Revisiting the minimum-norm problem

Abstract

AbstractThe design of optimal Magnetic Resonance Imaging (MRI) coils is modeled as a minimum-norm problem (MNP), that is, as an optimization problem of the form $\min_{x\in\mathcal{R}}\|x\|$ min x ∈ R ∥ x ∥ , where $\mathcal{R}$ R is a closed and convex subset of a normed space X. This manuscript is aimed at revisiting MNPs from the perspective of Functional Analysis, Operator Theory, and Banach Space Geometry in order to provide an analytic solution to the following MRI problem: $\min_{\psi\in\mathcal{R}}\|\psi\|_{2}$ min ψ ∈ R ∥ ψ ∥ 2 , where $\mathcal{R}:=\{\psi\in \mathbb{R}^{n}:\frac{\|A\psi-b\|_{\infty}}{\|b\|_{\infty}} \leq D\}$ R : = { ψ ∈ R n : ∥ A ψ − b ∥ ∞ ∥ b ∥ ∞ ≤ D } , with $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ A ∈ M m × n ( R ) , $D>0$ D > 0 , and $b\in\mathbb{R}^{m}\setminus\{0\}$ b ∈ R m ∖ { 0 } .