, we can look at them in the eyes and exploit the inherent data structure arising from the input cost and constraints. The first part of my lecture will focus on the notion of 'correlative sparsity', occurring when there are few correlations between the variables of the input problem. The second part will present a complementary framework, where we show how to exploit a distinct notion of sparsity, called 'term sparsity', occurring when there are a small number of terms involved in the input problem by comparison with the fully dense case. At last but not least, I will present a very recently developed type of sparsity that we call 'ideal-sparsity', which exploits the presence of equality constraints. Several illustrations will be provided on important applications arising from various fields, including computer arithmetic, robustness of deep networks, quantum entanglement, optimal power-flow, and matrix factorization ranks.

Keywords : optimisation polynomiale; exploitation de la parcimonie; programmation semi-définie

MSC Codes :
65F50 - Sparse matrices
 90C22 - Semidefinite programming
 90C23 - Polynomial optimization
Additional resources :
https://www.cirm-math.fr/RepOrga/2888/Slides/SparsePOPJNCF23.pdf
    Information on the Video

    Film maker : Hennenfent, Guillaume
    Language : English
    Available date : 20/03/2023
    Conference Date : 06/03/2023
    Subseries : Research School
    arXiv category : Optimization and Control
    Mathematical Area(s) : Control Theory & Optimization
    Format : MP4 (.mp4) - HD
    Video Time : 01:29:20
    Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2023-03-07_Magron.mp4
Information on the Event

Event Title : Francophone Computer Algebra Days / JNCF - Journées nationales de calcul formel
Event Organizers : Berthomieu, Jérémy ; Boito, Paola ; Guerrini, Eleonora ; Ollivier, François ; Spaenlehauer, Pierre-Jean
Dates : 06/03/2023 - 10/03/2023
Event Year : 2023
Event URL : https://conferences.cirm-math.fr/2888.html

Citation Data

DOI : 10.24350/CIRM.V.20013803
Cite this video as: Magron, Victor (2023). Exploiting sparsity in polynomial optimization - lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20013803
URI : http://dx.doi.org/10.24350/CIRM.V.20013803

See Also

Bibliography

  • MAGRON, Victor et WANG, Jie. Sparse Polynomial Optimization: Theory and Practice. arXiv preprint arXiv:2208.11158, 2022. - https://doi.org/10.48550/arXiv.2208.11158

  • ANDERSEN, Erling D., ROOS, Cornelis, et TERLAKY, Tamas. On implementing a primal-dual interior-point method for conic quadratic optimization. Mathematical Programming, 2003, vol. 95, p. 249-277. - http://dx.doi.org/10.1007/s10107-002-0349-3

  • BUGARIN, Florian, HENRION, Didier, et LASSERRE, Jean Bernard. Minimizing the sum of many rational functions. Mathematical Programming Computation, 2016, vol. 8, no 1, p. 83-111. - http://dx.doi.org/10.1007/s12532-015-0089-z

  • BODLAENDER, Hans L. et KOSTER, Arie MCA. Treewidth computations I. Upper bounds. Information and Computation, 2010, vol. 208, no 3, p. 259-275. - https://doi.org/10.1016/j.ic.2009.03.008

  • DUNNING, Iain, HUCHETTE, Joey, et LUBIN, Miles. JuMP: A modeling language for mathematical optimization. SIAM review, 2017, vol. 59, no 2, p. 295-320. - https://doi.org/10.1137/15M1020575

  • FAIRBANKS, James, BESANĘON, M., SIMON, S., et al. Juliagraphs/graphs. jl: an optimized graphs package for the julia programming language, 2021 - https://github. com/JuliaGraphs/Graphs. jl

  • GARSTKA, Michael, CANNON, Mark, et GOULART, Paul. COSMO: A conic operator splitting method for large convex problems. In : 2019 18th European Control Conference (ECC). IEEE, 2019. p. 1951-1956. - https://doi.org/10.23919/ECC.2019.8796161

  • HENRION, Didier, LASSERRE, Jean-Bernard, et LÖFBERG, Johan. GloptiPoly 3: moments, optimization and semidefinite programming. Optimization Methods & Software, 2009, vol. 24, no 4-5, p. 761-779. - https://doi.org/10.1080/10556780802699201

  • LOFBERG, Johan. YALMIP: A toolbox for modeling and optimization in MATLAB. In : 2004 IEEE international conference on robotics and automation (IEEE Cat. No. 04CH37508). IEEE, 2004. p. 284-289. - https://doi.org/10.1109/CACSD.2004.1393890

  • STURM, Jos F. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization methods and software, 1999, vol. 11, no 1-4, p. 625-653. - https://doi.org/10.1080/10556789908805766

  • WANG, Jie. ChordalGraph: A Julia Package to Handle Chordal Graphs. 2020. -


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