Revisiting the minimum-norm problem

Abstract

Abstract The 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 ∈ R ∥ x ∥ $\min_{x\in\mathcal{R}}\|x\|$ , where R $\mathcal{R}$ 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 ψ ∈ R ∥ ψ ∥ 2 $\min_{\psi\in\mathcal{R}}\|\psi\|_{2}$ , where R : = { ψ ∈ R n : ∥ A ψ − b ∥ ∞ ∥ b ∥ ∞ ≤ D } $\mathcal{R}:=\{\psi\in \mathbb{R}^{n}:\frac{\|A\psi-b\|_{\infty}}{\|b\|_{\infty}} \leq D\}$ , with A ∈ M m × n ( R ) $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ , D > 0 $D>0$ , and b ∈ R m ∖ { 0 } $b\in\mathbb{R}^{m}\setminus\{0\}$ .

Keywords

• Minimum norm
• Banach space
• Optimization