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Exact Solutions to the Maxmin Problem max ||Ax|| Subject to ||Bx||<= 1

Soledad Moreno-Pulido, Francisco Javier García-Pacheco, Clemente Cobos-Sánchez, Alberto Sánchez-Alzola

TL;DR

An exact solution to the maxmin problem, where A and B are real matrices, and a truly optimal minimum stored-energy Transcranian Magnetic Stimulation (TMS) coil is constructed, and an optimal geolocation involving statistical variables is found.

Abstract

In this manuscript we provide an exact solution to the maxmin problem max ||Ax|| subject to ||Bx||<= 1, where A and B are real matrices. This problem comes from a remodeling of max ||Ax|| subject to min ||Bx||, 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 parallel to Cx parallel to subject to parallel to x parallel to <= 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

Exact Solutions to the Maxmin Problem max ||Ax|| Subject to ||Bx||<= 1

TL;DR

An exact solution to the maxmin problem, where A and B are real matrices, and a truly optimal minimum stored-energy Transcranian Magnetic Stimulation (TMS) coil is constructed, and an optimal geolocation involving statistical variables is found.

Abstract

In this manuscript we provide an exact solution to the maxmin problem max ||Ax|| subject to ||Bx||<= 1, where A and B are real matrices. This problem comes from a remodeling of max ||Ax|| subject to min ||Bx||, 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 parallel to Cx parallel to subject to parallel to x parallel to <= 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
Paper Structure (22 sections, 14 theorems, 91 equations, 6 figures, 1 table)

This paper contains 22 sections, 14 theorems, 91 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Scope
  3. Novelties
  4. Preliminaries
  5. Characterizations of operators with null kernel
  6. Remodeling the original maxmin problem $\left\{\max \|T(x)\|\min \|S(x)\|\right.$
  7. The original maxmin problem has no solutions
  8. Equivalent reformulations for the original maxmin problem
  9. Solving the maxmin problem $\left\{\max \|T(x)\|\|S(x)\|\leqslant 1\right.$
  10. First case: $S$ is an isomorphism over its image
  11. The Moore-Penrose inverse
  12. Second case: $S$ is not an isomorphism over its image
  13. Characterizing when the finite dimensional reformulated maxmin has a solution
  14. Matrices on quotient spaces
  15. Conclusions: schematic summary
  16. ...and 7 more sections

Key Result

Proposition 2.1

A continuous linear operator $T:X\to Y$ between locally convex Hausdorff topological vector spaces $X$ and $Y$ verifies that $\ker(T)\neq \{0\}$ if and only if exists $S\in\mathcal{CL}(Y,X)\setminus\{0\}$ with $T\circ S=0$. In particular, if $X=Y$, then $\ker(T)\neq \{0\}$ if and only if $T\in \ell\

Figures (6)

  • Figure 1: a) Description of hemispherical surface where the optimal $\psi$ must been found along with the spherical regions of interest ROI and ROI2 where the electric field must be maximized and minimized respectively. b) Wirepaths with 18 turns of the TMS coil solution (red wires indicate reversed current flow with respect to blue).
  • Figure 2: a) Description of the two compartment scalp-brain model. b) E-field modulus induced at the surface of the brain by the designed TMS coil.
  • Figure 3: Geographic distribution of the sites considered in the study. 11 places are in the coastline of the region and 5 in the inner
  • Figure 4: Locations considering Ax and Bx axes. Group named $\it{A}$ represents the best places for the tourism rural inn, near Costa Tropical (Granada province). Sites on $\it{B}$ are also in the coastline of the region. Sites on $\it{C}$ are the worst locations considering the multiobjective problem, they are situated inside the region
  • Figure 5: a) Sites considering Ax and Bx and the function $y=-x$. The places with high values of Ax (max) and low values of Bx (min) are the best locations for the solution of the multiobjective problem (round). b) Multiobjective scores values obtained for each site projecting the point in the function $y=-x$. High values of this score indicate better places to locate the tourism rural inn.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Example 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 16 more