Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design

Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in...

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Vydané v:	Computational optimization and applications Ročník 73; číslo 2; s. 679 - 705
Hlavní autori:	Camino, Jean-Thomas, Artigues, Christian, Houssin, Laurent, Mourgues, Stéphane
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2019 Springer Nature B.V Springer Verlag
Predmet:	Communication satellites Computer Science Continuity (mathematics) Convex and Discrete Geometry Domains Integer programming Linear programming Linearization Management Science Mathematics Mathematics and Statistics Mixed integer Nonlinear programming Nonlinearity Operations Research Operations Research/Decision Theory Optimization Optimization and Control Satellites Statistics Mixed Integer Nonlinear Programming Multibeam satellites Mixed Integer Linear Programming center problem Euclidean norm linearization
ISSN:	0926-6003, 1573-2894
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Abstract	Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in the challenging Mixed Integer Nonlinear Programming (MINLP) class. For any MINLP where the nonlinearity is only present in the form of inequality constraints involving the Euclidean norm, we propose in this article an efficient methodology for linearizing the optimization problem at the cost of entirely controllable approximations even for non convex constraints. They make it possible to rely fully on Mixed Integer Linear Programming and all its strengths. We first empirically compare this linearization approach with a previously proposed linearization approach of the literature on the continuous k -center problem. This methodology is then successfully applied to a critical problem in the telecommunication satellite industry: the optimization of the beam layouts in multibeam satellite systems. We provide a proof of the NP-hardness of this very problem along with experiments on a realistic reference scenario.
AbstractList	Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in the challenging Mixed Integer Nonlinear Programming (MINLP) class. For any MINLP where the nonlinearity is only present in the form of inequality constraints involving the Euclidean norm, we propose in this article an efficient methodology for linearizing the optimization problem at the cost of entirely controllable approximations even for non convex constraints. They make it possible to rely fully on Mixed Integer Linear Programming and all its strengths. We first empirically compare this linearization approach with a previously proposed linearization approach of the literature on the continuous k -center problem. This methodology is then successfully applied to a critical problem in the telecommunication satellite industry: the optimization of the beam layouts in multibeam satellite systems. We provide a proof of the NP-hardness of this very problem along with experiments on a realistic reference scenario. Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in the challenging Mixed Integer Nonlinear Programming (MINLP) class. For any MINLP where the nonlinearity is only present in the form of inequality constraints involving the Euclidean norm, we propose in this article an efficient methodology for linearizing the optimization problem at the cost of entirely controllable approximations even for non convex constraints. They make it possible to rely fully on Mixed Integer Linear Programming and all its strengths. We first empirically compare this linearization approach with a previously proposed linearization approach of the literature on the continuous k-center problem. This methodology is then successfully applied to a critical problem in the telecommunication satellite industry: the optimization of the beam layouts in multibeam satellite systems. We provide a proof of the NP-hardness of this very problem along with experiments on a realistic reference scenario.
Author	Houssin, Laurent Camino, Jean-Thomas Mourgues, Stéphane Artigues, Christian
Author_xml	– sequence: 1 givenname: Jean-Thomas surname: Camino fullname: Camino, Jean-Thomas organization: LAAS-CNRS, Université de Toulouse, CNRS, UPS, Airbus Defence and Space, Space Systems, Telecommunication Systems Department – sequence: 2 givenname: Christian surname: Artigues fullname: Artigues, Christian organization: LAAS-CNRS, Université de Toulouse, CNRS, UPS – sequence: 3 givenname: Laurent orcidid: 0000-0001-5975-7639 surname: Houssin fullname: Houssin, Laurent email: laurent.houssin@laas.fr organization: LAAS-CNRS, Université de Toulouse, CNRS, UPS – sequence: 4 givenname: Stéphane surname: Mourgues fullname: Mourgues, Stéphane organization: Airbus Defence and Space, Space Systems, Telecommunication Systems Department
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Cites_doi	10.1016/j.comnet.2006.11.018 10.1007/s10589-012-9503-x 10.1137/0213014 10.1109/ICC.2006.255710 10.1007/BF03024331 10.1109/SYSCON.2016.7490613 10.1007/s11590-008-0095-4 10.1145/984622.984630 10.1016/j.jmva.2010.06.002 10.1007/s10589-013-9557-4 10.1145/377939.377946 10.1016/j.enganabound.2013.11.020 10.1109/ASMS-SPSC.2014.6934570 10.1007/978-3-319-68445-1_95 10.1016/j.patcog.2010.08.028 10.1023/A:1016083231326 10.1088/0305-4470/24/23/008 10.1137/120875909 10.1016/j.patrec.2013.05.001 10.1137/S1052623495288362 10.1287/moor.26.2.193.10561 10.1090/S0025-5718-1957-0690630-7
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Copyright	Springer Science+Business Media, LLC, part of Springer Nature 2019 Computational Optimization and Applications is a copyright of Springer, (2019). All Rights Reserved. Distributed under a Creative Commons Attribution 4.0 International License
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Keywords	Mixed Integer Nonlinear Programming Multibeam satellites Mixed Integer Linear Programming center problem Euclidean norm linearization
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References_xml	– reference: RabieHMEl-KhodaryIATharwatAAA particle swarm optimization algorithm for the continuous absolute p-center location problem with Euclidean distanceInt. J. Adv. Comput. Sci. Appl. (IJACSA)20134101106 – reference: Celebi, M., Celiker, F., Kingravi, H.: On Euclidean norm approximations. Pattern Recognition p. 278283 (2011) – reference: Glineur, F., De Houdain, R., Mons, B.: Computational experiments with a linear approximation of second-order cone optimization. Image Technical Report 0001, Mons (2000) – reference: KiatmanarojKArtiguesCHoussinLMessineFFrequency assignment in a SDMA satellite communication system with beam decentring featureComput. Optim. Appl.201356439455310187010.1007/s10589-013-9557-41318.78007 – reference: Camino, J.-T., Artigues, C., Houssin, L., Mourgues, S.: A greedy approach combined with graph coloring for non-uniform beam layouts under antenna constraints in multibeam satellite systems. In: Advanced Satellite Multimedia Systems Conference and the 13th Signal Processing for Space Communications Workshop (ASMS/SPSC) (2014) – reference: Xue, G., Rosen, J.B., Pardalos, P.M.: A polynomial time dual algorithm for the Euclidean multifacility location problem. In: Proceedings of Second Conference on Integer Programming and Combinatorial Optimization, pp. 227–236 (1992) – reference: Megiddo, N., Supowit, K.J.: On the complexity of some common geometric location problems. Society for Industrial and Applied Mathematics—SIAM J. Comput. 13(1):182–196 (1984) – reference: LibertiLMaculanNZhangYOptimal configuration of gamma ray machine radiosurgery units: the sphere covering subproblemOptim. Lett.200931109121245350910.1007/s11590-008-0095-41154.90347 – reference: BousquetMMaralGSatellite Communications Systems: Systems Techniques and Technology20095ChichesterWiley – reference: XueGYueYAn efficient algorithm for minimizing a sum of Euclidean norms with applicationsSIAM J. Optim.1997710171036147961210.1137/S1052623495288362 – reference: Biswas, P., Ye, Y.: Semidefinite programming for ad hoc wireless sensor network localization. In: Third International Symposium on Information Processing in Sensor Networks, pp. 46–54 (2004) – reference: CookJMRational formulae for the production of a spherically symmetric probability distributionMath. Comput.19571158818269063010.1090/S0025-5718-1957-0690630-70078.31403 – reference: FayedHAAtiyaAFA mixed breadth-depth first strategy for the branch and bound tree of Euclidean k-center problemsComput. Optim. Appl.2013543675703302931310.1007/s10589-012-9503-x1271.90094 – reference: D’Ambrosio, C., ans Liberti, L.: Distance geometry in linearizable norms. In: Geometric Science of Information, pp. 830–837. Springer International Publishing (2017) – reference: MullerMEA note on a method for generating points uniformly on n-dimensional spheresCommun. 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SubjectTerms	Communication satellites Computer Science Continuity (mathematics) Convex and Discrete Geometry Domains Integer programming Linear programming Linearization Management Science Mathematics Mathematics and Statistics Mixed integer Nonlinear programming Nonlinearity Operations Research Operations Research/Decision Theory Optimization Optimization and Control Satellites Statistics
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