The Study of Minimizing a Quadratic via Gram Matrix over Sphere with \(l^{1}\) Norm

Let \(H\) be a Hilbert space over the real numbers with inner product \(\langle \cdot,\cdot \rangle\)  and choose a finite set of \(linearly\)  \(independent\) vectors  \(\mathcal{X}= \{ x_{j} : j \in \mathbb{N}_{n}\}\) in \(H\).  The Gram matrix of the vectors in \(\mathcal{X}\) is \[\mathbf{G} := (\langle x_{j},x_{l} \rangle: j , l \in \mathbb{N}_{n}),\] which is a symmetric and positive definite matrix. In this paper,  we present the solution of the problem \[\min \Phi(e) := (e , \mathbf{G}^{-1}e ) - 2(e,\mathbf{d})\] subject to \(e \in S =\{ e \in \mathbb{R}^n : |e|_1 = 1 \},\) where \(\mathbf{d}\) is a known \(n-\)vector of real numbers and \((\cdot, \cdot)\) is the Euclidean inner product on \(\mathbb{R}^n\).

Furthermore, we apply the recent result  to improve  our previous work  on the extension of hypercircle inequality to  data error measured with \(l^1\) norm.