Supporting Vectors of Continuous  Linear Projections

García-Pacheco, Francisco Javier; Naranjo-Guerra, Enrique

doi:10.17654/fa009030085

Cited by 5 publications

(9 citation statements)

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“…The set of supporting vectors of a continuous linear operator 𝑇 ∶ 𝑋 → 𝑌 between normed spaces 𝑋, 𝑌 is defined as suppv(𝑇) ∶= {𝑥 ∈ 𝖲 𝑋 ∶ ‖𝑇(𝑥)‖ = ‖𝑇‖}. For a wider perspective on this topic, we refer the reader to [4,7], where the geometrical nature and topological structure of the set of supporting vectors is discussed.…”

Section: Extremal Structure Under 1-complementationmentioning

confidence: 99%

Compact convex sets free of inner points in infinite‐dimensional topological vector spaces

Campos‐Jiménez,

García‐Pacheco

2023

Mathematische Nachrichten

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Section: Extremal Structure Under 1-complementationmentioning

confidence: 99%

Compact convex sets free of inner points in infinite‐dimensional topological vector spaces

Campos‐Jiménez,

García‐Pacheco

2023

Mathematische Nachrichten

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“…The geometric and topological structure of supporting vectors can be consulted in [7][8][9]. On the other hand, generalized supporting vectors are defined and studied in [7,8].…”

Section: Scopementioning

confidence: 99%

“…For other perspective on supporting vectors and generalized supporting vectors, we refer the reader to [9].…”

Section: Scopementioning

confidence: 99%

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Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1

et al. 2020

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In this manuscript we provide an exact solution to the maxmin problem max ∥ A x ∥ subject to ∥ B x ∥ ≤ 1 , where A and B are real matrices. This problem comes from a remodeling of max ∥ A x ∥ subject to min ∥ B x ∥ , because the latter problem has no solution. Our mathematical method comes from the Abstract Operator Theory, whose strong machinery allows us to reduce the first problem to max ∥ C x ∥ subject to ∥ x ∥ ≤ 1 , which can be solved exactly by relying on supporting vectors. Finally, as appendices, we provide two applications of our solution: first, we construct a truly optimal minimum stored-energy Transcranian Magnetic Stimulation (TMS) coil, and second, we find an optimal geolocation involving statistical variables.

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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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Supporting Vectors of Continuous Linear Projections

Cited by 5 publications

References 0 publications

Compact convex sets free of inner points in infinite‐dimensional topological vector spaces

Compact convex sets free of inner points in infinite‐dimensional topological vector spaces

Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1

Pareto Optimality for Multioptimization of Continuous Linear Operators

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