Supporting vectors for the $$\ell _1$$-norm and the $$\ell _{\infty }$$-norm and an application

Sánchez‐Alzola, Alberto; García-Pacheco, Francisco Javier; Naranjo-Guerra, Enrique

doi:10.1007/s40096-021-00400-w

Cited by 5 publications

(2 citation statements)

References 16 publications

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“…We refer the reader to [2,21,22] for a topological and geometrical study of the set of supporting vectors of a continuous linear operator. Supporting vectors have been successfully applied to solve multiobjective optimization problems that typically arise in Bioengineering, Physics, and Statistics [14,15,[23][24][25], improving considerably the results obtained by means of other techniques, such as Heuristic methods [16,18,19].…”

Section: Supporting Vectorsmentioning

confidence: 99%

Analytical Solutions to Minimum-Norm Problems

et al. 2022

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For G∈Rm×n and g∈Rm, the minimization min∥Gψ−g∥2, with ψ∈Rn, is known as the Tykhonov regularization. We transport the Tykhonov regularization to an infinite-dimensional setting, that is min∥T(h)−k∥, where T:H→K is a continuous linear operator between Hilbert spaces H,K and h∈H,k∈K. In order to avoid an unbounded set of solutions for the Tykhonov regularization, we transform the infinite-dimensional Tykhonov regularization into a multiobjective optimization problem: min∥T(h)−k∥andmin∥h∥. We call it bounded Tykhonov regularization. A Pareto-optimal solution of the bounded Tykhonov regularization is found. Finally, the bounded Tykhonov regularization is modified to introduce the precise Tykhonov regularization: min∥T(h)−k∥with∥h∥=α. The precise Tykhonov regularization is also optimally solved. All of these mathematical solutions are optimal for the design of Magnetic Resonance Imaging (MRI) coils.

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Section: Supporting Vectorsmentioning

confidence: 99%

Analytical Solutions to Minimum-Norm Problems

et al. 2022

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“…The multiobjective optimization has been considerably analyzed and studied by many researchers, see for instance [1][2][3][4][5][6]. Multiobjective optimization problems play a crucial role in various fields like economics, engineering, management sciences [2,[7][8][9][10][11], and many more places in daily life.…”

Section: Introductionmentioning

confidence: 99%

Multiobjective Convex Optimization in Real Banach Space

Lai

Hassan

Maurya³

et al. 2021

Symmetry

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In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results obtained in the recent article by Cobos Sánchez et al. in 2021 on Pareto optimality for multiobjective optimization problems of continuous linear operators. The discussions in this paper are also related to second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones due to Mishra and Wang in 2005. Further, we establish Karush–Kuhn–Tucker optimality conditions using saddle point optimality conditions for the differentiable cases and present some examples to illustrate our results. The study in this article can also be seen and extended as symmetric results of necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds by Ruiz-Garzón et al. in 2019.

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Crossover detection based on variances of slope differences for multi-fractal detrended fluctuation analysis (MF-DFA)

Moreno-Pulido,

de la Torre,

Ruiz

et al. 2024

Nonlinear Dyn

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Multi-fractal detrended fluctuation analysis (MF-DFA) describes long-range correlations across scales in time series in terms of the linear fittings of the fluctuation functions $$F_q(s)$$ F q ( s ) for every moment q considered. It is relatively common to find time scales, also called crossovers, where different linearities are found at each side of the $$F_q(s)$$ F q ( s ) , leading to different (multi-)fractal properties. Generally, crossovers are manually identified by experts, subject then to their skills and possible bias. In this work, a novel methodology based on the variances of the slope differences (CDV-A) is proposed to automatically detect crossovers. A set of synthetic fluctuation functions with different mono-fractal and multi-fractal features, different noise levels and known crossovers is generated to assess the CDV-A. Moreover, the proposed methodology is also applied to state of the art fluctuation functions in order to compare the proposed manual and automated locations of the crossovers. Results show the proposed methodology is highly accurate, specially in low noise synthetic fluctuation functions and in most of the real cases.

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Supporting vectors for the $$\ell _1$$-norm and the $$\ell _{\infty }$$-norm and an application

Cited by 5 publications

References 16 publications

Analytical Solutions to Minimum-Norm Problems

Analytical Solutions to Minimum-Norm Problems

Multiobjective Convex Optimization in Real Banach Space

Crossover detection based on variances of slope differences for multi-fractal detrended fluctuation analysis (MF-DFA)

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