Linear combination of norms in improving approximation of Euclidean norm

Mukherjee, Jayanta

doi:10.1016/j.patrec.2013.05.001

Cited by 7 publications

(16 citation statements)

References 25 publications

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“…Note also that these two norms are much more naturally linearizable norms than the Euclidean norm. The most recent works ( [7], [19]) on the subject showed that in the Euclidean plane, the best maximum relative error of an optimized combination of the L 1 and L ∞ norms with respect to the L 2 norm is approximately equal to 5.6%. In the same paper the best combination of an overestimating norm, the euclidean Chamfering weighted distance in 2-D, and an underestimated norm, the Inverse square root weighted t-cost distance in 2-D provide the best empirical MRE of 1.29%.…”

Section: Linearization Of Euclidean Norm Dependent Constraints In Rmentioning

confidence: 99%

“…However, as we aim at modeling minimum or maximum distance constraints in industrial optimization problems, the linear combinations do not allow a strict enforcement of such constraints. In [19], exact MREs of such norms were provided. The MRE of the best overestimating norm is about 8.24 %, while the MRE of the best underestimating norm is about 7.61%.…”

Section: Linearization Of Euclidean Norm Dependent Constraints In Rmentioning

confidence: 99%

“…Mukherjee [19] presents approximations of the Euclidean norm based on linear combinations of other norms. He proposes a combination of the L ∞ and L 1 norms and also a combination of the euclidean chamfering weighted distance in n-D (CW D eu ) and the inverse square root weighted t-cost distance in n-D (WtD isr ).…”

Section: Extension Of These Principles To R 3 and Higher Dimensionsmentioning

confidence: 99%

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Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design

et al. 2019

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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.

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Section: Linearization Of Euclidean Norm Dependent Constraints In Rmentioning

confidence: 99%

Section: Linearization Of Euclidean Norm Dependent Constraints In Rmentioning

confidence: 99%

Section: Extension Of These Principles To R 3 and Higher Dimensionsmentioning

confidence: 99%

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Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design

et al. 2019

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“…In [16] , following a geometric approach good CWDs are obtained in 2D, 3D and 4D. More recently linear combination of different digital distances [15] are explored for the same purpose in arbitrary dimension. In [15] , we have shown how hyperspheres could be used for computing MREs and reported MREs of a few distances for approximating Euclidean metric in arbitrary dimensions ( ≤10).…”

Section: Introductionmentioning

confidence: 99%

“…More recently linear combination of different digital distances [15] are explored for the same purpose in arbitrary dimension. In [15] , we have shown how hyperspheres could be used for computing MREs and reported MREs of a few distances for approximating Euclidean metric in arbitrary dimensions ( ≤10). The analysis is further extended for the linear combination of WtD and CWD (called WtCWD) in [17] , and a very good approximation of Euclidean metric is reported in higher dimensional spaces ( ≤100).…”

Section: Introductionmentioning

confidence: 99%