Lineability of the set of supporting vectors

García-Pacheco, Francisco Javier

doi:10.1007/s13398-020-00981-6

Cited by 6 publications

(3 citation statements)

References 39 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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“…Multiobjective optimization problems (MOPs) appear quite often in all areas of pure and applied mathematics, for instance, in the geometry of Banach spaces [1][2][3], in operator theory [4][5][6][7], in lineability theory [8][9][10], in differential geometry [11][12][13][14], and in all areas of Experimental, Medical and Social Sciences [15][16][17][18][19][20]. By means of MOPs, many real-life situations can be modeled accurately.…”

Section: Introductionmentioning

confidence: 99%

Pareto Optimality for Multioptimization of Continuous Linear Operators

et al. 2021

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This manuscript determines the set of Pareto optimal solutions of certain multiobjective-optimization problems involving continuous linear operators defined on Banach spaces and Hilbert spaces. These multioptimization problems typically arise in engineering. In order to accomplish our goals, we first characterize, in an abstract setting, the set of Pareto optimal solutions of any multiobjective optimization problem. We then provide sufficient topological conditions to ensure the existence of Pareto optimal solutions. Next, we determine the Pareto optimal solutions of convex max–min problems involving continuous linear operators defined on Banach spaces. We prove that the set of Pareto optimal solutions of a convex max–min of form max∥T(x)∥, min∥x∥ coincides with the set of multiples of supporting vectors of T. Lastly, we apply this result to convex max–min problems in the Hilbert space setting, which also applies to convex max–min problems that arise in the design of truly optimal coils in engineering.

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“…A new module measure for quantum mechanical systems has been constructed whose physical meaning is still unclear (Lemma 4). It is very interesting to deeply study other measures similar to the one defined in Equation ( 2), such as, for instance, measures provided by the probability density operator [40,44] that describes the quantum state of a physical system.…”

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The Range of a Module Measure Defined on an Effect Algebra

García-Pacheco

2022

Symmetry

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Effect algebras are the main object of study in quantum mechanics. Module measures are those measures defined on an effect algebra with values on a topological module. Let R be a topological ring and M a topological R-module. Let L be an effect algebra. The range of a module measure μ:L→M is studied. Among other results, we prove that if L is an sRDP σ-effect algebra with a natural basis and μ:L→R is a countably additive measure, then μ has bounded variation.

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Lineability of the set of supporting vectors

Cited by 6 publications

References 39 publications

Analytical Solutions to Minimum-Norm Problems

Analytical Solutions to Minimum-Norm Problems

Pareto Optimality for Multioptimization of Continuous Linear Operators

The Range of a Module Measure Defined on an Effect Algebra

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